--- title: A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets tags: author: [Testing](https://docswell.com/user/3200568) site: [Docswell](https://www.docswell.com/) thumbnail: https://bcdn.docswell.com/page/GE5M21QDE4.jpg?width=480 description: A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets published: April 09, 26 canonical: https://docswell.com/s/3200568/KJW4EV-2026-04-09-005443 --- # Page. 1 ![Page Image](https://bcdn.docswell.com/page/GE5M21QDE4.jpg) Takaaki Fujita, Florentin Smarandache A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets # Page. 2 ![Page Image](https://bcdn.docswell.com/page/972941WMJR.jpg) Takaaki Fujita, Florentin Smarandache A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets Neutrosophic Science International Association (NSIA) Publishing House Gallup - Guayaquil United States of America – Ecuador 2025 ISBN 978-1-59973-842-0 # Page. 3 ![Page Image](https://bcdn.docswell.com/page/DJY4MZLP7M.jpg) Authors Takaaki Fujita Independent Researcher, Tokyo, Japan. Email: Takaaki.fujita060@gmail.com Florentin Smarandache University of New Mexico, Gallup Campus, NM 87301, USA. Email: fsmarandache@gmail.com Abstract Real-world phenomena frequently involve vagueness, partial truth, and incomplete information. To capture such uncertainty in a mathematically rigorous manner, numerous generalized set-theoretic frameworks have been introduced, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3, 4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutro-sophic Sets [8], PentaPartitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Within these frameworks, a vast number of concepts have been proposed and studied, especially in the contexts of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. This extensive body of work highlights both the importance of these theories and the breadth of their application domains. Consequently, many ideas, constructions, and structural patterns recur across these four families of uncertainty-oriented models. In this book, we present a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutro-sophic, and Plithogenic Sets. Our aim is to offer r eaders a s ystematic overview o f e xisting develop-ments and, through this unified exposition, to foster new insights, further conceptual extensions, and additional applications across a wide range of disciplines. Keywords: Fuzzy Set, Intuitionistic Fuzzy Set, Neutrosophic Set, Plithogenic Set. Editor: Neutrosophic Science International Association (NSIA) Publishing House https://fs.unm.edu/NSIA/ Division of Mathematics and Sciences University of New Mexico 705 Gurley Ave., Gallup Campus NM 87301, United States of America University of Guayaquil Av. Kennedy and Av. Delta “Dr. Salvador Allende” University Campus Guayaquil 090514, Ecuador # Page. 4 ![Page Image](https://bcdn.docswell.com/page/V7NYW34ME8.jpg) Table of Contents 1 Introduction 1.1 Uncertain Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Applied Area: Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic . . . . . . . . 1.3 Our Contribution in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 6 7 2 Preliminaries 2.1 Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Intuitionistic fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Neutrosophic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 10 11 13 16 3 Dynamic Reviews and Results of Uncertain Sets 21 3.1 Plithogenic Sets of Uncertain Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 m-Polar Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Complex Plithogenic Set (CPS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 SuperHyperPlithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Plithogenic Linguistic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 q-rung orthopair Plithogenic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7 Type-n Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 Iterative MultiPlithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9 Interval-Valued Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.10 Plithogenic OffSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.11 Plithogenic Cubic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.12 Plithogenic Soft, HyperSoft, and SuperHyperSoft Set . . . . . . . . . . . . . . . . . . . 65 3.13 Hesitant Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.14 Spherical Plithogenic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.15 T-Spherical Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.16 Plithogenic Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.17 Plithogenic soft rough set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.18 Linear Diophantine Plithogenic set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.19 TreePlithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.20 ForestPlithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.21 Plithogenic Soft Expert Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.22 Dynamic Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.23 Probabilistic Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3 # Page. 5 ![Page Image](https://bcdn.docswell.com/page/YJ9PX9QW73.jpg) 3.24 Triangular Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.25 Trapezoidal Plithogenic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.26 Nonstandard Plithogenic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.27 Refined Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.28 Subset–Valued Plithogenic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.29 Picture Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4 Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set 137 4.1 Uncertain Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.2 Functional Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.3 Other Uncertain Sets and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Appendix (List of Tables) 145 # Page. 6 ![Page Image](https://bcdn.docswell.com/page/GJ8D29GRJD.jpg) Chapter 1 Introduction 1.1 Uncertain Set Real-world phenomena often exhibit vagueness, partial truth, and incomplete information. To capture such uncertainty in a mathematically rigorous way, many generalized set-theoretic frameworks have been introduced, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3,4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutrosophic Sets [8], PentaPartitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Applications of Fuzzy Sets and their extensions—discussed in later sections—have been widely explored in fields such as decision science, chemistry, control systems, and machine learning [13]. Depending on the nature of the application and the number of uncertainty parameters required to characterize the underlying phenomena, an appropriate class of generalized sets is selected to model the problem effectively. In the classical fuzzy setting, each element x in the universe X is associated with a single membership degree µ(x) ∈ [0, 1], which expresses to what extent x belongs to the fuzzy set under consideration [1]. For an Intuitionistic Fuzzy Set, every element x is described by a pair (µ(x), ν(x)) of membership and non-membership functions µ, ν : X → [0, 1] satisfying 0 ≤ µ(x) + ν(x) ≤ 1, [2, 14]. A Neutrosophic Set refines this description by assigning to each element x a triple (T (x), I(x), F (x)), where T (x), I(x), and F (x) denote, respectively, the degrees of truth, indeterminacy, and falsity, typically taking values in [0, 1]. In contrast to the intuitionistic fuzzy case, these three values are not required to sum to 1, which allows one to encode incomplete, inconsistent, or even redundant information in a flexible way [14, 15]. 1 Neutrosophy highlights the central role of neutrality and indeterminacy, giving rise to neutrosophic set, logic, probability, statistics, measure, integral, and 1 Intuitionistic Fuzzy Sets neglect the role of indeterminacy, whereas Neutrosophic Fuzzy Operators assign inde- terminacy the same significance as truth-membership and falsehood-nonmembership [14, 16]. Moreover, it is widely recognized that the neutrosophic set generalizes the intuitionistic fuzzy set, the inconsistent intuitionistic fuzzy set (including picture fuzzy and ternary fuzzy sets), the Pythagorean fuzzy set, the spherical fuzzy set, and the q-rung orthopair fuzzy set; similarly, neutrosophication generalizes regret theory, grey system theory, and three-way decision theory [16]. 5 # Page. 7 ![Page Image](https://bcdn.docswell.com/page/LJLM2WG2ER.jpg) Chapter 1. Introduction related formalisms. These frameworks now find broad practical applications across numerous scientific and applied domains [13, 17]. Plithogenic Sets further generalize these notions by representing each element through its attribute values, together with the corresponding degrees of appurtenance, and by introducing a contradiction (or dissimilarity) function between distinct attribute values [10, 18, 19]. This additional structure supports context-sensitive aggregation of heterogeneous and possibly conflicting evaluations, thereby refining the classical fuzzy, intuitionistic fuzzy, and neutrosophic models (e.g. [17, 20]). For convenience, Table 1.1 summarizes the main data attached to each element in several well-known set extensions (notation harmonized for this book). Table 1.1: Representative set extensions and the canonical information stored per element. Set Type Fuzzy Set Intuitionistic Fuzzy Set Neutrosophic Set Plithogenic Set Canonical data attached to each element Membership mapping µ : X → [0, 1]. Membership µ and non-membership ν with µ(x) + ν(x) ≤ 1; the gap 1 − µ(x) − ν(x) models hesitation. Triple (T, I, F ) with T, I, F ∈ [0, 1] (truth, indeterminacy, falsity) considered as mutually independent coordinates. Tuple (P, v, P v, pdf, pCF) where pdf : P × P v → [0, 1]s encodes sdimensional appurtenance and pCF : P v × P v → [0, 1]t is a symmetric contradiction map in [0, 1]t . Within the plithogenic framework, one can recover plithogenic fuzzy, plithogenic intuitionistic fuzzy, and plithogenic neutrosophic models by choosing suitable dimensions s (for appurtenance) and t (for contradiction) [21–25]. In particular, scalar-contradiction cases with t = 1 yield natural extensions of the classical fuzzy, intuitionistic fuzzy, and neutrosophic paradigms; if, in addition, the contradiction function pCF is set identically to zero, one exactly recovers the corresponding non-plithogenic models (cf. [17]). These three frequently used plithogenic variants are summarized in Table 1.2. Table 1.2: Plithogenic scalar-contradiction variants (t = 1) and their classical limits. Variant s t Appurtenance vector (semantics) Limit when pCF ≡ 0 Plithogenic fuzzy 1 1 µ ∈ [0, 1] (single membership Classical fuzzy set degree) Plithogenic intuitionistic fuzzy 2 1 (µ, ν) ∈ [0, 1]2 with 1 − µ − ν ≥ 0 Intuitionistic fuzzy model Plithogenic neutrosophic 3 1 (T, I, F ) ∈ [0, 1]3 (truth, Neutrosophic model indeterminacy, falsity) 1.2 Applied Area: Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets play an essential role in modern science due to their mathematical depth, practical applicability, and capacity to model uncertainty effectively (cf. [13, 17]). Because of these properties, they have been widely studied and applied in numerous domains, including algebra, graph theory, hypergraph theory, probability, and decision-making. Tables 1.3, 1.4, and 1.5 provide a summarized overview of the extensions of classical concepts under the fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. From this perspective, it becomes clear that an exceptionally wide range of fields has explored both the applications and the underlying mathematical structures of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. These developments highlight not only the theoretical significance of such models but also their substantial contributions to real-world problem solving and practical decisionmaking. # Page. 8 ![Page Image](https://bcdn.docswell.com/page/47MY89Q97W.jpg) Chapter 1. Introduction Table 1.3: Parallel extensions of classical concepts in fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. Classical Concept Fuzzy Set [26] Fuzzy Set [1] Intuitionistic Fuzzy Intuitionistic Fuzzy Set [2] Relation Fuzzy Relation [28] Intuitionistic Fuzzy Relation [29] Function / Mapping Fuzzy Function [32] Intuitionistic Fuzzy Function [33] Graph [37] Fuzzy Graph [38] Intuitionistic Fuzzy Graph [39] Hypergraph [41, 42] Fuzzy Intuitionistic Fuzzy Hypergraph [43, 44] Hypergraph [45] SuperHyperFuzzy SuperHypergraph Intuitionistic Fuzzy graph [48–50] [51, 52] SuperHypergraph [53–55] Matrix / Linear Al- Fuzzy Matrix [62] / Intuitionistic Fuzzy gebra [61] Linear Algebra Matrix [63, 64] / Linear Algebra Algebra Fuzzy Algebra [68] (e.g., Intuitionistic Fuzzy (Group/Ring/…) Fuzzy Group) Algebra [69, 70] HyperAlgebra [74] Fuzzy Intuitionistic Fuzzy (HyperGroup HyperAlgebra [77] (e.g., HyperAlgebra [78, 79] [75]/HyperRing Fuzzy HyperGroup [75]) [76]/…) Topology [82] Fuzzy Topology [83, 84] Intuitionistic Fuzzy Topology [85] Measure / Probabil- Fuzzy Intuitionistic Fuzzy ity [89] Probability [90, 91] Probability [92, 93] Logic [98] Fuzzy Logic [1] Intuitionistic Fuzzy Logic [2] Optimization [101] / Fuzzy DecisionIntuitionistic Fuzzy Decision [102] Making [103, 104] DecisionMaking [105, 106] Clustering / Classi- Fuzzy Intuitionistic Fuzzy fication Clustering [110, 111] Clustering [112, 113] Numbers [116] Fuzzy Numbers [117] Intuitionistic Fuzzy Numbers [118] Neutrosophic Plithogenic Neutrosophic Set [27] Plithogenic Set [10] Neutrosophic Relation [30] Neutrosophic Function [34, 35] Neutrosophic Graph [4, 40] Neutrosophic Hypergraph [46] Neutrosophic SuperHypergraph [56, 57] Neutrosophic Matrix [63, 65] / Linear Algebra Neutrosophic Algebra [71, 72] Neutrosophic HyperAlgebra [80, 81] Plithogenic Relation (cf. [31]) Plithogenic Function [36] Plithogenic Graph [17] Neutrosophic Topology [86, 87] Neutrosophic Probability [94, 95] Neutrosophic Logic [99] Plithogenic Topology [88] Plithogenic Probability [96, 97] Plithogenic Logic [100] Neutrosophic DecisionMaking [107, 108] Plithogenic Decision-Making [109] Neutrosophic Clustering [114, 115] Neutrosophic Numbers [119] Plithogenic Clustering Plithogenic Hypergraph [47] Plithogenic SuperHypergraph [58–60] Plithogenic Matrix [66, 67] / Linear Algebra Plithogenic Algebra [15, 73] Plithogenic HyperAlgebra Plithogenic Numbers [120] 1.3 Our Contribution in This Book A vast number of concepts have been proposed and studied within the frameworks of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets, reflecting the importance of these theories and the diversity of their application domains (cf. [273]). Because of this richness, many ideas and structural patterns appear repeatedly across these four families of uncertainty-oriented models. In this book, we undertake a comprehensive and large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. Our aim is to provide readers with an organized overview of existing developments and, through this survey, to encourage new insights, novel conceptual extensions, and further applications in a wide range of disciplines. # Page. 9 ![Page Image](https://bcdn.docswell.com/page/P7R95G89E9.jpg) Chapter 1. Introduction Table 1.4: Part 2 — Additional concepts across classical, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. Classical Concept Fuzzy Intuitionistic Fuzzy Neutrosophic Plithogenic Metric Space [121] Fuzzy Metric Space [122] Fuzzy Measure Space [128] Neutrosophic Metric Space [124, 125] Neutrosophic Measure Space [131, 132] Neutrosophic Stochastic Process [137, 138] Neutrosophic Markov Chain [142–144] Neutrosophic Dynamical System [148] Neutrosophic Differential Equation [154, 155] Neutrosophic Optimization [159] Neutrosophic Control System [163, 164] Neutrosophic Automaton [169] Neutrosophic Lattice [173] Neutrosophic Category [178] Neutrosophic Time Series [184–186] Plithogenic Metric Space [126] Plithogenic Measure Space Neutrosophic Ontology [190, 191] Plithogenic Ontology Stochastic Process [133, 134] Fuzzy Stochastic Process [135] Markov Chain [140] Fuzzy Markov Chain [141] Intuitionistic Fuzzy Metric Space [123] Intuitionistic Fuzzy Measure Space [129, 130] Intuitionistic Fuzzy Stochastic Process [136] Intuitionistic Fuzzy Markov Chain [93] Dynamical System Fuzzy Dynamical System [146, 147] Intuitionistic Fuzzy Dynamical System Differential Equation [149] Fuzzy Differential Equation [150, 151] Optimization Problem [156] Fuzzy Optimization [157] Fuzzy Control System [161] Intuitionistic Fuzzy Differential Equation [152, 153] Intuitionistic Fuzzy Optimization [158] Intuitionistic Fuzzy Control System [162] Measure Space [127] Control System [160] Automaton [166] Lattice / Order [171] Category [175] Time Series [179] Ontology [187] Fuzzy Automaton [167] Fuzzy Lattice [172] Fuzzy Category [176] Fuzzy Time Series [180, 181] Fuzzy Ontology [188] Intuitionistic Fuzzy Automaton [168] Intuitionistic Fuzzy Lattice [69] Intuitionistic Fuzzy Category [177] Intuitionistic Fuzzy Time Series [182, 183] Intuitionistic Fuzzy Ontology [189] Plithogenic Stochastic Process [139] Plithogenic Markov Chain [145] Plithogenic Dynamical System Plithogenic Differential Equation Plithogenic Optimization Plithogenic Control System [165] Plithogenic Automaton [170] Plithogenic Lattice [174] Plithogenic Category Plithogenic Time Series Table 1.5: Part 3 — Further concepts across classical, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. Classical Concept Fuzzy Intuitionistic Fuzzy Neutrosophic Plithogenic Preference Relation [192] Fuzzy Preference Relation [193, 194] Aggregation Operator [200] Fuzzy Aggregation Operator [201] Entropy / Information Measure [207] Fuzzy Entropy / Information Measure [208] Similarity Measure [212] Fuzzy Similarity Measure [213] Game (Game Theory) [218] Fuzzy Game [219] Fuzzy Neural Network [223, 224] Regression Model [232] Fuzzy Regression Model [233, 234] Database [240] / Knowledge Base Fuzzy Database [241,242] / Knowledge Base [243, 244] Fuzzy Rule-Based System [250] Neutrosophic Preference Relation [197–199] Neutrosophic Aggregation Operator [204,205] Neutrosophic Entropy / Information Measure [210] Neutrosophic Similarity Measure [215, 216] Neutrosophic Game [221] Neutrosophic Neural Network [227–230] Neutrosophic Regression Model [237–239] Neutrosophic Database [246–249] / Knowledge Base Neutrosophic Rule-Based System [252, 253] Neutrosophic Matroid [258, 259] Plithogenic Preference Relation Plithogenic Aggregation Operator [18, 206] Plithogenic Entropy / Information Measure [211] Plithogenic Similarity Measure [217] Plithogenic Game Neural Network [222] Intuitionistic Fuzzy Preference Relation [195, 196] Intuitionistic Fuzzy Aggregation Operator [202,203] Intuitionistic Fuzzy Entropy / Information Measure [209] Intuitionistic Fuzzy Similarity Measure [214] Intuitionistic Fuzzy Game [220] Intuitionistic Fuzzy Neural Network [225, 226] Intuitionistic Fuzzy Regression Model [235, 236] Intuitionistic Fuzzy Database [245] / Knowledge Base Intuitionistic Fuzzy Rule-Based System [251] Intuitionistic Fuzzy Matroid [257] Intuitionistic Fuzzy Machine Learning Model [264, 265] Intuitionistic Fuzzy HyperStructure [79] Neutrosophic Machine Learning Model [266, 267] Plithogenic Machine Learning Model Neutrosophic HyperStructure [271, 272] Plithogenic HyperStructure [174] Rule-Based System Matroid [254] Fuzzy Matroid [255, 256] Machine Learning [261] Fuzzy Machine Learning Model [262, 263] HyperStructure [268, 269] Fuzzy HyperStructure [270] Plithogenic Neural Network (cf. [231]) Plithogenic Regression Model Plithogenic Database / Knowledge Base Plithogenic Rule-Based System Plithogenic Matroid [260] # Page. 10 ![Page Image](https://bcdn.docswell.com/page/PJXQKX837X.jpg) Chapter 2 Preliminaries This section gathers the background notions and notation required for the main results. Unless explicitly stated otherwise, every set that appears is assumed to be finite. 2.1 Fuzzy Set A Fuzzy Set assigns each element a single membership degree in [0, 1] [1,38,274]. The definitions and concrete examples are presented below. Definition 2.1.1 (Fuzzy Set). [1] Let X be a nonempty set. A fuzzy set A on X is characterized by its membership function µA : X → [0, 1]. That is, the fuzzy set A is defined as A = {(x, µA (x)) | x ∈ X}, where µA (x) represents the degree to which the element x ∈ X belongs to the set A. A brief concrete example of this concept is provided below. Example 2.1.2 (Comfortable room temperature). Let the universe be real temperatures X = R in degrees Celsius and define the fuzzy set A = “Comfortable temperature” by the triangular membership  0, t ≤ 16,     t − 16 t − 16     22 − 16 = 6 , 16 < t < 22, µA (t) = 28 − t 28 − t   = , 22 ≤ t < 28,    28 − 22 6    0, t ≥ 28. Concrete evaluations (numerically explicit): µA (18) = 18 − 16 2 1 = = ≈ 0.3333, 6 6 3 µA (22) = 1, µA (27) = 28 − 27 1 = ≈ 0.1667. 6 6 Hence t = 22◦ C is fully comfortable, 18◦ C is moderately comfortable, and 27◦ C is only slightly comfortable. 9 # Page. 11 ![Page Image](https://bcdn.docswell.com/page/3JK95WKNJD.jpg) Chapter 2. Preliminaries Example 2.1.3 (Premium customer by monthly spend). Let X = R≥0 denote monthly customer spending (USD). Define the fuzzy set P = “Premium customer” by the trapezoidal membership with breakpoints a = 200, b = 400, c = 1200, d = 1600:  0,      x−a x − 200   = ,    b − a 200   µP (x) = 1,    d−x 1600 − x    = ,   d − c 400     0, x ≤ a, a < x < b, b ≤ x ≤ c, c < x < d, x ≥ d. Concrete evaluations (step-by-step): µP (300) = 300 − 200 100 = = 0.5, 200 200 µP (900) = 1 µP (1400) = (on the plateau), 1600 − 1400 200 = = 0.5. 400 400 Thus a $300 spender is a premium customer to degree 0.5, $900 is fully premium, and $1400 declines to 0.5 as spending moves into the upper taper. 2.2 Intuitionistic fuzzy set An intuitionistic fuzzy set assigns each element membership and nonmembership degrees [2,275,276]. The definitions and concrete examples are presented below. Definition 2.2.1 (Intuitionistic fuzzy set). [277] Let E be a nonempty set. An intuitionistic fuzzy set (IFS) A on E is given by  A = hx, µA (x), νA (x)i : x ∈ E , where µA , νA : E −→ [0, 1] are, respectively, the membership and non–membership functions, and for every x ∈ E one has 0 ≤ µA (x) + νA (x) ≤ 1. The quantity πA (x) := 1 − µA (x) − νA (x) represents the hesitation degree at x. The usual fuzzy set notion is recovered in the special case νA (x) = 1 − µA (x) for all x ∈ E, that is, when πA (x) = 0 for every x. Example 2.2.2 (Medical diagnosis: intuitionistic fuzzy “high risk” class). Let E = {p1 , p2 , p3 , p4 } be a set of patients and consider the intuitionistic fuzzy concept A = “patient is at high cardiovascular risk”. We specify A by giving, for each patient pi , the membership degree µA (pi ) and the non–membership degree νA (pi ), with µA (pi ) + νA (pi ) ≤ 1. # Page. 12 ![Page Image](https://bcdn.docswell.com/page/LE3WK1ZZE5.jpg) Chapter 2. Preliminaries For instance, suppose the cardiologist assesses x µA (x) νA (x) p1 0.85 0.05 p2 0.60 0.20 p3 0.30 0.40 p4 0.10 0.70 Then, for each pi we have µA (pi ) + νA (pi ) ∈ {0.90, 0.80, 0.70, 0.80} ≤ 1, so these values define an intuitionistic fuzzy set. Here p1 has a high membership and very low non–membership (clearly high risk), while p4 has low membership and high non–membership (clearly not high risk). The remaining patients represent intermediate, uncertain cases with a nonzero “hesitation margin” 1 − µA (pi ) − νA (pi ). Example 2.2.3 (Customer satisfaction: intuitionistic fuzzy “satisfied” class). Let E = {Service A, Service B, Service C} denote three online services offered by a company. Consider the intuitionistic fuzzy notion B = “users are satisfied with the service”. The intuitionistic fuzzy set B is given by  B = hx, µB (x), νB (x)i : x ∈ E , where µB (x) is the degree of satisfaction and νB (x) is the degree of dissatisfaction. Assume that a survey yields the following aggregated assessments: x µB (x) νB (x) Then Service A 0.70 0.10 Service B 0.40 0.30 Service C 0.20 0.60    0.80 for Service A, µB (x) + νB (x) = 0.70 for Service B,   0.80 for Service C, all of which are ≤ 1, so B is an intuitionistic fuzzy set. Service A is mostly satisfactory, Service C is mostly unsatisfactory, and Service B lies in between. The remaining part 1 − µB (x) − νB (x) for each service measures the hesitation or lack of information in the survey responses. 2.3 Neutrosophic Set A Neutrosophic Set assigns to each element three independent membership degrees— truth (T ), indeterminacy (I), and falsity (F )—each taking values in [0, 1], thereby enabling flexible modeling of uncertainty [4, 14, 15, 278]. Neutrosophic Sets are widely recognized as generalizations of Fuzzy Sets and Intuitionistic Fuzzy Sets, and they offer a highly adaptable framework by explicitly accommodating the component I. The definitions and concrete examples are presented below. Definition 2.3.1 (Neutrosophic Set). [27,279] Let X be a non-empty set. A Neutrosophic Set (NS) A on X is characterized by three membership functions: TA : X → [0, 1], IA : X → [0, 1], FA : X → [0, 1], where for each x ∈ X, the values TA (x), IA (x), and FA (x) represent the degrees of truth, indeterminacy, and falsity, respectively. These values satisfy the following condition: 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3. # Page. 13 ![Page Image](https://bcdn.docswell.com/page/8EDK3XR47G.jpg) Chapter 2. Preliminaries A brief concrete example of this concept is provided below. Example 2.3.2 (Medical diagnosis under conflicting evidence: “Patient has influenza”). Medical diagnosis is the systematic process of identifying diseases or conditions from patient history, examinations, tests, and reasoning by clinicians (cf. [280, 281]). Let the universe be X = {patients}. For x ∈ X, suppose we observe: fever TC (x) in ◦ C, antigen test score a(x) ∈ [0, 1], and cough severity c(x) ∈ [0, 1]. Define the neutrosophic membership of the set A = “has influenza” by    T (x) − 37  C TA (x) = min 1, 0.5 a(x) + 0.3 max 0, + 0.2 c(x) , 2   37 − T (x)  C FA (x) = min 1, 0.6 (1 − a(x)) + 0.4 max 0, , 2     IA (x) = 1 − |2a(x) − 1| · 1 − min{1, |TC (x) − 37|/1.5} . Numerical instance. Take TC = 38.2, a = 0.7, c = 0.6 for a patient x∗ . Then TA (x∗ ) = min{1, 0.5 · 0.7 + 0.3 · (38.2 − 37)/2 + 0.2 · 0.6} = min{1, 0.35 + 0.3 · 0.6 + 0.12} = min{1, 0.35 + 0.18 + 0.12} = min{1, 0.65} = 0.65, FA (x∗ ) = min{1, 0.6 · (1 − 0.7) + 0.4 · max(0, (37 − 38.2)/2)} = min{1, 0.6 · 0.3 + 0} = min{1, 0.18} = 0.18,    IA (x∗ ) = 1 − |1.4 − 1| · 1 − min{1, 1.2/1.5} = (1 − 0.4) · (1 − 0.8) = 0.6 · 0.2 = 0.12. Hence (TA , IA , FA )(x∗ ) = (0.65, 0.12, 0.18) and TA + IA + FA = 0.95 ≤ 3 as required. Example 2.3.3 (Logistics ETA assessment: “Shipment arrives on time”). Let X = {shipments}. For x ∈ X, let r(x) ∈ [0, 1] be the carrier on-time reliability, µ(x) > 0 the predicted remaining transit time (days), s(x) > 0 the remaining slack until the promised date (days), and u(x) ∈ [0, 1] an external-uncertainty score (e.g., weather/customs). Define membership for B = “arrives on time” by g(x) := s(x) ∈ (0, 1), s(x) + µ(x) TB (x) = min{1, 0.6 r(x) + 0.4 g(x)}, FB (x) = min{1, 0.6 (1 − r(x)) + 0.4 (1 − g(x))}, IB (x) = u(x). Numerical instance. Let r = 0.85, µ = 1.8, s = 2.0, u = 0.25 for a shipment x† . Then 2.0 2.0 g(x† ) = = ≈ 0.5263, 2.0 + 1.8 3.8 TB (x† ) = min{1, 0.6 · 0.85 + 0.4 · 0.5263} = min{1, 0.51 + 0.2105} = 0.7205, FB (x† ) = min{1, 0.6 · 0.15 + 0.4 · (1 − 0.5263)} = min{1, 0.09 + 0.1895} = 0.2795, IB (x† ) = 0.25. Thus (TB , IB , FB )(x† ) = (0.7205, 0.25, 0.2795) and TB + IB + FB = 1.2500 ≤ 3, satisfying the neutrosophic bounds. # Page. 14 ![Page Image](https://bcdn.docswell.com/page/V7PK4PWVJ8.jpg) Chapter 2. Preliminaries 2.4 Rough Set A Rough Set approximates a subset using lower and upper bounds based on equivalence classes, capturing certainty and uncertainty in membership [282–285]. The definitions and concrete examples are presented below. Definition 2.4.1 (Rough Set Approximation). [286] Let X be a non-empty universe of discourse, and let R ⊆ X × X be an equivalence relation (or indiscernibility relation) on X. The equivalence relation R partitions X into disjoint equivalence classes, denoted by [x]R for x ∈ X, where: [x]R = {y ∈ X | (x, y) ∈ R}. For any subset U ⊆ X, the lower approximation U and the upper approximation U of U are defined as follows: 1. Lower Approximation U : U = {x ∈ X | [x]R ⊆ U }. The lower approximation U includes all elements of X whose equivalence classes are entirely contained within U . These are the elements that definitely belong to U . 2. Upper Approximation U : U = {x ∈ X | [x]R ∩ U 6= ∅}. The upper approximation U contains all elements of X whose equivalence classes have a nonempty intersection with U . These are the elements that possibly belong to U . The pair (U , U ) forms the rough set representation of U , satisfying the relationship: U ⊆ U ⊆ U. A brief concrete example of this concept is provided below. Example 2.4.2 (Email spam filtering with rough approximations). Email spam filtering automatically detects, classifies, and separates unsolicited or malicious messages from legitimate emails using rules and machine learning (cf. [287, 288]). Consider a mailbox with ten emails X = {e1 , e2 , . . . , e10 }. Define an indiscernibility (equivalence) relation R that groups emails by two coarse features only: (sender domain category ∈ {known, unknown} and subject contains the keyword “free” ∈ {yes, no}). This yields the following R-equivalence classes (blocks): B1 = {e1 , e2 } (unknown domain, “free” in subject), B2 = {e3 , e4 , e5 } B3 = {e6 , e7 } (unknown domain, no “free”), (known domain, “free”), B4 = {e8 , e9 , e10 } (known domain, no “free”). Let the target concept be U = {emails that are actually spam} = {e1 , e2 , e3 , e6 , e7 }. By definition of rough sets (with respect to R): # Page. 15 ![Page Image](https://bcdn.docswell.com/page/2JVVX28RJQ.jpg) Chapter 2. Preliminaries 1) Lower approximation  U = x ∈ X : [x]R ⊆ U = B1 ∪ B3 = {e1 , e2 , e6 , e7 }. Explanation: B1 ⊆ U and B3 ⊆ U ; hence all elements of these blocks are certainly spam. 2) Upper approximation  U = x ∈ X : [x]R ∩ U 6= ∅ = B1 ∪ B2 ∪ B3 = {e1 , e2 , e3 , e4 , e5 , e6 , e7 }. Explanation: B2 intersects U (it contains e3 ), so all of B2 are possibly spam. Block B4 does not intersect U . 3) Boundary, positive, and negative regions BNDR (U ) = U \ U = {e3 , e4 , e5 }, POSR (U ) = U = {e1 , e2 , e6 , e7 }, NEGR (U ) = X \ U = {e8 , e9 , e10 }. 4) Numerical indices (with explicit values) |U | = 4, |U | = 7, |X| = 10. Pawlak accuracy of approximation: αR (U ) = |U | 4 = ≈ 0.5714. 7 |U | Coverage (certainty rate in the universe): κR (U ) = |U | 4 = = 0.4. |X| 10 Interpretation. Emails in B1 and B3 are certainly spam under these coarse features; emails in B2 are ambiguous (boundary); emails in B4 are certainly non-spam. Example 2.4.3 (Factory quality control with sensor-based indiscernibility). Factory quality control monitors production processes and outputs, inspecting samples, detecting defects, and ensuring products meet safety and performance standards (cf. [289]). A factory produces twelve items X = {p1 , . . . , p12 }. Two coarse sensors are used: surface scratch flag ∈ {0, 1} and thickness bin ∈ {thin, thick}. Items are indiscernible if they share the same ordered pair (scratch, thickness). This induces the R-equivalence classes E1 = {p1 , p2 , p3 } E2 = {p4 , p5 } (1, thin), (1, thick), E3 = {p6 , p7 , p8 } (0, thin), E4 = {p9 , p10 , p11 , p12 } (0, thick). Let the true concept be U = {defective items} = {p1 , p2 , p3 , p4 , p9 } (obtained after a detailed downstream inspection). # Page. 16 ![Page Image](https://bcdn.docswell.com/page/5EGLVR56JL.jpg) Chapter 2. Preliminaries 1) Lower approximation  U = x ∈ X : [x]R ⊆ U = E1 = {p1 , p2 , p3 }. Explanation: all of E1 are truly defective; other blocks contain a mix. 2) Upper approximation  U = x ∈ X : [x]R ∩ U 6= ∅ = E1 ∪ E2 ∪ E4 = {p1 , p2 , p3 , p4 , p5 , p9 , p10 , p11 , p12 }. 3) Boundary, positive, and negative regions BNDR (U ) = U \ U = {p4 , p5 , p9 , p10 , p11 , p12 }, POSR (U ) = U = {p1 , p2 , p3 }, NEGR (U ) = X \ U = {p6 , p7 , p8 }. 4) Numerical indices (computed explicitly) |U | = 3, |U | = 9, |X| = 12. Pawlak accuracy of approximation: αR (U ) = |U | 3 1 = = ≈ 0.3333. 9 3 |U | Coverage (certainty rate in the universe): κR (U ) = |U | 3 1 = = = 0.25. |X| 12 4 Interpretation. Items in E1 are certainly defective under the coarse sensors; E2 and E4 are ambiguous (boundary); E3 is certainly non-defective. Rough approximations separate what can be guaranteed (lower), what is only possible (upper), and what is impossible (negative) using only the coarse sensor information. Table 2.1 lists the extended rough–set families. Table 2.1: Concise comparison of extended rough–set families Concept One–line summary Refs. HyperRough Sets Rough approximations over hyperrelations/hypergraphs (multiway neighborhoods). Hierarchical (super/hyper) powerset rough models with multilayer approximations. Approximations induced by dominance (preference) relations for MCDA. Bayes risk thresholds yield three-way decisions (accept/defer/reject). Rough approximations extended to multisets with element multiplicities. Unified rough models combining multiple relations/approximation operators. [290, 291] SuperHyperRough Sets Dominance-Based Rough Sets (DRSA) Decision-Theoretic Rough Sets (DTRS) Rough Multisets Composite Rough Sets [292–295] [296–298] [299–301] [302, 303] [304–306] # Page. 17 ![Page Image](https://bcdn.docswell.com/page/4JQY6VZ27P.jpg) Chapter 2. Preliminaries 2.5 Soft set A Soft Set is a parameterized family of subsets used to handle uncertainty, introduced by Molodtsov in 1999 for decision-making problems [307, 308]. The definitions and concrete examples are presented below. Definition 2.5.1 (Soft Set [308]). Let U be a universe set and E be a set of parameters. Let A ⊆ E and denote by P(U ) the power set of U . A pair (F, A) is called a soft set over U if F : A → P(U ). For each parameter  ∈ A, the set F () is called the -approximation of the soft set (F, A). In other words, a soft set over U is a parameterized family of subsets of U . A brief concrete example of this concept is provided below. Example 2.5.2 (Apartment Selection (Tokyo Rental Case)). Apartment selection evaluates multiple rental options using criteria like location, rent, size, amenities, and suitability for residents and lifestyles preferences (cf. [309]). Let the universe U = {A1 , A2 , A3 , A4 } denote four available apartments. Let the parameter set be E = {near_station, pet_friendly, under_U120,000, twoLDK_or_more, built_after_2015}, and take A = E. Define the soft set (F, A) over U by listing, for each parameter  ∈ A, the subset F () ⊆ U of apartments satisfying : F (near_station) = {A1 , A3 , A4 }, F (pet_friendly) = {A2 , A3 }, F (under_U120,000) = {A1 , A2 }, F (twoLDK_or_more) = {A1 , A4 }, F (built_after_2015) = {A3 , A4 }. Interpretation. The mapping encodes, for each practical requirement, which apartments meet it. For a renter who requires “near_station” and “pet_friendly”, the feasible candidates are F (near_station) ∩ F (pet_friendly) = {A3 }. Example 2.5.3 (Laptop Purchase under Practical Preferences). Laptop purchase is selecting a notebook computer balancing performance, portability, battery life, budget, brand support, future needs, and upgrades (cf. [310]). Let the universe U = {L1 , L2 , L3 , L4 } denote four laptop models under consideration. Let the parameter set be E = {lightweight(≤ 1.2 kg), long_battery(≥ 10 h), budget(≤ U100,000), ram16GB, screen14in}, and take A = E. Define the soft set (F, A) over U by F (lightweight) = {L1 , L3 }, F (long_battery) = {L1 , L2 , L4 }, F (budget) = {L2 , L3 }, F (ram16GB) = {L1 , L4 }, F (screen14in) = {L1 , L3 , L4 }. # Page. 18 ![Page Image](https://bcdn.docswell.com/page/K74W4M3PE1.jpg) Chapter 2. Preliminaries Interpretation. Each parameter captures a user preference, and F () lists laptops meeting it. A buyer who insists on “budget” and “screen14in” filters to F (budget) ∩ F (screen14in) = {L3 }. If the buyer also prefers “long_battery”, then F (budget) ∩ F (screen14in) ∩ F (long_battery) = ∅, signaling that no single item simultaneously satisfies all three preferences and that trade-offs are required. Related concepts of the Soft Set include HyperSoft Set [311, 312], IndetermSoft Set [313–315], SuperHyperSoft Set [316, 317], TreeSoft Set [318–321], ForestSoft Set [322–325], Bipolar Soft Set [326, 327], and Double-Framed Soft Set [328,329], all of which extend the classical Soft Set framework in different directions. The definitions of the HyperSoft Set and SuperHyperSoft Set are given as follows. Note that table 2.2 presents a concise comparison of the Soft Set, Hypersoft Set, and SuperHyperSoft Set. Table 2.2: Concise comparison of Soft Set, Hypersoft Set, and SuperHyperSoft Set. Here P(U ) denotes the power set of U . Soft Set Hypersoft Set SuperHyperSoft Set Universe Parameter domain U A ⊆ E (single attribute) U C = A1 × · · · × Am (fixed m attributes) Input (query key) ∈A Mapping Granularity F : A → P(U ) Single parameter value γ = (γ1 , . . . , γm ) with γi ∈ A i G : C → P(U ) Exact m-tuple of values Expressiveness Reductions Low (one attribute at a time) — Medium (multi-attribute conjunction) m=1 ⇒ Soft Set Typical query “cars with color=red” “laptops with (CPU = i7, RAM = 16, SSD = 512)” U C = P(A1 ) × · · · × P(An ) (subset–valued per attribute) α = (α1 , . . . , αn ) with αi ⊆ Ai F : C → P(U ) Sets of admissible values per attribute (“any of these”) High (multi-attribute with set-level choices) All αi singletons ⇒ Hypersoft; n=1 and singleton ⇒ Soft “trips with season∈ {Spring, Autumn}, budget∈ {Low, Mid}, type∈ {Solo, Business}” Definition 2.5.4 (Hypersoft Set). [312] Let U be a universal set, and let A1 , A2 , . . . , Am be attribute domains. Define C = A1 × A2 × · · · × Am , the Cartesian product of these domains. A hypersoft set over U is a pair (G, C), where G : C → P(U ). The hypersoft set is expressed as: (G, C) = {(γ, G(γ)) | γ ∈ C, G(γ) ∈ P(U )}. For an m-tuple γ = (γ1 , γ2 , . . . , γm ) ∈ C, where γi ∈ Ai for i = 1, 2, . . . , m, G(γ) represents the subset of U corresponding to the combination of attribute values γ1 , γ2 , . . . , γm . Example 2.5.5 (Hypersoft Set — Laptop selection by multi-attribute profile). Let the universe of laptops be U = {L1 , L2 , L3 , L4 , L5 , L6 }. Assign each item its (CPU, RAM in GB, SSD in GB): L1 : (i5, 8, 256), L2 : (i5, 16, 512), L3 : (i7, 8, 512), L4 : (i7, 16, 256), L5 : (i7, 16, 512), L6 : (i5, 8, 512). # Page. 19 ![Page Image](https://bcdn.docswell.com/page/LJ1Y481XEG.jpg) Chapter 2. Preliminaries Let the attribute domains be A1 = {i5, i7}, A2 = {8, 16}, A3 = {256, 512}, and let C = A1 × A2 × A3 . Define the hypersoft mapping G : C → P(U ) by G(γ1 , γ2 , γ3 ) = { L ∈ U : (CPU(L), RAM(L), SSD(L)) = (γ1 , γ2 , γ3 ) }. Then, concretely, G(i7, 16, 512) = {L5 }, G(i5, 8, 512) = {L6 }, G(i5, 16, 256) = ∅ (no laptop matches). This hypersoft set captures the real-life query “laptops with a given CPU–RAM–SSD triple,” returning exactly the subset of products in U that satisfy the chosen attribute combination. Definition 2.5.6 (SuperHyperSoft Set). [330] Let U be a universal set, and let P(U ) denote the power set of U . Consider n distinct attributes a1 , a2 , . . . , an , where n ≥ 1. Each attribute ai is associated with a set of attribute values Ai , satisfying the property Ai ∩ Aj = ∅ for all i 6= j. Define P(Ai ) as the power set of Ai for each i = 1, 2, . . . , n. Then, the Cartesian product of the power sets of attribute values is given by: C = P(A1 ) × P(A2 ) × · · · × P(An ). A SuperHyperSoft Set over U is a pair (F, C), where: F : C → P(U ), and F maps each element (α1 , α2 , . . . , αn ) ∈ C (with αi ∈ P(Ai )) to a subset F (α1 , α2 , . . . , αn ) ⊆ U . Mathematically, the SuperHyperSoft Set is represented as: (F, C) = {(γ, F (γ)) | γ ∈ C, F (γ) ∈ P(U )}. Here, γ = (α1 , α2 , . . . , αn ) ∈ C, where αi ∈ P(Ai ) for i = 1, 2, . . . , n, and F (γ) corresponds to the subset of U defined by the combined attribute values α1 , α2 , . . . , αn . Example 2.5.7 (SuperHyperSoft Set — Travel package filtering by subset-valued criteria). A travel package bundles transport, accommodation, activities, and services into one pre-arranged offer at a combined, often discounted price overall (cf. [331]). Let the universe of travel packages be U = {p1 , p2 , p3 , p4 , p5 }. Attribute value sets: A1 = {Spring, Summer, Autumn, Winter}, A2 = {Low, Mid, High}, A3 = {Solo, Family, Business}. Each package carries one concrete value per attribute: p1 : (Summer, High, Family), p2 : (Winter, Mid, Family), p3 : (Autumn, Mid, Solo), p4 : (Autumn, High, Business), Form the SuperHyperSoft domain C = P(A1 ) × P(A2 ) × P(A3 ), p5 : (Spring, Low, Solo). # Page. 20 ![Page Image](https://bcdn.docswell.com/page/GJWGXZ8K72.jpg) Chapter 2. Preliminaries and define F : C → P(U ) by, for γ = (α1 , α2 , α3 ), F (α1 , α2 , α3 ) = { p ∈ U : Season(p) ∈ α1 , Budget(p) ∈ α2 , Type(p) ∈ α3 }. Concrete queries with subset-valued criteria: α1 = {Summer, Autumn}, α2 = {High}, α3 = {Family, Business} : F (α1 , α2 , α3 ) = {p1 , p4 }; α1 = {Winter}, α2 = {Mid}, α3 = {Family} : F (α1 , α2 , α3 ) = {p2 }; α1 = {Spring, Autumn}, α2 = {Low, Mid}, α3 = {Solo} : F (α1 , α2 , α3 ) = {p3 , p5 }. Allowing subsets αi ⊆ Ai in the query domain enables practical filters such as “any of these seasons,” “budget up to Mid,” and “Solo or Business travelers,” producing the matching subset of packages in U. # Page. 21 ![Page Image](https://bcdn.docswell.com/page/4EZL618N73.jpg) Chapter 2. Preliminaries # Page. 22 ![Page Image](https://bcdn.docswell.com/page/Y76W2LP97V.jpg) Chapter 3 Dynamic Reviews and Results of Uncertain Sets This chapter surveys the relationships among Plithogenic Sets, Fuzzy Sets, Intuitionistic Fuzzy Sets, and Neutrosophic Sets. Plithogenic Sets are known to generalize Fuzzy Sets, Intuitionistic Fuzzy Sets, and Neutrosophic Sets by incorporating attribute-based membership and contradiction management. Table 3.1and3.2 presents an overview of the uncertain-set families discussed in this chapter. 3.1 Plithogenic Sets of Uncertain Values The Plithogenic Set is a mathematical framework designed to integrate multi-valued degrees of appurtenance and contradiction, making it particularly effective for addressing complex decision-making scenarios [10,18]. Numerous studies have explored the properties and applications of Plithogenic Sets, as highlighted in works such as [332–335]. Definition 3.1.1 (Plithogenic Set). [18, 100] Let S be a universal set, and P ⊆ S. A Plithogenic Set P S is defined as: P S = (P, v, P v, pdf, pCF ) where: • v is an attribute. • P v is the range of possible values for the attribute v. • pdf : P × P v → [0, 1]s is the Degree of Appurtenance Function (DAF) 1 • pCF : P v × P v → [0, 1]t is the Degree of Contradiction Function (DCF). 1 It is important to note that the definition of the Degree of Appurtenance Function varies across different papers. Some studies define this concept using the power set, while others simplify it by avoiding the use of the power set [336]. The author has consistently defined the Classical Plithogenic Set without employing the power set. 21 # Page. 23 ![Page Image](https://bcdn.docswell.com/page/G75M21KD74.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.1: Overview of the uncertain-set families in this chapter (Part I). Uncertain-set family Brief overview Plithogenic Set Attribute-based framework with appurtenance degrees and a contradiction function, unifying fuzzy, intuitionistic, neutrosophic and related sets. Splits each attribute value into m semantic poles (e.g. positive / neutral / negative) with pole-level contradiction. Uses complex-valued degrees; modulus measures strength, argument encodes extra features such as time, context, or direction. Plithogenic structure on hyper- and superhyperobjects, with nested degrees on superhyper-elements in higher-order settings. Employs linguistic labels (“low”, “medium”, “high”, etc.) as values with plithogenic aggregation and labelwise contradiction. Combines q-rung orthopair bounds on truth/falsity with plithogenic attributes and contradiction-sensitive operators. Family indexed by n organizing multiple membership components and collecting multi-valued plithogenic variants. Iterated, multi-level plithogenic construction for hierarchical or staged decision and evaluation processes. Represents appurtenance by intervals to capture measurement imprecision and disagreement between sources or experts. Allows over- and under-membership, modeling paradoxical, inconsistent, or extreme information in a plithogenic way. Combines a crisp component with an interval-valued one, both treated plithogenically, for two-layer uncertainty. Soft-like parameter, hyper-parameter, and superhyperparameter families equipped with plithogenic degrees and contradiction. Assigns a finite set of plausible plithogenic degrees to each element–attribute pair, explicitly modeling hesitation. Uses spherical-type constraints on positive, neutral, and negative components inside a plithogenic structure. m-Polar Plithogenic Set Complex Plithogenic Set SuperHyperPlithogenic Set Plithogenic Linguistic Set q-rung orthopair Plithogenic Set Type-n Plithogenic Set Iterative MultiPlithogenic Set Interval-Valued Plithogenic Set Plithogenic OffSet Plithogenic Cubic Set Plithogenic Soft / HyperSoft / SuperHyperSoft Set Hesitant Plithogenic Set Spherical Plithogenic Set These functions satisfy the following axioms for all a, b ∈ P v: 1. Reflexivity of Contradiction Function: pCF (a, a) = 0 2. Symmetry of Contradiction Function: pCF (a, b) = pCF (b, a) We now present concrete examples of the concept. Example 3.1.2 (Apartment Rental under Contradictory Criteria). Apartment rental is a housing arrangement where tenants pay regular rent to use residential units owned by landlords for living (cf. [337]). # Page. 24 ![Page Image](https://bcdn.docswell.com/page/9J29416MER.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.2: Overview of the uncertain-set families in this chapter (Part II). Uncertain-set family Brief overview T-Spherical Plithogenic Set T-spherical (or q-rung spherical) refinement with powered-sum bounds on truth, indeterminacy, and falsity. Rough lower/upper approximations constructed from plithogenic membership and contradiction-aware similarity. Plithogenic sets on tree-structured universes or attributes, where aggregation follows the tree hierarchy. Extension of TreePlithogenic sets to several disjoint hierarchies (a forest) in one plithogenic model. Soft expert system in which each expert’s opinions form a plithogenic soft set with attribute-wise contradiction. Time- or state-dependent plithogenic degrees and contradiction, evolving across scenarios or time steps. Integrates probability distributions with plithogenic degrees or attributes, mixing probabilistic and plithogenic uncertainty. Uses hyperreal, near-[0, 1] membership vectors with infinitesimal over/under-membership under plithogenic contradiction. Plithogenic degrees constrained by linear Diophantine relations on parameters, preserving attribute-based contradiction. Assigns each item–attribute pair a subset of membership vectors in [0, 1]s , aggregated via contradiction-weighted plithogenic operators. Plithogenic extension of triangular fuzzy / intuitionistic / neutrosophic numbers with triangular membership shapes. Plithogenic extension of trapezoidal fuzzy, intuitionistic fuzzy, and neutrosophic numbers using trapezoidal profiles. Splits components (e.g. truth, indeterminacy, falsity) into several subcomponents within one plithogenic object. Plithogenic analogue of picture fuzzy sets with acceptance, neutrality, rejection, and refusal under contradiction. Plithogenic Rough Set TreePlithogenic Set ForestPlithogenic Set Plithogenic Soft Expert Set Dynamic Plithogenic Set Probabilistic Plithogenic Set Nonstandard Plithogenic Set Linear Diophantine Plithogenic Set Subset-Valued Plithogenic Set Triangular Plithogenic Set Trapezoidal Plithogenic Set Refined Plithogenic Set Picture Plithogenic Set Let the universe be three candidate apartments P = {A1 , A2 , A3 }. Consider one attribute v = rental_criterion with value-set P v = {low_rent, near_station, large_space}. We take a plithogenic Degree of Appurtenance Function (DAF) with s = 3 components (truth T , indeterminacy I, falsity F ) and a scalar Degree of Contradiction Function (DCF) pCF : P v × P v → [0, 1] (t = 1). The contradiction matrix (symmetric, zeros on the diagonal) is pCF low_rent near_station large_space low_rent 0 0.30 0.70 near_station 0.30 0 0.50 large_space 0.70 0.50 0 and the DAF pdf : P × P v → [0, 1]3 (entries are (T, I, F )) is specified by pdf (A1 , low_rent) = (0.85, 0.05, 0.10), pdf (A1 , near_station) = (0.90, 0.05, 0.10), pdf (A1 , large_space) = (0.40, 0.20, 0.60), pdf (A2 , low_rent) = (0.60, 0.15, 0.35), pdf (A2 , near_station) = (0.50, 0.20, 0.50), pdf (A2 , large_space) = (0.75, 0.10, 0.25), pdf (A3 , low_rent) = (0.45, 0.25, 0.55), pdf (A3 , near_station) = (0.80, 0.10, 0.20), pdf (A3 , large_space) = (0.55, 0.15, 0.45). Interpretation. The decision maker may select a dominant value (e.g., near_station); the contradictions pCF quantify how much other values conflict with the dominant one during plithogenic aggregation. # Page. 25 ![Page Image](https://bcdn.docswell.com/page/DEY4MZ9PJM.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.1.3 (Personalized Diet Planning (Clinical Setting)). Personalized diet planning designs nutrition plans tailored to an individual’s health goals, preferences, restrictions, and lifestyle using data-driven adjustments continuously (cf. [338, 339]). Let the universe be three patients P = {p1 , p2 , p3 }. Consider the attribute v = diet_style with value-set P v = {ketogenic, low_fat, vegetarian}. Use a scalar DCF with the following matrix: pCF ketogenic low_fat vegetarian ketogenic 0 0.60 0.90 low_fat 0.60 0 0.20 vegetarian 0.90 0.20 0 and define the DAF (triples (T, I, F )) by pdf (p1 , ketogenic) = (0.70, 0.10, 0.30), pdf (p1 , low_fat) = (0.40, 0.20, 0.60), pdf (p1 , vegetarian) = (0.30, 0.20, 0.70), pdf (p2 , ketogenic) = (0.20, 0.20, 0.80), pdf (p2 , low_fat) = (0.65, 0.15, 0.35), pdf (p2 , vegetarian) = (0.55, 0.15, 0.45), pdf (p3 , ketogenic) = (0.50, 0.25, 0.50), pdf (p3 , low_fat) = (0.45, 0.20, 0.55), pdf (p3 , vegetarian) = (0.75, 0.10, 0.25). Interpretation. Strong contradictions (e.g., ketogenic vs vegetarian at 0.90) reflect dietary principles that rarely co-exist; the (T, I, F ) entries encode how well each patient aligns with each style under medical, lifestyle, and ethical constraints. Example 3.1.4 (Sustainable Car Choice (Powertrain Trade-offs)). A sustainable car choice prioritizes low emissions, high fuel efficiency, lifecycle impact, and ethics while still meeting everyday mobility needs (cf. [340]). Let the universe be three cars P = {C1 , C2 , C3 }. Consider the attribute v = powertrain with value-set P v = {electric, hybrid, gasoline}. Take the scalar DCF pCF electric hybrid gasoline electric 0 0.30 1.00 hybrid 0.30 0 0.60 gasoline 1.00 0.60 0 and the DAF (triples (T, I, F )) as pdf (C1 , electric) = (0.90, 0.05, 0.10), pdf (C1 , hybrid) = (0.70, 0.10, 0.30), pdf (C1 , gasoline) = (0.10, 0.20, 0.90), pdf (C2 , electric) = (0.40, 0.20, 0.60), pdf (C2 , hybrid) = (0.80, 0.10, 0.20), pdf (C2 , gasoline) = (0.35, 0.20, 0.65), pdf (C3 , electric) = (0.20, 0.15, 0.80), pdf (C3 , hybrid) = (0.50, 0.20, 0.50), pdf (C3 , gasoline) = (0.85, 0.05, 0.15). Interpretation. The contradiction pCF (electric, gasoline) = 1.00 models fully opposed values (zero compatibility) in environmental terms, while electric vs hybrid has only mild contradiction 0.30. The (T, I, F ) vectors encode each car’s appurtenance to the plithogenic set of “sustainable choices” under these value interactions. # Page. 26 ![Page Image](https://bcdn.docswell.com/page/VJNYW3LM78.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Tables 3.3 and 3.4 present examples of sets that can be generalized by the Plithogenic Set framework (cf. [341]). Various concepts for handling uncertainty are being continuously defined and studied in the scientific community. Table 3.3: A catalogue of Plithogenic Set families by number of components s. s 1 2 t 0 0 3 0 4 0 5 0 6 7 0 0 8 9 n 0 0 0 2n 3n 1 2 3 4 5 6 7 8 9 1 2 3 0 0 1 1 1 1 1 1 1 1 1 2 2 2 Representative type(s) Fuzzy Set [1, 44]; N-Set [342]; Shadowed Set [343–345] Intuitionistic Fuzzy Set [2, 346]; Vague Set [5, 347]; Bipolar Fuzzy Set [348]; Intuitionistic Evidence Set [349–351]; Variable Fuzzy Set [352–354]; Paraconsistent Fuzzy Set [355, 356]; Bifuzzy Set [357, 358] Neutrosophic Set(a) [278,279]; Hesitant Fuzzy Set [6,359]; Tripolar Fuzzy Set [360– 362]; Three-way Fuzzy Set [363, 364]; Picture Fuzzy Set [7, 365]; Spherical Fuzzy Set [366, 367]; Inconsistent Intuitionistic Fuzzy Set [368, 369]; Ternary Fuzzy Set [334,370]; Neutrosophic Fuzzy Set [371,372]; (Kleene three-valued logic [373,374];) Neutrosophic Vague Set [375, 376] Quadripartitioned Neutrosophic Set [8, 377]; Double-Valued Neutrosophic Set [378, 379]; Dual hesitant fuzzy sets [380, 381]; Ambiguous Set(b) [382–384]; LocalNeutrosophic Set [385]; Support Neutrosophic Set [386]; (Belnap four-valued logic [387, 388];) Turiyam Neutrosophic Set(c) [389–392] Pentapartitioned Neutrosophic Set [393–395]; Triple-valued Neutrosophic Set [396–398] Hexapartitioned Neutrosophic Set; Quadruple-Valued Neutrosophic Set [397, 399] Heptapartitioned Neutrosophic Set; Quintuple-Valued Neutrosophic Set [397, 400, 401] Octapartitioned Neutrosophic Set [402] Nonapartitioned Neutrosophic Set [402] n-Refined Fuzzy Set [403, 404]; (n-valued (Łukasiewicz) logic [405];) Multi-valued (Fuzzy) Sets [406]; MultiFuzzy Set [407] n-Refined Intuitionistic Fuzzy Set [404]; Multi-Intuitionistic Fuzzy Set [407] n-Refined Neutrosophic Set [404]; Multi-Neutrosophic Set [407, 408] Plithogenic Fuzzy Set [23, 409, 410] Plithogenic Intuitionistic Fuzzy Set [411] Plithogenic Neutrosophic Set [24, 25, 412] Plithogenic Quadripartitioned Neutrosophic Set Plithogenic Pentapartitioned Neutrosophic Set Plithogenic Hexapartitioned Neutrosophic Set Plithogenic Heptapartitioned Neutrosophic Set Plithogenic Octapartitioned Neutrosophic Set Plithogenic Nonapartitioned Neutrosophic Set Double-valued Plithogenic Fuzzy Set [413] Double-valued Plithogenic Intuitionistic Fuzzy Set [413] Double-valued Plithogenic Neutrosophic Set [413] (a) It is widely recognized that the neutrosophic set generalizes the intuitionistic fuzzy set, the inconsistent intuitionistic fuzzy set (including picture fuzzy and ternary fuzzy sets), the Pythagorean fuzzy set, the spherical fuzzy set, and the q-rung orthopair fuzzy set; similarly, neutrosophication generalizes regret theory, grey system theory, and three-way decision theory [16]. (b) Ambiguous Sets are known to form a subclass of Quadripartitioned Neutrosophic Sets [8, 377] as well as of DoubleValued Neutrosophic Sets [384]. (c) Turiyam Neutrosophic Sets are known to constitute a subclass of the existing Quadripartitioned Neutrosophic Sets [414]. 3.2 m-Polar Plithogenic Set The m-polar (multipolar) structure is used in various contexts (e.g. [415]) and can also be applied within Fuzzy, Neutrosophic, and Plithogenic frameworks. An m-polar plithogenic set models elements with m contrasting poles, appurtenance vectors, and value/pole contradiction-controlled aggregation for complex decision contexts. The m-polar plithogenic set generalizes the Multipolar Fuzzy Set [416, 417], the Multipolar Intuitionistic Fuzzy Set [418, 419], and the Multipolar Neutrosophic Set [420–423]. # Page. 27 ![Page Image](https://bcdn.docswell.com/page/YE9PX94WJ3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.4: Overview of Plithogenic Sets by the contradiction dimension t in the Degree of Contradiction Function pCF : P v × P v → [0, 1]t . t DCF space [0, 1]t Interpretation of pCF Reduction / Containment Recovers classical multivalued families: fuzzy (s=1), intuitionistic/vague (s=2), neutrosophic (s=3), etc. Typical use or example 0 [0, 1]0 ≡ {0} No contradiction modeling (or a constant-zero DCF). Aggregation relies only on the Degree of Appurtenance Function (DAF) pdf . 1 [0, 1] Single scalar contradiction per pair of values; typically used to attenuate pdf relative to a dominant value or context. The “classical” plithogenic set. Operations weight/penalize memberships via (1 − pCF ) or related schemes. One principal tension axis (e.g., laptop choice: lightweight vs performance; purchasing: price vs quality). 2 [0, 1]2 Two-dimensional contradiction vector (e.g., structural vs contextual conflict). Requires a vector-to-scalar reducer (weighted sum, norm, or lexicographic rule) during aggregation. Double-valued plithogenic variants; strictly more expressive than t=1 and reduces to t=1 by projecting/ignoring one component. Modeling dual antagonisms (e.g., treatment planning: efficacy vs side-effects and cost; networking: latency vs energy). When attribute values do not meaningfully conflict; e.g., ranking items using only performance scores without trade-off penalties. Definition 3.2.1 (m-Polar Plithogenic Set). Let S be a universe and P ⊆ S a domain of interest. Let v be a (fixed) attribute with value set Pv . Let Π = {π1 , . . . , πm } be a set of m poles (for instance, m = 2 for “positive/negative”, m = 3 for “positive/neutral/negative”). Fix integers s ≥ 1 and t ≥ 0. An m-polar plithogenic set (abbrev. mPPS) on P with respect to v is a tuple mPPS :=  P, v, Pv , Π, pdfΠ , pCFv , pCFΠ , Agg , where • the m-polar plithogenic degree of appurtenance function pdfΠ : P × Pv −→ [0, 1]s m assigns to each pair (x, α) ∈ P × Pv an m-tuple of s-component vectors, pdfΠ (x, α) =  µ1 (x, α), . . . , µm (x, α) , µr (x, α) ∈ [0, 1]s (r = 1, . . . , m), where each µr (x, α) encodes the s plithogenic components (e.g. s = 1 fuzzy grade, s = 2 intuitionistic pair, s = 3 neutrosophic triple) at pole πr for the value α; • pCFv : Pv × Pv → [0, 1]t is the value-level Degree of Contradiction Function (DCF), symmetric and reflexive: pCFv (α, α) = 0, pCFv (α, β) = pCFv (β, α) (∀ α, β ∈ Pv ); • pCFΠ : Π × Π → [0, 1]t is the pole-level DCF, also symmetric and reflexive: pCFΠ (π, π) = 0, pCFΠ (π, π 0 ) = pCFΠ (π 0 , π) (∀ π, π 0 ∈ Π); # Page. 28 ![Page Image](https://bcdn.docswell.com/page/GE8D29QRED.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets • Agg is a fixed contradiction-aware reduction operator that, for any chosen dominant context (d, δ) ∈ Pv × Π, produces an s-component aggregated degree  m  pdf(d,δ) (x, α) := Agg pdfΠ (x, α), pCFv (α, d), pCFΠ (πr , δ) r=1 ∈ [0, 1]s . A canonical choice is the componentwise weighted mean based on compatibility weights   wr (α | d, δ) := 1 − Φv (pCFv (α, d)) 1 − ΦΠ (pCFΠ (πr , δ)) , where Φv , ΦΠ : [0, 1]t → [0, 1] are monotone fusions of the t-dimensional contradiction vectors. Writing  µr (x, α) = µr,1 (x, α), . . . , µr,s (x, α) , the aggregated degree is then given componentwise by m X (d,δ) pdfj (x, α) := r=1 wr (α | d, δ) µr,j (x, α) m X , j = 1, . . . , s, (3.1) wr (α | d, δ) r=1 with the usual convention that 0/0 := 0.  The triple pdfΠ , pCFv , pCFΠ describes how each element is evaluated at multiple poles and how contradictions act both between attribute values and between poles. The operator Agg specifies how a dominant context (d, δ) attenuates or amplifies polar components to obtain a single s-component plithogenic degree when required. Remark 3.2.2. If s = 1, then pdfΠ (x, α) ∈ ([0, 1]1 )m ∼ = [0, 1]m and each pole carries a single fuzzy-grade component. For s = 2 and s = 3 one recovers, respectively, intuitionistic-type and neutrosophic-type m-polar structures within the same plithogenic framework. Tables 3.5 and 3.6 present a selection of concepts that can be generalized within the framework of the Multipolar Plithogenic Set. We now present concrete examples of the concept. Example 3.2.3 (Hiring under Multi-Polar Evaluation (m = 3)). Let the universe of candidates be P = {c1 , c2 , c3 }. Consider the attribute v = experience_level with value set Pv = {junior, mid, senior}. Take the pole set Π = {π1 , π2 , π3 } = {pro, neutral, contra} so that m = 3. We work in the special case s = 1, hence we identify ([0, 1]1 )3 ∼ = [0, 1]3 and write each appurtenance vector as a triple of real numbers in [0, 1]. The value–level DCF pCFv : Pv × Pv → [0, 1] and pole–level DCF pCFΠ : Π × Π → [0, 1] are given (in matrix form) by pCFv junior mid senior junior 0 0.30 0.80 mid 0.30 0 0.40 senior 0.80 0.40 0 # Page. 29 ![Page Image](https://bcdn.docswell.com/page/LELM2WX27R.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.5: Cases for m ∈ {1, 2, 3, ...} (polar level), t = 0 (no DCF), and s ∈ {1, 2, 3, 4, 5}. Here m=1 = classical, m=2 = bipolar, m=3 = tripolar. m 1 1 1 2 2 t 0 0 0 0 0 s 1 2 3 1 2 2 0 3 2 0 3 2 0 3 2 0 4 2 0 5 3 0 1 3 0 2 3 0 3 m 0 1 m 0 2 m 0 3 m 0 3 m 0 3 m 0 4 Representative name Fuzzy Set Intuitionistic Fuzzy Set Neutrosophic Set Bipolar Fuzzy Set [348, 424] Bipolar Intuitionistic Fuzzy Set [425–427] (Bipolar vague sets [428, 429]) Bipolar Neutrosophic Set [430–432] Bipolar Picture Fuzzy Set [433, 434]/Bipolar Hesitant Fuzzy Set [435, 436] Bipolar Spherical Fuzzy Set [437] Bipolar Quadripartitioned Neutrosophic Set [438–440] Bipolar Pentapartitioned Neutrosophic Set [441, 442] Tripolar Fuzzy Set [360–362, 443] Tripolar Intuitionistic Fuzzy Set Tripolar Neutrosophic Set [444, 445] m-polar Fuzzy Set [417, 446, 447] m-polar Intuitionistic Fuzzy Set [418, 419] m-polar Neutrosophic Set [420, 421, 448] m-polar Picture Fuzzy Set [449, 450]/m-polar Hesitant Fuzzy Set [451, 452] m-polar Spherical Fuzzy Set [453, 454] m-polar Quadripartitioned Neutrosophic Set [455, 456] pCFΠ pro neutral contra Brief note (poles / comment) Single pole {π+ }. Single pole; stores (µ, ν). Single pole; stores (T, I, F ). Poles {π+ , π− }; positive vs. negative degrees. Bipolar analogue of intuitionistic fuzzy (incl. bipolar vague). Bipolar analogue of neutrosophic. Bipolar analogue of picture / hesitant fuzzy. Bipolar analogue of spherical fuzzy. Bipolar analogue of quadripartitioned neutrosophic. Bipolar analogue of pentapartitioned neutrosophic. Poles {π+ , π0 , π− }; positive/neutral/negative. Tripolar intuitionistic model. Tripolar neutrosophic model. Generic m poles; fuzzy case. Generic m poles; intuitionistic case. Generic m poles; neutrosophic case. Generic m poles; picture / hesitant fuzzy case. Generic m poles; spherical fuzzy case. Generic m poles; quadripartitioned neutrosophic case. pro neutral 0 0.20 0.20 0 1.00 0.20 contra 1.00 0.20 0 which are symmetric and satisfy pCFv (α, α) = 0, pCFΠ (π, π) = 0. Define the m–polar appurtenance map pdfΠ : P × Pv −→ ([0, 1]1 )3 ∼ = [0, 1]3 # Page. 30 ![Page Image](https://bcdn.docswell.com/page/4JMY89L9JW.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.6: Cases for m (polar level), t = 1 (single DCF), and s ∈ {1, 2, 3, 4, 5}. Here m=1 = classical, m=2 = bipolar, m=3 = tripolar. m t s 1 1 1 1 1 2 1 1 3 2 1 1 2 1 2 2 1 3 3 1 1 3 1 2 3 1 3 Representative name Plithogenic Fuzzy Set Plithogenic Intuitionistic Fuzzy Set Plithogenic Neutrosophic Set Bipolar Plithogenic Fuzzy Set Bipolar Plithogenic Intuitionistic Set Bipolar Plithogenic Neutrosophic Set Tripolar Plithogenic Fuzzy Set Tripolar Plithogenic Intuitionistic Set Tripolar Plithogenic Neutrosophic Set Brief note (poles / DCF) Single pole; one value-level DCF modulates µ. Single pole; DCF attenuates (µ, ν). Single pole; DCF acts over (T, I, F ). Bipolar poles; one value-level DCF; optional pole-level handling. Bipolar, DCF-aware intuitionistic. Bipolar, DCF-aware neutrosophic. Tripolar poles; one value-level DCF; optional pole interactions. Tripolar, DCF-aware intuitionistic. Tripolar, DCF-aware neutrosophic. whose entries are written as pdfΠ (x, α) = (µpro , µneutral , µcontra ). For the three candidates we set c1 : pdfΠ (c1 , junior) = (0.60, 0.30, 0.10), pdfΠ (c1 , mid) = (0.70, 0.20, 0.10), pdfΠ (c1 , senior) = (0.40, 0.20, 0.40); c2 : pdfΠ (c2 , junior) = (0.30, 0.20, 0.50), pdfΠ (c2 , mid) = (0.50, 0.30, 0.20), pdfΠ (c2 , senior) = (0.80, 0.10, 0.10); c3 : pdfΠ (c3 , junior) = (0.20, 0.30, 0.50), pdfΠ (c3 , mid) = (0.40, 0.30, 0.30), pdfΠ (c3 , senior) = (0.60, 0.20, 0.20). Fix the dominant context (d, δ) = (senior, pro), meaning that the decision maker is primarily interested in the value senior and the pole pro. For each (α, πr ) ∈ Pv × Π we use the compatibility weights   wr (α | d, δ) := 1 − pCFv (α, d) 1 − pCFΠ (πr , δ) , r = 1, 2, 3, and aggregate by the normalized weighted mean 3 X pdf (d,δ) (x, α) := wr (α | d, δ) µr (x, α) r=1 3 X r=1 ∈ [0, 1], wr (α | d, δ) # Page. 31 ![Page Image](https://bcdn.docswell.com/page/PJR95GK979.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets with the convention 0/0 := 0. We illustrate the computation for candidate c1 . (1) Case α = mid. We have 1 − pCFv (mid, senior) = 1 − 0.40 = 0.60, and, against δ = pro, 1 − pCFΠ (π1 , δ) = 1 − 0 = 1, 1 − pCFΠ (π2 , δ) = 1 − 0.20 = 0.8, 1 − pCFΠ (π3 , δ) = 1 − 1.00 = 0. Thus the weight vector is  w(mid | d, δ) = 0.60, 0.60 × 0.8, 0.60 × 0 = (0.60, 0.48, 0), and 3 X wr (mid | d, δ) = 0.60 + 0.48 + 0 = 1.08. r=1 Using pdfΠ (c1 , mid) = (0.70, 0.20, 0.10), we obtain 0.60 · 0.70 + 0.48 · 0.20 + 0 · 0.10 1.08 0.42 + 0.096 0.516 = = ≈ 0.478. 1.08 1.08 pdf(d,δ) (c1 , mid) = (2) Case α = senior. Now 1 − pCFv (senior, senior) = 1 − 0 = 1, so the weights are determined purely by the poles: 3 X  w(senior | d, δ) = 1, 0.8, 0 , wr (senior | d, δ) = 1 + 0.8 + 0 = 1.8. r=1 Using pdfΠ (c1 , senior) = (0.40, 0.20, 0.40), 1 · 0.40 + 0.8 · 0.20 + 0 · 0.40 1.8 0.40 + 0.16 0.56 = = ≈ 0.311. 1.8 1.8 pdf(d,δ) (c1 , senior) = These computations make explicit how contradictions at both the value level (pCFv ) and the pole level (pCFΠ ) jointly attenuate or emphasize the components of pdfΠ in order to produce a single plithogenic score pdf(d,δ) (x, α) for each candidate–value pair. Example 3.2.4 (Vendor Selection with Four Poles (m = 4)). Vendor selection is the critical process where organizations evaluate suppliers and choose those meeting cost, quality, reliability, and compliance requirements (cf. [457]). Let the universe of vendors be P = {v1 , v2 }. Consider the attribute v = logistics_objective with value set Pv = {low_cost, fast_delivery, high_reliability}. # Page. 32 ![Page Image](https://bcdn.docswell.com/page/PEXQKXL3JX.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets We use four poles Π = {benefit, neutral, risk, compliance}, so m = 4. Again we take s = 1, so pdfΠ : P × Pv → [0, 1]4 . The value–level DCF pCFv and pole–level DCF pCFΠ are pCFv low_cost fast_delivery high_reliability low_cost 0 0.50 0.60 pCFΠ benefit neutral risk compliance fast_delivery 0.50 0 0.20 benefit 0 0.20 0.90 0.10 high_reliability 0.60 0.20 0 neutral risk compliance 0.20 0.90 0.10 0 0.20 0.20 0.20 0 0.80 0.20 0.80 0 Both matrices are symmetric with zero diagonal entries, so they are valid DCFs. For vendor v1 , the 4–polar appurtenance vectors pdfΠ (v1 , α) = (µbenefit , µneutral , µrisk , µcompliance ) are chosen as pdfΠ (v1 , low_cost) = (0.65, 0.15, 0.30, 0.40), pdfΠ (v1 , fast_delivery) = (0.70, 0.10, 0.20, 0.50), pdfΠ (v1 , high_reliability) = (0.60, 0.20, 0.10, 0.90). Fix the dominant context (d, δ) = (high_reliability, benefit). For each (α, πr ) ∈ Pv × Π the weights are again   wr (α | d, δ) := 1 − pCFv (α, d) 1 − pCFΠ (πr , δ) , r = 1, . . . , 4, and the scalarized membership is given by 4 X pdf (d,δ) (x, α) := wr (α | d, δ) µr (x, α) r=1 4 X ∈ [0, 1]. wr (α | d, δ) r=1 First note that, for the fixed pole δ = benefit,  1 − pCFΠ (·, δ) = 1, 0.8, 0.1, 0.9 corresponding to {benefit, neutral, risk, compliance}. (1) Case α = fast_delivery. Here 1 − pCFv (fast_delivery, high_reliability) = 1 − 0.20 = 0.80. Thus w(fast_delivery | d, δ) = 0.80 · (1, 0.8, 0.1, 0.9) = (0.80, 0.64, 0.08, 0.72), # Page. 33 ![Page Image](https://bcdn.docswell.com/page/3EK95WLNED.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets and 4 X wr (fast_delivery | d, δ) = 0.80 + 0.64 + 0.08 + 0.72 = 2.24. r=1 With pdfΠ (v1 , fast_delivery) = (0.70, 0.10, 0.20, 0.50), the numerator is num = 0.80 · 0.70 + 0.64 · 0.10 + 0.08 · 0.20 + 0.72 · 0.50 = 0.56 + 0.064 + 0.016 + 0.36 = 1.000, so pdf(d,δ) (v1 , fast_delivery) = 1.000 ≈ 0.4464. 2.24 (2) Case α = high_reliability. Now 1 − pCFv (high_reliability, high_reliability) = 1 − 0 = 1, so the weights coincide with the pole–compatibility vector: w(high_reliability | d, δ) = (1, 0.8, 0.1, 0.9), 4 X wr (high_reliability | d, δ) = 2.8. r=1 With pdfΠ (v1 , high_reliability) = (0.60, 0.20, 0.10, 0.90), num = 1 · 0.60 + 0.8 · 0.20 + 0.1 · 0.10 + 0.9 · 0.90 = 0.60 + 0.16 + 0.01 + 0.81 = 1.58, and hence pdf(d,δ) (v1 , high_reliability) = 1.58 ≈ 0.5643. 2.8 These values show how the four–pole structure, together with the contradiction degrees pCFv and pCFΠ , allows the decision maker to emphasize “benefit” in a high_reliability–oriented context, while still accounting for neutrality, risk, and compliance through the weights wr (α | d, δ). Example 3.2.5 (Smart-City Intersection Design (m = 3)). A smart-city intersection integrates sensors, adaptive signals, and connected vehicles to optimize traffic flow, safety, emissions, and pedestrian experience citywide (cf. [458]). Let P = {A, B} be two competing design options for a city intersection. Consider the attribute v = objective with value set Pv = {safety, throughput, cost}. We take the three poles Π = {pro, neutral, contra}, so that m = 3, and again work with s = 1, so that pdfΠ (x, α) ∈ [0, 1]3 . The value–level contradiction degrees pCFv and pole–level contradiction degrees pCFΠ are defined by pCFv safety throughput cost safety 0 0.50 0.30 throughput 0.50 0 0.40 cost 0.30 0.40 0 # Page. 34 ![Page Image](https://bcdn.docswell.com/page/L73WK13Z75.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets pCFΠ pro neutral contra pro neutral 0 0.20 0.20 0 1.00 0.20 contra 1.00 0.20 0 which are symmetric with zero diagonal entries, hence valid DCFs. For each design option X ∈ {A, B} and each α ∈ Pv , the triple pdfΠ (X, α) = (µpro (X, α), µneutral (X, α), µcontra (X, α)) encodes the 3–polar evaluation of X with respect to objective α. We choose A: pdfΠ (A, safety) = (0.85, 0.10, 0.05), pdfΠ (A, throughput) = (0.60, 0.20, 0.20), pdfΠ (A, cost) = (0.55, 0.25, 0.20); B: pdfΠ (B, safety) = (0.70, 0.20, 0.10), pdfΠ (B, throughput) = (0.80, 0.10, 0.10), pdfΠ (B, cost) = (0.40, 0.30, 0.30). We again use the weighted mean reduction 3 X pdf (d,δ) (x, α) := wr (α | d, δ) µr (x, α) r=1 3 X , wr (α | d, δ) r=1 where wr (α | d, δ) := 1 − pCFv (α, d) We fix the dominant context   1 − pCFΠ (πr , δ) . (d, δ) = (safety, pro), representing a “safety-first, pro” viewpoint. Then 1 − pCFΠ (·, δ) = 1, 0.8, 0  for the poles {pro, neutral, contra}. (1) Design B, α = throughput. We have 1 − pCFv (throughput, safety) = 1 − 0.50 = 0.50, so w(throughput | d, δ) = 0.50 · (1, 0.8, 0) = (0.50, 0.40, 0), with 3 X wr (throughput | d, δ) = 0.50 + 0.40 + 0 = 0.90. r=1 Using pdfΠ (B, throughput) = (0.80, 0.10, 0.10), 0.50 · 0.80 + 0.40 · 0.10 + 0 · 0.10 0.90 0.40 + 0.04 0.44 = = ≈ 0.489. 0.90 0.90 pdf(d,δ) (B, throughput) = # Page. 35 ![Page Image](https://bcdn.docswell.com/page/87DK3X44JG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets (2) Design A, α = safety. Here 1 − pCFv (safety, safety) = 1 − 0 = 1, so the weights are simply w(safety | d, δ) = (1, 0.8, 0), 3 X wr (safety | d, δ) = 1.8. r=1 Using pdfΠ (A, safety) = (0.85, 0.10, 0.05), 1 · 0.85 + 0.8 · 0.10 + 0 · 0.05 1.8 0.85 + 0.08 0.93 = = ≈ 0.517. 1.8 1.8 pdf(d,δ) (A, safety) = These scalarized scores pdf(d,δ) (B, throughput) and pdf(d,δ) (A, safety) explicitly show how a “safetyfirst, pro” context downweights contradictory poles and objectives, while still preserving the full multi-polar information encoded in pdfΠ . The m–polar plithogenic machinery thus provides a mathematically precise, contradiction–aware comparison of intersection designs in terms of safety, throughput, and cost. 3.3 Complex Plithogenic Set (CPS) A complex plithogenic set uses complex-valued memberships and contradiction functions to aggregate attributes under context-sensitive multidimensional uncertainty and conflict-aware reasoning [459]. Definition 3.3.1 (Complex Plithogenic Set (CPS)). [459] Let P be a nonempty set (the universe under study), v be an attribute with a nonempty finite set of values Pv , and let s ∈ N and t ∈ N0 denote, respectively, the number of appurtenance components and of contradiction channels. (i) Complex degrees of appurtenance. Let D := {z ∈ C : |z| ≤ 1}. A complex degree of appurtenance function (CDAF) is  cpdf : P × Pv −→ D s , cpdf(x, α) = γ1 (x, α), . . . , γs (x, α) , with each component represented in amplitude–phase form γi (x, α) = pi (x, α) eiϕi (x,α) where pi (x, α) ∈ [0, 1] and ϕi (x, α) ∈ [0, 2π). (ii) Degrees of contradiction (DCF). For each j = 1, . . . , t let pCFj : Pv × Pv −→ [0, 1] be symmetric and reflexive, i.e., pCFj (a, a) = 0 and pCFj (a, b) = pCFj (b, a) for all a, b ∈ Pv . (iii) Fusion of contradictions and compatibility weights. Fix a contradiction fusion Φ : [0, 1]t → [0, 1] that is symmetric, monotone (componentwise), and satisfies Φ(0, . . . , 0) = 0. (If t = 0, set Φ(·) ≡ 0.) Given a dominant value δ ∈ Pv , define the compatibility weight  w(a | δ) := 1 − Φ pCF1 (a, δ), . . . , pCFt (a, δ) ∈ [0, 1] (a ∈ Pv ). # Page. 36 ![Page Image](https://bcdn.docswell.com/page/VJPK4PMVE8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets (iv) Complex plithogenic aggregation (relative to δ). For each x ∈ P , define the δ–relative complex plithogenic degree Γ(δ) (x) ∈ D s componentwise by the weighted complex mean  P w(a | δ) γi (x, a)   P   a∈Pv P , if (δ) a∈Pv w(a | δ) > 0, w(a | δ) Γi (x) := i = 1, . . . , s.  a∈Pv    0, otherwise, The Complex Plithogenic Set (CPS) associated with (P, v, Pv ), s, t, cpdf, {pCFj }, and Φ is the tuple CPS :=  P, v, Pv , s, t, cpdf, {pCFj }tj=1 , Φ, Γ(δ) , where Γ(δ) supplies the contradiction-aware complex membership (as an s–vector) relative to the dominant value δ ∈ Pv . Table 3.7 presents the summary of the Complex Plithogenic Set (CPS). Table 3.7: Summary of Complex Plithogenic Set (CPS) cases for t ∈ {0, 1} and s ∈ {1, 2, 3}. Here Γ(δ) is the contradiction-aware complex membership (Def. 3.3.1), γi ∈ D are CDAF components, and w(a | δ) = 1 − Φ(pCF1 (a, δ), . . . , pCFt (a, δ)). For t = 1 we take Φ(z) = z; for t = 0 we set Φ ≡ 0. s t Specification (weights and aggregation) / Interpretation No contradiction channel (t = 0): Φ ≡ 0 so w(a | δ) = 1 for all a ∈ Pv (independent of δ). 1 P (δ) 1 0 Γ1 (x) = γ1 (x, a) (uniform complex mean). Single-component CPS (complex |Pv | a∈Pv fuzzy set [460–462]–like). 2 0 Γ(δ) (x) ∈ D2 with each component the uniform complex mean over Pv . Two-component CPS (complex intuitionistic fuzzy set [463–465] and complex vague set [466, 467]–like). 3 0 Γ(δ) (x) ∈ D3 with componentwise uniform complex means. Three-component CPS (complex neutrosophic set [468–470],Complex picture fuzzy set [471,472], Complex Hesitant fuzzy [473–475], and Complex Spherical fuzzy [476, 477]-like). 4 0 Γ(δ) (x) ∈ D4 with componentwise uniform complex means. Four-component CPS (complex quadripartitioned neutrosophic set [478, 479]-like). 5 0 Γ(δ) (x) ∈ D5 with componentwise uniform complex means. Five-component CPS (complex pentapartitioned neutrosophic set [480, 481]-like). Single contradiction channel (t = 1): Φ(z) = z, so w(a | δ) = 1 − pCF(a, δ).  P 1 − pCF(a, δ) γ1 (x, a) a∈P (δ) v  P 1 1 Γ1 (x) = . Single-component CPS with DCF-based 1 − pCF(a, δ) a∈Pv 2 1 3 1 reweighting (complex plithogenic fuzzy–like). Same weighting; Γ(δ) (x) ∈ D2 computed componentwise by the above weighted complex mean. Same weighting; Γ(δ) (x) ∈ D3 computed componentwise by the above weighted complex mean. Notes. (i) If all phases vanish (ϕi ≡ 0), the table reduces to the real-valued plithogenic counterparts. (ii) For t = 0, Γ(δ) does not depend on δ; for t = 1, larger pCF(a, δ) downweights the contribution of value a. We now present concrete examples of the concept. Example 3.3.2 (CPS in Smart-Grid EV Charging (tariff-aware dispatch)). Smart-grid EV charging coordinates electric vehicle charging with grid signals, optimizing energy use, reducing costs, balancing demand and renewables integration (cf. [482]). # Page. 37 ![Page Image](https://bcdn.docswell.com/page/2EVVX29REQ.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Consider EV charging decisions for one car x = EV1 . Universe P = {EV1 }; attribute v = tariff period with values Pv = {offpeak, shoulder, peak}. Take s = 2 complex components (grid benefit, stability) and t = 2 contradiction channels (price, transformer load). Let the dominant value be δ = peak. For the DCFs (symmetric, reflexive) we only need the pairs (a, δ): pCF1 (offpeak, peak) = 0.8, pCF1 (shoulder, peak) = 0.4, pCF2 (offpeak, peak) = 0.5, pCF1 (peak, peak) = 0, pCF2 (shoulder, peak) = 0.2, pCF2 (peak, peak) = 0. 2 Fuse contradictions by the mean Φ(c1 , c2 ) = c1 +c 2 , hence the compatibility weights  0.8+0.5  = 0.35,  w(offpeak | peak) = 1 − 2  0.4+0.2 w(a | δ) = 1 − Φ pCF1 (a, δ), pCF2 (a, δ) ⇒ w(shoulder | peak) = 1 − 2 = 0.70,   w(peak | peak) = 1. Complex degrees (amplitude–phase): for component 1 (grid benefit) and 2 (stability), γ (1) (EV1 , offpeak) = 0.60 ei 0 = (0.600 + 0.000 i), ◦ γ (1) (EV1 , shoulder) = 0.80 ei 30 = (0.800 cos 30◦ + 0.800 sin 30◦ i) = (0.693 + 0.400 i), ◦ γ (1) (EV1 , peak) = 0.40 ei 60 = (0.200 + 0.346 i); ◦ γ (2) (EV1 , offpeak) = 0.50 ei 90 = (0.000 + 0.500 i), ◦ γ (2) (EV1 , shoulder) = 0.70 ei 60 = (0.350 + 0.606 i), ◦ γ (2) (EV1 , peak) = 0.30 ei 90 = (0.000 + 0.300 i). ◦ Weighted complex mean (denominator D = 0.35 + 0.70 + 1 = 2.05): Num(1) = 0.35(0.600 + 0.000i) + 0.70(0.693 + 0.400i) + 1(0.200 + 0.346i) = (0.8950 + 0.6264 i), (peak) Γ1 (EV1 ) = ◦ Num(1) = (0.2373 + 0.4778 i) = 0.5329 ei 34.99 ; D Num(2) = 0.35(0.000 + 0.500i) + 0.70(0.350 + 0.606i) + 1(0.000 + 0.300i) = (0.2450 + 0.8994 i), (peak) Γ2 (EV1 ) = ◦ Num(2) = (0.1195 + 0.4384 i) = 0.4547 ei 74.83 . D Thus the CPS yields a contradiction-aware complex membership vector Γ(peak) (EV1 ) = (0.5329ei35.0 , 0.4547ei74.8 ) ◦ ◦ . Example 3.3.3 (CPS in Medical Imaging Triage (CT/MRI/Ultrasound fusion)). Medical image triage prioritizes radiology or diagnostic images for review, flagging urgent abnormalities using automated or AI-assisted analysis to clinicians [483]. Patient x = A, attribute v = imaging modality with Pv = {CT, US, MRI}. Take one complex component (s = 1) representing diagnostic confidence, and t = 2 DCF channels: radiation risk vs. diagnostic yield. Dominant value δ = MRI. Let pCF1 (CT, MRI) = 0.9, pCF2 (CT, MRI) = 0.2, Weights: pCF1 (US, MRI) = 0.0, pCF2 (US, MRI) = 0.6, w(CT | MRI) = 1 − 0.9+0.2 = 0.45, 2 w(MRI | MRI) = 1, 2 Φ(c1 , c2 ) = c1 +c 2 . w(US | MRI) = 1 − 0.0+0.6 = 0.70, 2 D = 0.45 + 0.70 + 1 = 2.15. # Page. 38 ![Page Image](https://bcdn.docswell.com/page/57GLVRZ6EL.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets CDAF (amplitude–phase in degrees): γ(A, CT) = 0.65 ei 10 = (0.640 + 0.113 i), γ(A, US) = 0.50 ei 70 = (0.171 + 0.470 i), ◦ ◦ γ(A, MRI) = 0.85 ei 20 = (0.798 + 0.291 i). ◦ Weighted sum and aggregation: Num = 0.45(0.640 + 0.113i) + 0.70(0.171 + 0.470i) + 1(0.798 + 0.291i) = (1.2066 + 0.6700 i), ◦ Num Γ(MRI) (A) = = (0.5612 + 0.3118 i) = 0.6420 ei 29.06 . D Hence MRI-dominant fusion produces a high complex confidence with phase ≈ 29◦ (moderate uncertainty skew). Example 3.3.4 (CPS in E-Commerce Multi-Tier Product Positioning). Product x = LaptopA, attribute v = tier with Pv = {budget, balanced, premium}. Use s = 2 complex components: (1) user sentiment, (2) delivery satisfaction. Let t = 2 DCF channels (price gap, feature gap), dominant tier δ = balanced. Contradictions (only pairs to δ): pCF1 (budget, balanced) = 0.4, pCF2 (budget, balanced) = 0.6, Weights: pCF1 (premium, balanced) = 0.5, pCF2 (premium, balanced) = 0.2, 2 Φ(c1 , c2 ) = c1 +c 2 . w(budget | balanced) = 1 − 0.4+0.6 = 0.50, 2 w(premium | balanced) = 1 − 0.5+0.2 = 0.65, 2 w(balanced | balanced) = 1, D = 2.15. CDAFs (amplitude–phase; rectangular shown to three decimals): γ (1) (LaptopA, budget) = 0.55 ei15 = (0.532 + 0.142 i), ◦ γ (1) (LaptopA, balanced) = 0.75 ei25 = (0.680 + 0.317 i), ◦ γ (1) (LaptopA, premium) = 0.70 ei35 = (0.574 + 0.402 i); ◦ γ (2) (LaptopA, budget) = 0.60 ei80 = (0.104 + 0.591 i), ◦ γ (2) (LaptopA, balanced) = 0.65 ei50 = (0.418 + 0.498 i), ◦ γ (2) (LaptopA, premium) = 0.55 ei40 = (0.421 + 0.354 i). ◦ Weighted complex means (piecewise numerators shown; then divide by D = 2.15): Num(1) = 0.50(0.532 + 0.142i) + 1(0.680 + 0.317i) + 0.65(0.574 + 0.402i) = (1.3181 + 0.6491 i), (balanced) Γ1 (LaptopA) = (1.3181 + 0.6491 i)/2.15 = (0.6131 + 0.3019 i) = 0.6834 ei 26.22 ; ◦ Num(2) = 0.50(0.104 + 0.591i) + 1(0.418 + 0.498i) + 0.65(0.421 + 0.354i) = (0.7438 + 1.0232 i), (balanced) Γ2 (LaptopA) = (0.7438 + 1.0232 i)/2.15 = (0.3459 + 0.4759 i) i 53.99◦ = 0.5883 e . So the CPS reports a balanced-tier, contradiction-aware complex appraisal vector Γ(balanced) (LaptopA) = ◦ ◦ (0.6834ei26.22 , 0.5883ei53.99 ). # Page. 39 ![Page Image](https://bcdn.docswell.com/page/4EQY6VL2JP.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.4 SuperHyperPlithogenic Set A (m, n)-SuperhyperPlithogenic Set assigns m-level parameter-subsets and their attribute values to n-level fuzzy-contradiction degree-sets, capturing multi-faceted membership and inter-attribute conflicts and uncertainty patterns [484–486]. The definition of the (m, n)-SuperhyperPlithogenic Set is presented below. Definition 3.4.1 ((m, n)-SuperhyperPlithogenic Set). [485] Let X be a nonempty set and let V = {v1 , . . . , vk } be a finite set of attributes. For each v ∈ V , let Pv be the set of its possible values. Fix positive integers m, n and positive dimensions s, t. Define the m-th nested powerset [81, 487] of X by  P 0 (X) = X, P r (X) = P P r−1 (X) (r ≥ 1),  and similarly P n [0, 1]s for the s-dimensional unit cube. An (m, n)-SuperhyperPlithogenic Set over X is the quintuple   ˜ (m,n) }v∈V , pCF (m,n) , SHP (m,n) = P m (X), V, {Pv }v∈V , {pdf v where (i) P m (X) is the domain of “super-elements” of level m. (ii) For each v ∈ V , Pv is the finite set of its values. (iii) The Hyper Degree of Appurtenance Function ˜ (m,n) : P m (X) × Pv −→ P n [0, 1]s pdf v  (m,n) ˜ assigns to each (A, a) with A ∈ P m (X) and a ∈ Pv a nonempty subset pdf v representing all possible membership‐degree vectors of dimension s. (A, a) ⊆ [0, 1]s (iv) The Degree of Contradiction Function [  [  pCF (m,n) : Pv × Pv −→ [0, 1]t v∈V v∈V satisfies for all a, b: (a) pCF (m,n) (a, a) = 0 (reflexivity), (b) pCF (m,n) (a, b) = pCF (m,n) (b, a) (symmetry). Tables 3.8 and 3.9 present the associated concepts related to (m, n)-SuperhyperPlithogenic Sets. We now present concrete examples of the concept. Example 3.4.2 ((m,n)=(2,1): Hospital “team-of-teams” triage with hyper–neutrosophic memberships). Let X = {ER, ICU, OR, RAD} be hospital units. Work on the superlevel P 2 (X); consider the super–element  A = {ER, ICU}, {OR, RAD} ∈ P 2 (X). Use a single attribute v =“clinical risk bracket” with values Pv = {low, moderate, high}. Choose s = 3 (neutrosophic (T, I, F )), n = 1 (each membership is a finite subset of [0, 1]3 ), and t = 1. # Page. 40 ![Page Image](https://bcdn.docswell.com/page/KJ4W4MDP71.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.8: Specializations of (m, n)-SuperhyperPlithogenic Sets Parameter choice Domain level (0, 0)SuperhyperPlithogenic Set P 0 (X) = X (no superlevel) (0, n)SuperhyperPlithogenic Set P 0 (X) = X (no superlevel) (m, 0)SuperhyperPlithogenic Set P m (X) (super / hyperlevel on the universe) Membership / contradiction structure Single plithogenic DAF pdf : X × P v → [0, 1]s and DCF pCF : P v × P v → [0, 1]t DAF takes values in P n ([0, 1]s ): plithogenic memberships are hyper/nested Usual plithogenic DAF/DCF on superelements (no hyper–membership nesting) Resulting model Plithogenic Set HyperPlithogenic Set (cf. [488–490]) SuperPlithogenic Set (cf. [491]) Table 3.9: HyperPlithogenic Set specializations by number of DAF components Model Hyper–Plithogenic Fuzzy Set Hyper–Plithogenic Intuitionistic Fuzzy Set Hyper–Plithogenic Neutrosophic Set DAF s s=1 dim. Membership / contradiction description Each super/hyper–element has a single plithogenic membership in [0, 1], modulated by a value–level DCF over attribute values. Each super/hyper–element carries a pair (membership, non–membership) in [0, 1]2 , aggregated via the plithogenic contradiction function. Each super/hyper–element stores (T, I, F ) ∈ [0, 1]3 ; the plithogenic DCF weights them according to value–contradiction. s=2 s=3 Resulting structure HyperFuzzy (plithogenic) Set [492–495] Hyper–Intuitionistic Fuzzy Set Hyper–Neutrosophic Set [291, 496–499] Hyper Degree of Appurtenance (HDAF) for A: ˜ (2,1) (A, low) = {(0.30, 0.50, 0.40), (0.40, 0.40, 0.50)}, pdf ˜ (2,1) (A, moderate) = {(0.50, 0.30, 0.40), (0.60, 0.30, 0.30)}, pdf ˜ (2,1) (A, high) = {(0.80, 0.10, 0.10), (0.70, 0.20, 0.20)}. pdf Centroids (componentwise means) per value: ū(low) = (0.35, 0.45, 0.45), ū(moderate) = (0.55, 0.30, 0.35), ū(high) = (0.75, 0.15, 0.15). Contradiction (symmetric, reflexive, only off–diagonals shown): pCF (2,1) (low, high) = 0.7, pCF (2,1) (moderate, high) = 0.3, pCF (2,1) (low, moderate) = 0.4. For a “dominant” value δ = high, set compatibility weights w(a | δ) = 1 − pCF (2,1) (a, δ): w(low | high) = 0.3, w(moderate | high) = 0.7, w(high | high) = 1, D = 0.3 + 0.7 + 1 = 2. A plithogenic reduction (weighted mean of centroids) yields µ(high) (A) = 0.3 ū(low) + 0.7 ū(moderate) + 1 ū(high) . 2 Coordinatewise calculations: 0.3 · 0.35 + 0.7 · 0.55 + 1 · 0.75 0.105 + 0.385 + 0.75 1.240 T : = = = 0.620, 2 2 2 0.3 · 0.45 + 0.7 · 0.30 + 1 · 0.15 0.135 + 0.210 + 0.150 0.495 I: = = = 0.2475, 2 2 2 0.3 · 0.45 + 0.7 · 0.35 + 1 · 0.15 0.135 + 0.245 + 0.150 0.530 F : = = = 0.265. 2 2 2 # Page. 41 ![Page Image](https://bcdn.docswell.com/page/LE1Y48ZX7G.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Therefore, under “high–risk” dominance, the super–team A has reduced hyper–neutrosophic degree µ(high) (A) = (0.620, 0.2475, 0.265). Example 3.4.3 ((m,n)=(1,2): Urban multi–modal trip bundle with nested (scenario–clustered) feasibility). A multi-modal trip combines multiple transport modes, such as walking, buses, trains, and bikes, within one integrated journey for travelers [500]. Let X be trip segments (bus links, rail legs, walk connectors). On P 1 (X) = P(X) choose the super–element A = {BusSeg1 , RailSeg3 }. Use the attribute v =“congestion level” with Pv = {free, moderate, heavy}. Take s = 1 (scalar feasibility in [0, 1]), n = 2 (a set of sets of feasibilities), and t = 1. HDAF as scenario clusters (each inner set is a scenario; values are segment–level feasibilities aggregated to A):  ˜ (1,2) (A, free) = {0.90, 0.80}, {0.85, 0.75} , pdf  ˜ (1,2) (A, moderate) = {0.70, 0.60}, {0.75, 0.55} , pdf  ˜ (1,2) (A, heavy) = {0.50, 0.30}, {0.45, 0.35} . pdf Inner means (per scenario), then outer mean (per value):  1 ū(free) = 12 0.90+0.80 + 0.85+0.75 = (0.85 + 0.80) = 0.825, 2 2  21 1 0.70+0.60 0.75+0.55 ū(moderate) = 2 + = (0.65 + 0.65) = 0.65, 2 2  21 1 0.50+0.30 0.45+0.35 ū(heavy) = 2 + = 2 (0.40 + 0.40) = 0.40. 2 2 Contradiction (symmetric, reflexive): pCF (1,2) (free, heavy) = 0.8, pCF (1,2) (moderate, heavy) = 0.5, pCF (1,2) (free, moderate) = 0.4. With dominant δ = heavy, weights w(a | δ) = 1 − pCF (1,2) (a, δ): w(free | heavy) = 0.2, w(moderate | heavy) = 0.5, w(heavy | heavy) = 1, D = 0.2+0.5+1 = 1.7. Reduced feasibility: µ(heavy) (A) = 0.2 · 0.825 + 0.5 · 0.65 + 1 · 0.40 0.165 + 0.325 + 0.400 0.890 = = ≈ 0.5235. 1.7 1.7 1.7 Thus, for rush–hour dominance, the super–trip A remains feasible at ≈ 0.524 after contradiction–aware nesting. Example 3.4.4 ((m,n)=(2,2): Multi–tier supplier portfolio with clustered neutrosophic appraisals). A multi-tier supplier portfolio organizes vendors across primary, secondary, and tertiary levels to diversify risk, ensure resilience, and optimize costs (cf. [501]). Let X be base–level suppliers. On the domain P 2 (X), consider  A = {Tier1A, Tier1B}, {Tier2C} . Attribute v =“sourcing strategy” with Pv = {single, dual, multi}. Choose s = 3 (neutrosophic (T, I, F )), n = 2 (clustered scenario sets), t = 1. Clustered HDAF for A (each inner brace is a scenario cluster; vectors are (T, I, F )):  ˜ (2,2) (A, single) = {(0.70, 0.20, 0.30), (0.60, 0.30, 0.30)}, {(0.75, 0.20, 0.25)} , pdf  ˜ (2,2) (A, dual) = {(0.80, 0.15, 0.15), (0.78, 0.15, 0.12)}, {(0.75, 0.18, 0.18)} , pdf  ˜ (2,2) (A, multi) = {(0.72, 0.20, 0.18), (0.70, 0.22, 0.18)}, {(0.68, 0.25, 0.20)} . pdf # Page. 42 ![Page Image](https://bcdn.docswell.com/page/GEWGXZ9KJ2.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Inner (scenario) centroids, then outer (value) centroids: Single:  c1 = 12 (0.70, 0.20, 0.30) + (0.60, 0.30, 0.30) = (0.65, 0.25, 0.30), c2 = (0.75, 0.20, 0.25), Dual: ū(single) = 12 (c1 + c2 ) = (0.70, 0.225, 0.275).  c1 = 12 (0.80, 0.15, 0.15) + (0.78, 0.15, 0.12) = (0.79, 0.15, 0.135), c2 = (0.75, 0.18, 0.18), Multi: ū(dual) = 12 (c1 + c2 ) = (0.77, 0.165, 0.1575).  c1 = 12 (0.72, 0.20, 0.18) + (0.70, 0.22, 0.18) = (0.71, 0.21, 0.18), c2 = (0.68, 0.25, 0.20), ū(multi) = 12 (c1 + c2 ) = (0.695, 0.23, 0.19). Contradiction (symmetric, reflexive): pCF (2,2) (single, dual) = 0.4, pCF (2,2) (multi, dual) = 0.3, pCF (2,2) (single, multi) = 0.7. With dominant δ = dual, weights w(a | δ) = 1 − pCF (2,2) (a, δ): w(single | dual) = 0.6, w(dual | dual) = 1, w(multi | dual) = 0.7, D = 0.6 + 1 + 0.7 = 2.3. Reduced neutrosophic degree (componentwise): µ(dual) (A) = 0.6 ū(single) + 1 ū(dual) + 0.7 ū(multi) . 2.3 Calculations: 0.6 · 0.70 + 1 · 0.77 + 0.7 · 0.695 0.420 + 0.770 + 0.4865 1.6765 = = ≈ 0.7298, 2.3 2.3 2.3 0.6 · 0.225 + 1 · 0.165 + 0.7 · 0.230 0.135 + 0.165 + 0.161 0.461 I: = = ≈ 0.2004, 2.3 2.3 2.3 0.6 · 0.275 + 1 · 0.1575 + 0.7 · 0.190 0.165 + 0.1575 + 0.133 0.4555 F : = = ≈ 0.1980. 2.3 2.3 2.3 T : Hence, under “dual–sourcing” dominance, the super–portfolio A attains µ(dual) (A) ≈ (0.7298, 0.2004, 0.1980) . 3.5 Plithogenic Linguistic Set A plithogenic linguistic set assigns linguistic-term memberships and contradiction-aware weights to objects, aggregating hesitant or single-term evaluations under a dominant context. The plithogenic linguistic set is closely related to the Hesitant Fuzzy Linguistic Set [502–504] and the Linguistic Neutrosophic Set [505–509]. Definition 3.5.1 (Plithogenic Linguistic Set (single-term and hesitant variants)). Let P be a nonempty finite universe of objects and let S = {s0 , s1 , . . . , sg } be a finite linguistic term set endowed with its natural total order s0 ≺ s1 ≺ · · · ≺ sg . Write ι : S → {0, 1, . . . , g} for the (order-preserving) index map ι(si ) = i, and define the normalized linguistic distance dS (si , sj ) := |ι(si ) − ι(sj )| ∈ [0, 1]. g (A) Linguistic contradiction (on terms and on hesitant sets). # Page. 43 ![Page Image](https://bcdn.docswell.com/page/47ZL619NJ3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets (A1) (Term-level DCF ) A linguistic degree of contradiction is any map pCF : S × S → [0, 1] that is symmetric and reflexive: pCF (a, a) = 0 and pCF (a, b) = pCF (b, a). A canonical choice is pCF (a, b) := dS (a, b). (A2) (Lift to hesitant sets) For nonempty H, K ⊆ S define the lifted contradiction n o [ (H, K) := max max min pCF (a, b), max min pCF (a, b) ∈ [0, 1], pCF a∈H b∈K b∈K a∈H [ (∅, ∅) := 0 and pCF [ (H, ∅) = pCF [ (∅, K) := 1 otherwise. and set pCF (B) Value domain and linguistic appurtenance. Fix s ∈ N (the number of appurtenance components). We consider two admissible value domains:  Pv ∈ S (single-term variant), P ? (S) := P(S) \ {∅} (hesitant variant) . A linguistic degree of appurtenance function (LDAF) is pdf : P × Pv −→ [0, 1]s ,  pdf (x, α) = µ1 (x, α), . . . , µs (x, α) , with each component µi nonnegative and bounded by 1. (C) Fusion of contradiction channels (optional). For t ∈ N0 let {pCFj }tj=1 be t (term-level) contradiction maps as in (A1) and let Φ : [0, 1]t → [0, 1] be a symmetric, coordinatewise-monotone fusion with Φ(0, . . . , 0) = 0. (If t = 0, we set Φ(·) ≡ 0.) (D) Compatibility weights and plithogenic aggregation. Fix a dominant linguistic value δ ∈ Pv . Define a compatibility weight w(· | δ) ∈ [0, 1] by  single-term: w(a | δ) := 1 − Φ pCF1 (a, δ), . . . , pCFt (a, δ) (a ∈ S), hesitant: [ 1 (H, δ), . . . , pCF [ t (H, δ) w(H | δ) := 1 − Φ pCF  (H ∈ P ? (S)). For x ∈ P , the δ–relative plithogenic linguistic degree Γ(δ) (x) ∈ [0, 1]s is defined componentwise by the weighted mean  P w(α | δ) µi (x, α)   P   α∈Pv P , if (δ) α∈Pv w(α | δ) > 0, w(α | δ) Γi (x) := i = 1, . . . , s.  α∈Pv    0, otherwise, The Plithogenic Linguistic Set (PLS) associated with (P, S, Pv ), s, t, pdf , {pCFj }, and Φ is the tuple PLS :=  P, v, S, Pv , s, t, pdf, {pCFj }tj=1 , Φ, Γ(δ) , where v designates the (linguistic) attribute and Γ(δ) yields the contradiction-aware aggregated linguistic membership relative to the dominant value δ. Table 3.10 summarizes the reductions of the Plithogenic Linguistic Set. We now present concrete examples of the concept. # Page. 44 ![Page Image](https://bcdn.docswell.com/page/YJ6W2LK9JV.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.10: Reductions of the Plithogenic Linguistic Set Parameter setting t = 0 (no contradiction channel) Pv = S, s = 1, t = 1, Φ(z) = z Pv = P ? (S), t = 0 Reduction / meaning w(α | δ) ≡ 1; Γ(δ) becomes the simple unweighted average over Pv (independent of δ). Standard plithogenic linguistic weighting: Γ(δ) (x) is a contradiction-based weighted mean w.r.t. the dominant term δ. Hesitant fuzzy linguistic case [436, 510]: nonempty subsets of S act as hesitant term sets, aggregated by unweighted means. Example 3.5.2 (Procurement: Supplier quality under a “Very Good” policy (single–term PLS)). Let the linguistic term set be S = {s0 = Very Poor, s1 = Poor, s2 = Fair, s3 = Good, s4 = Very Good, s5 = Excellent}, i j with index map ι(si ) = i and g = 5. Define the linguistic distance dS (si , sj ) = . Take g t = 1 and set pCF (a, b) = dS (a, b). The dominant linguistic value is δ = s4 (“Very Good”), hence the compatibility weights are w(si | δ) = 1 − dS (si , s4 ):  w(s0 ), w(s1 ), w(s2 ), w(s3 ), w(s4 ), w(s5 ) = (0.2, 0.4, 0.6, 0.8, 1.0, 0.8), P and i w(si | δ) = 3.8. |ι(s )−ι(s )| For supplier A (the universe P = {A}, single–term value domain Pv = S), suppose the linguistic appurtenance (for s = 1) is  µ(A, s0 ), µ(A, s1 ), µ(A, s2 ), µ(A, s3 ), µ(A, s4 ), µ(A, s5 ) = (0.02, 0.08, 0.20, 0.40, 0.22, 0.08). The δ–relative plithogenic linguistic degree is P5 w(si | δ) µ(A, si ) (δ) Γ (A) = i=0 . P5 i=0 w(si | δ) Compute the numerator explicitly: 0.2 · 0.02 + 0.4 · 0.08 + 0.6 · 0.20 + 0.8 · 0.40 + 1.0 · 0.22 + 0.8 · 0.08 = 0.004 + 0.032 + 0.120 + 0.320 + 0.220 + 0.064 = 0.760. Hence 0.760 = 0.2000. 3.8 Thus, under a “Very Good” policy, supplier A’s contradiction–aware aggregated quality is 0.20. Γ(δ) (A) = Example 3.5.3 (Restaurant service: hesitant evaluations under a “Very Good” target (hesitant PLS)). Use the same S, ι, g = 5, dS , and pCF = dS as above. Work in the hesitant variant with value domain a finite subfamily of nonempty term sets Pv = {H1 , H2 , H3 }, H1 = {s3 , s4 }, H2 = {s2 , s3 }, H3 = {s4 , s5 }. Let the dominant value be δ = {s4 }. The lifted contradiction from Definition (A2) gives [ (H, δ) = max dS (a, s4 ). pCF a∈H Therefore the compatibility weights are w(H1 | δ) = 1 − max{dS (s3 , s4 ), dS (s4 , s4 )} = 1 − max{0.2, 0} = 0.8, # Page. 45 ![Page Image](https://bcdn.docswell.com/page/GJ5M21PDJ4.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets w(H2 | δ) = 1 − max{dS (s2 , s4 ), dS (s3 , s4 )} = 1 − max{0.4, 0.2} = 0.6, w(H3 | δ) = 1 − max{dS (s4 , s4 ), dS (s5 , s4 )} = 1 − max{0, 0.2} = 0.8, so P w = 0.8 + 0.6 + 0.8 = 2.2. For restaurant R (universe P = {R}) on the attribute “service speed”, take (with s = 1) µ(R, H1 ) = 0.60, µ(R, H2 ) = 0.25, µ(R, H3 ) = 0.35. Then Γ(δ) (R) = 0.8 · 0.60 + 0.6 · 0.25 + 0.8 · 0.35 0.480 + 0.150 + 0.280 0.910 = = ≈ 0.4136. 0.8 + 0.6 + 0.8 2.2 2.2 Thus, against a “Very Good” target, the hesitant plithogenic aggregation yields an overall service–speed score ≈ 0.414. Example 3.5.4 (IT helpdesk: contradiction–aware (T, I, F ) aggregation (multi–component PLS)). An IT helpdesk provides users with technical support, troubleshooting, system guidance, and issue resolution to maintain smooth daily computer and network operations (cf. [511]). Keep the same S, ι, g = 5, dS , pCF = dS , and take the dominant value δ = s4 . Let s = 3 encode (T, I, F ) = (truth, indeterminacy, falsity) components of satisfaction. For a resolved ticket X on attribute “user satisfaction”, specify single–term appurtenances for each si ∈ S: term si T = µ1 (X, si ) I = µ2 (X, si ) F = µ3 (X, si ) s0 0.05 0.10 0.80 s1 0.10 0.15 0.70 s2 0.30 0.20 0.50 s3 0.55 0.20 0.30 s4 0.70 0.15 0.20 s5 0.60 0.15 0.25 With δ = s4 , the weights are  w(s0 ), w(s1 ), w(s2 ), w(s3 ), w(s4 ), w(s5 ) = (0.2, 0.4, 0.6, 0.8, 1.0, 0.8), X w = 3.8. Aggregate componentwise: (δ) ΓT (X) = = (δ) ΓI (X) = = (δ) ΓF (X) = = 0.2 · 0.05 + 0.4 · 0.10 + 0.6 · 0.30 + 0.8 · 0.55 + 1.0 · 0.70 + 0.8 · 0.60 3.8 0.01 + 0.04 + 0.18 + 0.44 + 0.70 + 0.48 1.85 = ≈ 0.4868, 3.8 3.8 0.2 · 0.10 + 0.4 · 0.15 + 0.6 · 0.20 + 0.8 · 0.20 + 1.0 · 0.15 + 0.8 · 0.15 3.8 0.02 + 0.06 + 0.12 + 0.16 + 0.15 + 0.12 0.63 = ≈ 0.1658, 3.8 3.8 0.2 · 0.80 + 0.4 · 0.70 + 0.6 · 0.50 + 0.8 · 0.30 + 1.0 · 0.20 + 0.8 · 0.25 3.8 0.16 + 0.28 + 0.30 + 0.24 + 0.20 + 0.20 1.38 = ≈ 0.3632. 3.8 3.8 Therefore, under a “Very Good” dominant context, the contradiction–aware aggregated triplet is Γ(δ) (X) ≈ (T, I, F ) = (0.4868, 0.1658, 0.3632). # Page. 46 ![Page Image](https://bcdn.docswell.com/page/LE3WK16QE5.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.6 q-rung orthopair Plithogenic sets A q-Rung n-Tuple Plithogenic Set models elements with multi-attribute memberships and contradictions, generalizing q-rung orthopair fuzzy sets [512, 513]. Notation 3.6.1. Fix an integer n ≥ 2 and q ∈ N with q ≥ 1. Define the q-orthant constraint set in [0, 1]n by n n o X Oq,n := (m1 , . . . , mn ) ∈ [0, 1]n : miq ≤ n − 1 . i=1 Definition 3.6.2 (q-Rung n-Tuple Plithogenic Set (general form)). Let S be a universe and let P ⊆ S be nonempty. Fix an attribute v with a nonempty value set P v. A q-rung n-tuple plithogenic set is a quintuple PSq,n = (P, v, P v, pdfq,n , pCF), where • pdfq,n : P × P v → [0, 1]n is the degree of appurtenance function that returns an n-tuple  pdfq,n (x, a) = ma,1 (x), . . . , ma,n (x) , and satisfies the q-rung constraint n X q ma,i (x) ≤ n−1 for all (x, a) ∈ P × P v; i=1 equivalently, pdfq,n (x, a) ∈ Oq,n for all (x, a); • pCF : P v × P v → [0, 1] is a degree of contradiction satisfying pCF(a, a) = 0 and pCF(a, b) = pCF(b, a) for all a, b ∈ P v. When n = 2 we recover the orthopair case; for general n, we obtain an orthovector of appurtenance degrees constrained by the q-sum bound n − 1. Table 3.11 provides an overview of the naming map for q-rung n-tuple plithogenic sets PSq,n . We now present concrete examples of the concept. Example 3.6.3 (Medical triage (Fermatean orthopair, q = 3, n = 2; attribute = symptom severity)). Let the value set be P v = {s0 = none, s1 = mild, s2 = moderate, s3 = severe}, indexed by ι(si ) = i. Define the plithogenic contradiction by the normalized distance pCF(si , sj ) = |ι(si ) − ι(sj )| ∈ [0, 1], 3 and fix the dominant value δ = s3 (severe). Then the compatibility weights are   w(si | δ) = 1 − pCF(si , δ) = 1 − 33 , 1 − 23 , 1 − 13 , 1 − 0 = (0, 13 , 23 , 1), with P i w(si | δ) = 2. For a patient x, suppose the Fermatean (orthopair) degrees for “has the disease” vs “does not have the disease” are a s0 s1 s2 s3 ma (x) = µ(x, a) 0.05 0.35 0.70 0.88 na (x) = ν(x, a) 0.98 0.90 0.80 0.38 # Page. 47 ![Page Image](https://bcdn.docswell.com/page/8EDK3XVW7G.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.11: Naming map for q-rung n-tuple plithogenic sets PSq,n n (tuple size) q=2 q=3 n=2 Pythagorean fuzzy set [514–516] embedded in PS2,2 Fermatean fuzzy set [517–519] embedded in PS3,2 Pythagorean neutrosophic set [334, 522, 523], Pythagorean hesitant fuzzy set [524, 525], and Pythagorean picture fuzzy set [526–528] embedded in PS2,3 Fermatean neutrosophic set [529– 531], Fermatean Picture Fuzzy set [532, 533], and Fermatean Hesitant Fuzzy set [534, 535] embedded in PS3,3 q-rung orthopair fuzzy sets [520, 521] in PSq,2 n=3 q-rung orthopair neutrosophic sets [536, 537], q-rung orthopair Picture Fuzzy Set [538–540], and q-rung orthopair Hesitant Fuzzy Set [541, 542] embedded in PSq,3 Notes. (1) “Pythagorean” ⇔ q = 2; “Fermatean” ⇔ q = 3. (2) n = 2 corresponds to the orthopair fuzzy case; n = 3 to the orthopair neutrosophic case. and each pair satisfies the q-rung constraint µ3 + ν 3 ≤ 1 (e.g., for s3 : 0.883 + 0.383 = 0.681472 + 0.054872 = 0.736344 ≤ 1). Plithogenic aggregation (weighted by w) gives µ(δ) (x) = 0 · 0.05 + 13 · 0.35 + 23 · 0.70 + 1 · 0.88 0 + 0.116666 . . . + 0.466666 . . . + 0.88 = = 0.731666 . . . , 2 2 0 · 0.98 + 13 · 0.90 + 23 · 0.80 + 1 · 0.38 0 + 0.30 + 0.533333 . . . + 0.38 = = 0.606666 . . . . 2 2 Constraint check after aggregation: ν (δ) (x) = (µ(δ) )3 + (ν (δ) )3 ≈ 0.7316663 + 0.6066673 ≈ 0.392 + 0.223 = 0.615 < 1.  Thus the contradiction–aware Fermatean orthopair for triage under the “severe” context is µ(δ) , ν (δ) ≈ (0.732, 0.607). Example 3.6.4 (Credit risk (Pythagorean neutrosophic orthovector, q = 2, n = 3; attribute = income stability)). Let P v = {a0 = low, a1 = medium, a2 = high} with ι(ai ) = i. Set pCF(ai , aj ) = |ι(ai ) − ι(aj )| , 2 w(a | δ) = 1 − pCF(a, δ), Hence w(a0 | δ) = 0, w(a1 | δ) = 12 , w(a2 | δ) = 1 and P δ = a2 . w = 1.5. For an applicant x, let the Pythagorean neutrosophic triple (T, I, F ) (creditworthy, indeterminate, not creditworthy) be a a0 a1 a2 T = µ1 (x, a) 0.20 0.55 0.85 I = µ2 (x, a) 0.20 0.25 0.10 F = µ3 (x, a) 0.90 0.55 0.35 P3 2 2 2 2 Each satisfies the q-rung constraint i=1 µi ≤ n − 1 = 2 (e.g., for a2 : 0.85 + 0.10 + 0.35 = 0.7225 + 0.01 + 0.1225 = 0.855 ≤ 2). Aggregate componentwise: T (δ) (x) = 0 · 0.20 + 12 · 0.55 + 1 · 0.85 0.275 + 0.85 1.125 = = = 0.75, 1.5 1.5 1.5 # Page. 48 ![Page Image](https://bcdn.docswell.com/page/V7PK4PVXJ8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 0 · 0.20 + 12 · 0.25 + 1 · 0.10 0.125 + 0.10 0.225 = = = 0.15, 1.5 1.5 1.5 0 · 0.90 + 12 · 0.55 + 1 · 0.35 0.275 + 0.35 0.625 F (δ) (x) = = = = 0.416666 . . . . 1.5 1.5 1.5 Constraint check after aggregation: I (δ) (x) = (0.75)2 + (0.15)2 + (0.416666 . . .)2 = 0.5625 + 0.0225 + 0.1736 . . . ≈ 0.7586 < 2. Thus the contradiction–aware Pythagorean neutrosophic assessment under the “high income stability” context is  T (δ) , I (δ) , F (δ) = (0.75, 0.15, 0.4167). Example 3.6.5 (Consumer electronics reliability (orthopair, q = 4, n = 2; attribute = warranty length)). Consumer electronics are everyday electronic devices that individuals buy for personal use, including phones, laptops, televisions, audio equipment, and wearables (cf. [543]). Let P v = {b0 = 1 yr, b1 = 2 yr, b2 = 3 yr} with ι(bi ) = i and |ι(bi ) − ι(bj )| , δ = b2 . 2 P Then w(b0 | δ) = 0, w(b1 | δ) = 12 , w(b2 | δ) = 1 and w = 1.5. pCF(bi , bj ) = For a laptop model x, consider the q-rung (q = 4) orthopair for “reliable” vs “not reliable”: b µ(x, b) ν(x, b) b0 0.20 0.98 b1 0.65 0.60 b2 0.90 0.30 Each pair respects µ4 + ν 4 ≤ 1 (e.g., for b2 : 0.904 + 0.304 = 0.6561 + 0.0081 = 0.6642 ≤ 1). Aggregate: µ(δ) (x) = 0 · 0.20 + 12 · 0.65 + 1 · 0.90 0.325 + 0.90 1.225 = = = 0.816666 . . . , 1.5 1.5 1.5 0 · 0.98 + 12 · 0.60 + 1 · 0.30 0.30 + 0.30 0.60 = = = 0.4. 1.5 1.5 1.5 Constraint check after aggregation: ν (δ) (x) = (µ(δ) )4 + (ν (δ) )4 ≈ 0.8166664 + 0.44 ≈ 0.4449 + 0.0256 = 0.4705 < 1. Therefore, under the “3-year warranty” dominant context, the contradiction–aware q-rung (q = 4)  orthopair is µ(δ) , ν (δ) ≈ (0.8167, 0.4000). 3.7 Type-n Plithogenic Set A type-n plithogenic set recursively nests plithogenic memberships and contradictions across n levels to model hierarchical uncertain attributes and relations [544]. Definition 3.7.1 (Type-n Plithogenic Set). [544] Let S be a universal set, and let P ⊆ S. Let v be an attribute taking values in a set P v. Let s ≥ 1 and t ≥ 1 be fixed integers (the dimensions for the Degree of Appurtenance Function and the Degree of Contradiction Function, respectively). A Type-n Plithogenic Set of dimension (s, t), denoted by  P S (n,s,t) = P, v, P v, pdf, pCF , is defined recursively as follows: # Page. 49 ![Page Image](https://bcdn.docswell.com/page/2JVVX2Z3JQ.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.12: Naming for Type-n Plithogenic Sets when t = 0 (no explicit contradiction dimension) Appurtenance dim. s t Standard name for P S (n,s,t) 1 2 3 4 5 0 0 0 0 0 Type-n Fuzzy Set [545–547] Type-n Intuitionistic Fuzzy Set Type-n Neutrosophic Set [496, 544] Type-n Quadripartitioned Neutrosophic Set Type-n Pentapartitioned Neutrosophic Set Note. Here t = 0 indicates that no Degree of Contradiction Function is carried at the top level. This is a degenerate (t = 0) variant of Definition 3.7.1, interpreted as “no explicit contradiction channel”. 1. Base Case (n = 1): A Type-1 Plithogenic Set is simply a classical Plithogenic Set (cf. [10]), i.e. pdf : P × P v → [0, 1]s , pCF : P v × P v → [0, 1]t , subject to the axioms: (a) Reflexivity of Contradiction: pCF (a, a) = 0, ∀ a ∈ P v, (b) Symmetry of Contradiction: pCF (a, b) = pCF (b, a), ∀ a, b ∈ P v. (c) Hence, a Type-1 Plithogenic Set is P S (1,s,t) =  P, v, P v, pdf, pCF , where pdf and pCF map into [0, 1]s and [0, 1]t , respectively. 2. Recursive Case (n > 1): A Type-n Plithogenic Set is given by (s,t) pdf : P × P v −→ Mn−1 ([0, 1]), (s,t) pCF : P v × P v −→ Mn−1 ([0, 1]), where each Mk ([0, 1]) denotes the set of all Type-k Plithogenic Sets (of dimension (s, t)) on the unit interval [0, 1] for their degree of appurtenance and contradiction values.2 (s,t) Concretely, if (n − 1) = 1, then the recursion bottoms out in a classical Plithogenic Set; if (n − 1) > 1, the recursion continues. Thus, a Type-n Plithogenic Set is an (n-level) hierarchical structure where each membership entry is itself a Type-(n − 1) Plithogenic object. Tables 3.12 and 3.13 present the associated concepts of Type-n Plithogenic Sets. We now present concrete examples of the concept. Example 3.7.2 (Healthcare treatment planning — Type-2 Plithogenic Set (n = 2, neutrosophic s = 3, one contradiction channel t = 1)). Healthcare treatment planning designs patient-specific interventions by integrating diagnosis, risks, goals, and resources to optimize clinical outcomes and minimize complications (cf. [569]). Let the universe P contain a patient x. The top-level attribute is therapy intensity with values P v = {L = low, M = medium, H = high}, indexed by ι(L) = 0, ι(M ) = 1, ι(H) = 2. Define the top-level contradiction by normalized distance pCF(a, b) = |ι(a) − ι(b)| ∈ [0, 1], 2 w(a | δ) = 1 − pCF(a, δ), δ = H. 2 More precisely, at each recursion level, the range [0, 1]s for the DAF becomes replaced by a set of dimension-s plithogenic membership objects, each of which is itself a Type-(n − 1) Plithogenic Set. Likewise, the range [0, 1]t for the DCF is replaced by Type-(n − 1) plithogenic contradiction objects. # Page. 50 ![Page Image](https://bcdn.docswell.com/page/5EGLVR4YJL.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.13: Concrete instantiations for n = 2 and n = 3 with t = 0 n s t Standard name 2 2 2 2 2 2 2 2 2 2 1 1 2 2 3 3 3 3 4 5 0 0 0 0 0 0 0 0 0 0 Type-2 Fuzzy Set [548–550] Type-2 Shadowed Set [551, 552] Type-2 Intuitionistic Fuzzy Set [553, 554] Type-2 Vague Set [555, 556] Type-2 Neutrosophic Set [191, 557] Type-2 Picture Fuzzy Set [558, 559] Type-2 Hesitant Fuzzy Set [560, 561] Type-2 Spherical Fuzzy Set [562] Type-2 Quadripartitioned Neutrosophic Set Type-2 Pentapartitioned Neutrosophic Set 3 3 3 3 3 1 2 3 4 5 0 0 0 0 0 Type-3 Fuzzy Set [161, 262, 563, 564] Type-3 Intuitionistic Fuzzy Set [565–567] Type-3 Neutrosophic Set [568] Type-3 Quadripartitioned Neutrosophic Set Type-3 Pentapartitioned Neutrosophic Set Reading. Increasing n raises the type level (depth of iteration of plithogenic-valued memberships). Choosing s ∈ {1, 2, 3, 4, 5} selects the membership tuple shape: fuzzy (s = 1), intuitionistic fuzzy (s = 2), neutrosophic (s = 3), ...; setting t = 0 suppresses the contradiction dimension at the top level. Hence w(L | H) = 0, w(M | H) = 12 , w(H | H) = 1 and P w = 1.5. At each top value a ∈ {L, M, H}, the membership object is a Type-1 plithogenic (neutrosophic) triple (T, I, F ) ∈ [0, 1]3 aggregated from a micro-attribute adherence with values Q = {q0 = poor, q1 = d i , qj ) = |i − j|/2, and micro-weights w(q | q2 ) = fair, q2 = good}, (qi ) = i, micro-DCF pCF(q d 1 − pCF(q, q2 ), i.e. (0, 12 , 1) with sum 1.5. Micro-level data (neutrosophic (T, I, F )) for x: intensity a L M H q0 (poor) (0.35, 0.25, 0.60) (0.45, 0.25, 0.55) (0.52, 0.20, 0.48) q1 (fair) (0.55, 0.20, 0.45) (0.65, 0.18, 0.40) (0.78, 0.14, 0.26) q2 (good) (0.70, 0.15, 0.35) (0.82, 0.12, 0.28) (0.90, 0.10, 0.18) Micro-aggregation to obtain the Type-1 neutrosophic triple at each a (weights 0, 12 , 1; sum 1.5): (T, I, F )L =  0·0.35+ 1 ·0.55+1·0.70 2 1.5 ,  1 1 0·0.25+ 2 ·0.20+1·0.15 0·0.60+ 2 ·0.45+1·0.35 , 1.5 1.5 = (0.6500, 0.1667, 0.3833), (T, I, F )M = (0.7633, 0.1400, 0.3200), (T, I, F )H = (0.8600, 0.1133, 0.2067). Top-level Type-2 aggregation (weights 0, 12 , 1; sum 1.5) relative to δ = H: 0 · 0.6500 + 12 · 0.7633 + 1 · 0.8600 = 0.8278, 1.5 0 · 0.1667 + 12 · 0.1400 + 1 · 0.1133 I (δ) (x) = = 0.1222, 1.5 0 · 0.3833 + 12 · 0.3200 + 1 · 0.2067 F (δ) (x) = = 0.2444. 1.5 T (δ) (x) = # Page. 51 ![Page Image](https://bcdn.docswell.com/page/4JQY6VG67P.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Thus, under the dominant context “high intensity”, the Type-2 neutrosophic evaluation is (T, I, F ) ≈ (0.828, 0.122, 0.244). Example 3.7.3 (Global logistics reliability — Type-3 Plithogenic Set (n = 3, fuzzy s = 1, t = 1)). Global logistics reliability measures how consistently international supply chains deliver goods on time, intact, and as contracted across worldwide markets (cf. [570]). We assess a vendor x along three nested levels. Level 3 (top): attribute vendor scope V = {v0 = local, v1 = regional, v2 = global}, indices ι(vi ) = i, DCF pCF(vi , vj ) = |i − j|/2, dominant δV = v2 , weights wV = (0, 12 , 1), sum 1.5. Level 2 (mid): attribute delivery mode D = {d0 = road, d1 = air, d2 = sea}, indices 0, 1, 2, DCF |i − j|/2, dominant δD = d1 , weights wD = ( 12 , 1, 0), sum 1.5. Level 1 (bottom): attribute weather W = {w0 = clear, w1 = rain, w2 = storm}, indices 0, 1, 2, DCF |i − j|/2, dominant δW = w0 , weights wW = (1, 12 , 0), sum 1.5. Bottom-level fuzzy reliability scores µ ∈ [0, 1] (by delivery mode and weather): d road air sea µ(clear) µ(rain) µ(storm) 0.80 0.60 0.25 0.90 0.75 0.30 0.85 0.55 0.20 Level 1 aggregation (weights 1, 12 , 0; sum 1.5): µroad = 1 1·0.80+ 2 ·0.60+0·0.25 = 0.7333, 1.5 1 1·0.90+ 2 ·0.75+0·0.30 = 0.8500, 1.5 1 1·0.85+ 2 ·0.55+0·0.20 µsea = = 0.7500. 1.5 µair = Level 2 aggregation per vendor scope (dominant d1 ; weights 12 , 1, 0; sum 1.5). For illustration, suppose local/regional/global slightly differ in these level-1 results: scope local (v0 ) regional (v1 ) global (v2 ) Then µroad 0.7333 0.7000 0.7600 µair 0.8500 0.8800 0.9200 µsea 0.7500 0.7200 0.7800 1 ·0.7333+1·0.8500+0·0.7500 = 0.8111, 1.5 1 ·0.7000+1·0.8800 µv1 = 2 = 0.8200, 1.5 1 ·0.7600+1·0.9200 µ v2 = 2 = 0.8667. 1.5 µ v0 = 2 Level 3 aggregation (dominant v2 ; weights 0, 12 , 1; sum 1.5): µ(δV ) (x) = 0 · 0.8111 + 12 · 0.8200 + 1 · 0.8667 0.4100 + 0.8667 = = 0.8511. 1.5 1.5 Thus, the Type-3 fuzzy plithogenic reliability under the “global scope” context is µ ≈ 0.851. # Page. 52 ![Page Image](https://bcdn.docswell.com/page/K74W4M5ME1.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.7.4 (University admission suitability — Type-2 Plithogenic Set (n = 2, intuitionistic fuzzy s = 2, t = 1)). University admission suitability evaluates how well an applicant’s academic record, abilities, and background match a university’s entry requirements and expectations (cf. [571]). Top attribute program intensity P v = {L = light, B = balanced, I = intensive}, indices 0, 1, 2, DCF pCF(a, b) = |i − j|/2, dominant δ = I, weights w(L | I) = 0, w(B | I) = 12 , w(I | I) = 1, sum 1.5. At each a ∈ P v, the membership object is Type-1 intuitionistic fuzzy (µ, ν) aggregated from microattribute study habits Q = {q0 = poor, q1 = average, q2 = strong}, with micro-DCF |i − j|/2, dominant q2 , micro-weights (0, 12 , 1), sum 1.5. All pairs satisfy µ + ν ≤ 1. Micro-level data and aggregation: a L B I q0 (0.30, 0.60) (0.40, 0.55) (0.45, 0.50) q1 (0.55, 0.35) (0.65, 0.30) (0.72, 0.22) q2 (0.70, 0.20) (0.80, 0.15) (0.90, 0.08) Level-1 aggregation (weights 0, 12 , 1; sum 1.5): (µ, ν)L =  0·0.30+ 1 ·0.55+1·0.70 2 1.5 (µ, ν)B = (0.7500, 0.2000), ,  1 0·0.60+ 2 ·0.35+1·0.20 = (0.6500, 1.5 0.2500), (µ, ν)I = (0.8400, 0.1267). (Check: 0.65+0.25 = 0.90 ≤ 1, 0.75+0.20 = 0.95 ≤ 1, 0.84+0.1267 ≈ 0.9667 ≤ 1.) Top-level Type-2 aggregation (weights 0, 12 , 1; sum 1.5): µ(δ) = 0 · 0.65 + 12 · 0.75 + 1 · 0.84 0.375 + 0.84 = = 0.8100, 1.5 1.5 0 · 0.25 + 12 · 0.20 + 1 · 0.1267 0.10 + 0.1267 = = 0.1511. 1.5 1.5 Hence, under the “intensive program” context, the Type-2 intuitionistic fuzzy assessment is (µ, ν) ≈ (0.810, 0.151) with µ + ν ≈ 0.961 ≤ 1. ν (δ) = 3.8 Iterative MultiPlithogenic Set A MultiPlithogenic Set models elements with multi-component degrees and contradiction among attribute values for rich, flexible uncertainty representation in applications. An Iterative MultiPlithogenic Set recursively assigns plithogenic-valued memberships, layering appurtenance and contradiction across levels for hierarchical, multi-attribute, robust uncertainty modeling [407]. Notation 3.8.1. Fix a nonempty universe P (of objects to be evaluated), a single attribute v with a finite value set Pv , and two finite index sets: I = {1, . . . , k} for DAF components, J = {1, . . . , `} for DCF components. Let {Li }i∈I and {L0j }j∈J be complete lattices (typically Li = L0j = [0, 1]). # Page. 53 ![Page Image](https://bcdn.docswell.com/page/LJ1Y48WYEG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.8.2 (Iterative MultiPlithogenic codomains). [407] Define the level-0 plithogenic fiber as Y L(0) := Li . i∈I Recursively, for n ≥ 1 set n L(n) (P, Pv ) := o Φ : P × Pv −→ L(n−1) (P, Pv ) . (Thus an element of L(n) is a plithogenic-valued mapping on P × Pv whose values are level-(n − 1) objects.) Definition 3.8.3 (Iterative MultiPlithogenic Set (order n)). [407] Let n ∈ N, n ≥ 1. An Iterative MultiPlithogenic Set of order n on P is a tuple   (n) IMPS(n) := P, v, Pv , { PDFi }i∈I , { pCFj }j∈J , where • for each i ∈ I, the i-th iterative Degree of Appurtenance Function (DAF) is PDFi (n) : P × Pv −→ L(n−1) (P, Pv ); • for each j ∈ J , the Degree of Contradiction Function (DCF) pCFj : Pv × Pv −→ L0j is symmetric and reflexive: pCFj (α, α) = 0, pCFj (α, β) = pCFj (β, α) for all α, β ∈ Pv . When n = 1 we get the classical (non-iterative) case PDFi MultiPlithogenic DAFs. (1) : P × Pv → L(0) = Q i∈I Li , i.e., ordinary Table 3.14 presents the taxonomy of Iterative MultiPlithogenic Sets by order and instance. We now present concrete examples of the proposed concept. Example 3.8.4 (Smart-home HVAC mode selection — IMPS of order 2 with two fuzzy components). We evaluate a dwelling x ∈ P under attribute HVAC mode with values Pv = {E = Eco, N = Normal, T = Turbo}. Define ι(E) = 0, ι(N ) = 1, ι(T ) = 2 and the term-level contradiction pCF(a, b) = |ι(a) − ι(b)| ∈ [0, 1]. 2 Fix the dominant context δ = T so that the compatibility weights are w(E | T ) = 0, w(N | T ) = 12 , w(T | T ) = 1, P w = 1.5. This is an Iterative MultiPlithogenic Set of order 2 with two DAF components: (i) cost-saving c ∈ [0, 1] (higher is cheaper), and (ii) comfort u ∈ [0, 1] (higher is more comfortable). At level 1 (the inner plithogenic objects), each mode a ∈ Pv aggregates over the micro-attribute outdoor temperature Q = {Mild, Warm, Hot}, with indices 0, 1, 2, micro-DCF |i − j|/2, dominant micro-context “Hot,” and micro-weights wQ (Mild) = 0, wQ (Warm) = 12 , wQ (Hot) = 1 (sum 1.5). # Page. 54 ![Page Image](https://bcdn.docswell.com/page/GJWGXZ6172.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.14: Taxonomy of Iterative MultiPlithogenic Sets by order and instance Order General family Fuzzy instance Intuitionistic fuzzy instance Neutrosophic stance 1 MultiPlithogenic Set MultiFuzzy Set [407, 572, 573] MultiIntuitionistic Fuzzy Set [574–576] (cf. MultiVague Set [577, 578]) n≥2 Iterative MultiPlithogenic Set (order n) Iterative MultiFuzzy Set (order n) [407] Iterative MultiIntuitionistic Fuzzy Set (order n) [407] MultiNeutrosophic Set [408, 579–581] (cf.Multi-Hesitant Fuzzy Set [582–584] and Multi-Picture Fuzzy Set [585]) Iterative MultiNeutrosophic Set (order n) [407] DAF codomain and typical component shapes. At order 1: PDFi choices: (1) MultiFuzzy: MultiIntuitionistic Fuzzy: MultiNeutrosophic: Q i∈I Li . Typical Li = [0, 1] ⇒ (µi )i∈I ∈ [0, 1]k , Li = [0, 1]2 ⇒ (µi , νi ) with µi + νi ≤ 1, Li = [0, 1]3 ⇒ (Ti , Ii , Fi ) ∈ [0, 1]3 . At order n ≥ 2: PDFi : P × Pv → L(n−1) (P, Pv ). mode a E : (c, u) N : (c, u) T : (c, u) Warm (0.75, 0.55) (0.60, 0.80) (0.45, 0.92) (n) Level-1 data for x: : P × Pv → L(0) = in- Mild (0.85, 0.60) (0.70, 0.75) (0.55, 0.88) Hot (0.65, 0.50) (0.50, 0.85) (0.35, 0.95) Level-1 aggregation (weights 0, 12 , 1; denominator 1.5): (c, u)E =  0·0.85+ 1 ·0.75+1·0.65 2 1.5 (c, u)N = (0.5333, 0.8333), ,  1 0·0.60+ 2 ·0.55+1·0.50 = (0.6833, 1.5 0.5167), (c, u)T = (0.3833, 0.9400). Top-level (order 2) aggregation relative to δ = T (weights 0, 12 , 1; denominator 1.5): 1 0·0.6833+ 2 ·0.5333+1·0.3833 = 0.2667+0.3833 = 0.4333, 1.5 1.5 1 0·0.5167+ 2 ·0.8333+1·0.9400 u(δ) (x) = = 0.4167+0.9400 = 0.9044. 1.5 1.5 c(δ) (x) = Hence, under the dominant “Turbo” context, the order-2 multi-fuzzy evaluation is (c, u) ≈ (0.4333, 0.9044). Example 3.8.5 (Fraud-alert triage — IMPS of order 2 with two fuzzy components). Let Pv = {I = Ignore, M = Monitor, B = Block} with ι(I) = 0, ι(M ) = 1, ι(B) = 2, pCF(a, b) = |ι(a) − ι(b)|/2, dominant δ = B giving w(I | B) = 0, w(M | B) = 12 , w(B | B) = 1 (sum 1.5). Two components are recorded at the inner level: precision p ∈ [0, 1] and recall r ∈ [0, 1]. Each action a aggregates over micro-attribute evidence severity Q = {` = low, m = mid, h = high} with indices 0, 1, 2, micro-DCF |i − j|/2, dominant h, and micro-weights (0, 12 , 1) (sum 1.5). Level-1 data: a I M B (p, r)` (p, r)m (0.95, 0.20) (0.85, 0.35) (0.90, 0.50) (0.82, 0.70) (0.80, 0.60) (0.72, 0.80) (p, r)h (0.70, 0.50) (0.75, 0.85) (0.65, 0.95) # Page. 55 ![Page Image](https://bcdn.docswell.com/page/4EZL61YX73.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Level-1 aggregation (weights 0, 12 , 1; denominator 1.5): (p, r)I =  0·0.95+ 1 ·0.85+1·0.70 2 1.5 (p, r)M = (0.7733, 0.8000), ,  1 0·0.20+ 2 ·0.35+1·0.50 = (0.7500, 1.5 0.4500), (p, r)B = (0.6733, 0.9000). Top-level (order 2) aggregation relative to δ = B (weights 0, 12 , 1; denominator 1.5): 1 0·0.7500+ 2 ·0.7733+1·0.6733 = 0.3867+0.6733 = 0.7066, 1.5 1.5 1 0·0.4500+ 2 ·0.8000+1·0.9000 r(δ) (x) = = 0.4000+0.9000 = 0.8667. 1.5 1.5 p(δ) (x) = Thus, in the dominant “Block” context, the order-2 multi-fuzzy assessment is (p, r) ≈ (0.7066, 0.8667). Example 3.8.6 (Public transit planning — IMPS of order 2 with two neutrosophic components). Public transit planning designs, schedules, and optimizes buses, trains, and routes to move people efficiently, affordably, safely, and sustainably citywide (cf. [586, 587]). We choose service frequency Pv = {L = low, M = medium, H = high}, ι(L) = 0, ι(M ) = 1, ι(H) = 2, pCF(a, b) = |ι(a) − ι(b)|/2, dominant δ = H, so w(L | H) = 0, w(M | H) = 12 , w(H | H) = 1 (sum 1.5). This order-2 MultiNeutrosophic instance has two components, each a neutrosophic triple (T, I, F ): component 1 = reliability, component 2 = low crowding (comfort). At level 1, each frequency a aggregates by the time-of-day micro-attribute Q = {OP = off-peak, PK = peak, LN = late-night}, with indices 0, 1, 2, micro-DCF |i − j|/2, dominant micro-context PK, micro-weights wQ (OP) = 1 1 2 , wQ (PK) = 1, wQ (LN) = 2 (sum 2.0). Level-1 data for x: Reliability (T, I, F ) a L M H OP (0.70, 0.20, 0.30) (0.78, 0.18, 0.25) (0.82, 0.15, 0.22) PK (0.55, 0.25, 0.45) (0.68, 0.20, 0.35) (0.80, 0.14, 0.20) LN (0.65, 0.22, 0.35) (0.72, 0.18, 0.28) (0.78, 0.16, 0.22) OP (0.80, 0.15, 0.20) (0.75, 0.18, 0.25) (0.70, 0.22, 0.30) PK (0.40, 0.30, 0.60) (0.55, 0.25, 0.45) (0.50, 0.28, 0.50) LN (0.75, 0.18, 0.25) (0.70, 0.20, 0.30) (0.65, 0.24, 0.35) Low-crowding (T, I, F ) a L M H Level-1 aggregation (weights 12 , 1, 12 ; denominator 2.0): Reliability (T, I, F )L =  0.5·0.70+1·0.55+0.5·0.65 0.5·0.20+1·0.25+0.5·0.22 0.5·0.30+1·0.45+0.5·0.35 , , 2 2 2  = (0.6125, 0.2300, 0.3875), (T, I, F )M = (0.7150, 0.1900, 0.3075), (T, I, F )H = (0.8000, 0.1475, 0.2100). # Page. 56 ![Page Image](https://bcdn.docswell.com/page/Y76W2LDP7V.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Low-crowding (T, I, F )L =  0.5·0.80+1·0.40+0.5·0.75 0.5·0.15+1·0.30+0.5·0.18 0.5·0.20+1·0.60+0.5·0.25 , , 2 2 2  = (0.5875, 0.2325, 0.4125), (T, I, F )M = (0.6375, 0.2200, 0.3625), (T, I, F )H = (0.5875, 0.2550, 0.3958). Top-level (order 2) aggregation relative to δ = H (weights 0, 12 , 1; denominator 1.5): Reliability 1 0·0.6125+ 2 ·0.7150+1·0.8000 = 0.3575+0.8000 = 0.7717, 1.5 1.5 1 0·0.2300+ 2 ·0.1900+1·0.1475 I (δ) = = 0.0950+0.1475 = 0.1617, 1.5 1.5 1 0·0.3875+ 2 ·0.3075+1·0.2100 F (δ) = = 0.1538+0.2100 = 0.2425. 1.5 1.5 T (δ) = Low-crowding 1 0·0.5875+ 2 ·0.6375+1·0.5875 = 0.3188+0.5875 = 0.6042, 1.5 1.5 1 0·0.2325+ 2 ·0.2200+1·0.2550 I (δ) = = 0.1100+0.2550 = 0.2433, 1.5 1.5 1 0·0.4125+ 2 ·0.3625+1·0.3958 F (δ) = = 0.1813+0.3958 = 0.3958. 1.5 1.5 T (δ) = Therefore, under the dominant “high frequency” context, the order-2 multi-neutrosophic outputs are: Reliability (T, I, F ) ≈ (0.7717, 0.1617, 0.2425) and Low-crowding (T, I, F ) ≈ (0.6042, 0.2433, 0.3958). 3.9 Interval-Valued Plithogenic Set An Interval-Valued Plithogenic Set assigns each object interval-valued memberships and contradiction degrees, handling uncertainty with bounded ranges instead of single numeric values [588]. Definition 3.9.1 (Interval–Valued Plithogenic Set). [588] Let S be a universal set and let P ⊆ S be a nonempty subset. Let v be an attribute whose set of possible values is P v. Fix two positive integers s ≥ 1 and t ≥ 1. Denote by I([0, 1]) := { [`, u] ⊆ [0, 1] | 0 ≤ ` ≤ u ≤ 1 } the family of all closed subintervals of [0, 1]. An Interval–Valued Plithogenic Set (IVPS) of dimension (s, t) is a tuple  (s,t) P SIV = P, v, P v, pdfIV , pCFIV , where 1. the interval–valued degree of appurtenance function (interval–valued DAF) s pdfIV : P × P v −→ I([0, 1]) assigns to every pair (x, a) ∈ P × P v an s–tuple of membership–intervals pdfIV (x, a) =  [α1L (x, a), α1U (x, a)], . . . , [αsL (x, a), αsU (x, a)] , where 0 ≤ αiL (x, a) ≤ αiU (x, a) ≤ 1 for all i = 1, . . . , s; # Page. 57 ![Page Image](https://bcdn.docswell.com/page/G75M213P74.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 2. the interval–valued degree of contradiction function (interval–valued DCF) t pCFIV : P v × P v −→ I([0, 1]) assigns to every pair (a, b) ∈ P v × P v a t–tuple of contradiction–intervals  [β1L (a, b), β1U (a, b)], . . . , [βtL (a, b), βtU (a, b)] , pCFIV (a, b) = where 0 ≤ βjL (a, b) ≤ βjU (a, b) ≤ 1 for all j = 1, . . . , t, and which satisfies the usual plithogenic axioms: • (Reflexivity) for all a ∈ P v, pCFIV (a, a) =  [0, 0], . . . , [0, 0] ; • (Symmetry) for all a, b ∈ P v, pCFIV (a, b) = pCFIV (b, a). If, for every (x, a) and every (a, b), the intervals are degenerate, i.e. [αiL (x, a), αiU (x, a)] = [αi (x, a), αi (x, a)], then P SIV (s,t) [βjL (a, b), βjU (a, b)] = [βj (a, b), βj (a, b)], reduces to the classical (crisp–valued) plithogenic set of dimension (s, t). The contents of Table 3.15 describe how typical interval-valued fuzzy and neutrosophic families arise as special cases of the Interval-Valued Plithogenic Set (IVPS). Table 3.15: Typical interval–valued fuzzy/neutrosophic families captured as special cases of the Interval–Valued Plithogenic Set (IVPS) by choosing s equal to the number of interval components and t = 0 (no DCF) or t ≥ 1 (DCF–aware). Interval–valued set Interval–Valued Fuzzy Set [589–591] Interval–Valued Intuitionistic Fuzzy Set [592–595] (Intervalvalued vague sets [596, 597]) Interval valued pythagorean fuzzy sets [598, 599] Interval–Valued Neutrosophic Set [600–603] Interval-valued picture fuzzy sets [604–606] Interval-valued hesitant fuzzy sets [607, 608] Interval-valued spherical fuzzy sets [609, 610] Interval–Valued Quadripartitioned Neutrosophic Set [611, 612] Interval–Valued Pentapartitioned Neutrosophic Set [9, 613, 614] Generalization by IVPS Single interval membership is pdfIV (·, ·) ∈ I([0, 1]); contradiction optional. Two interval components (membership, nonmembership) become s=2 plithogenic intervals. s 1 t 0/1 2 0/1 2 0/1 Two interval components become s=2 plithogenic intervals. 3 0/1 3 0/1 3 0/1 3 0/1 4 0/1 Three interval components (truth, indeterminacy, falsity) become s=3 plithogenic intervals. Three interval components become s=3 plithogenic intervals. Three interval components become s=3 plithogenic intervals. Three interval components become s=3 plithogenic intervals. Four interval components embedded as s=4 in IVPS. 5 0/1 Five interval components embedded as s=5 in IVPS. We now present concrete examples of the proposed concept. # Page. 58 ![Page Image](https://bcdn.docswell.com/page/9J2941ZZER.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.9.2 (E–commerce shipping choice — Interval–Valued Fuzzy Plithogenic (s=1, t=1)). E-commerce shipping choice is selecting delivery options balancing speed, cost, reliability, tracking, environmental impact, customer convenience, and overall price tradeoffs (cf. [615]). Consider a product x ∈ P to be recommended with respect to attribute v = “shipping option” and P v = {E = Economy, S = Standard, X = Express}. Let the interval–valued DAF (suitability) for x be pdfIV (x, E) = [0.50, 0.70], pdfIV (x, S) = [0.60, 0.85], pdfIV (x, X) = [0.40, 0.65]. Assume an interval–valued DCF (larger means more contradictory) given by pCFIV (E, S) = [0.20, 0.30], pCFIV (X, S) = [0.30, 0.50], pCFIV (E, X) = [0.60, 0.80], pCFIV (a, a) = [0, 0]. Fix the dominant value δ = S. Define compatibility intervals w(a | δ) := 1 − pCFIV (a, δ): w(E | S) = [0.70, 0.80], w(S | S) = [1, 1], w(X | S) = [0.50, 0.70], so the total weight interval is X W := w = [0.70 + 1 + 0.50, 0.80 + 1 + 0.70] = [2.20, 2.50]. Using interval arithmetic with non–negative bounds: [α, β] · [γ, δ] = [αγ, βδ] and [A, B] + [C, D] = [A + C, B + D], the numerator interval is N = w(E | S) pdfIV (x, E) + w(S | S) pdfIV (x, S) + w(X | S) pdfIV (x, X) = [0.70, 0.80]·[0.50, 0.70] + [1, 1]·[0.60, 0.85] + [0.50, 0.70]·[0.40, 0.65] = [0.35, 0.56] + [0.60, 0.85] + [0.20, 0.455] = [1.15, 1.865]. For division by a positive interval [c, d] we use [A, B]/[c, d] = [A/d, B/c]. Hence the aggregated (Standard–relative) suitability interval is h 1.15 1.865 i N Γ(S) (x) = = , = [0.4600, 0.8477]. W 2.50 2.20 Interpretation: under the dominant “Standard” context, x has overall IV–fuzzy suitability in [0.4600, 0.8477]. Example 3.9.3 (Hiring decision (candidate y) — Interval–Valued Intuitionistic Plithogenic (s=2, t=1)). Hiring decision selects candidates based on skills, experience, cultural fit, and organizational needs, balancing fairness, cost, and long-term potential carefully (cf. [616]). Let v = “contract type” with P v = {I = Intern, F = Fixed-term, P = Permanent}. For candidate y, take interval–valued intuitionistic appurtenances (µ, ν): pdfIV (y, I) = ( [0.55, 0.75], [0.10, 0.25] ), pdfIV (y, F ) = ( [0.65, 0.80], [0.10, 0.20] ), pdfIV (y, P ) = ( [0.50, 0.65], [0.20, 0.35] ), (each pair chosen so that max µ + max ν ≤ 1). Let the DCF intervals be pCFIV (I, P ) = [0.50, 0.70], pCFIV (F, P ) = [0.20, 0.30], pCFIV (a, a) = [0, 0], and set the dominant value δ = P . Compatibility intervals: w(I | P ) = [0.30, 0.50], w(F | P ) = [0.70, 0.80], w(P | P ) = [1, 1], W = [2.00, 2.30]. Aggregating µ (membership) by interval arithmetic: Nµ = [0.30, 0.50]·[0.55, 0.75] + [0.70, 0.80]·[0.65, 0.80] + [1, 1]·[0.50, 0.65] = [0.165, 0.375] + [0.455, 0.640] + [0.50, 0.65] = [1.120, 1.665], # Page. 59 ![Page Image](https://bcdn.docswell.com/page/DEY4MZRNJM.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets µ(P ) (y) = h 1.120 1.665 i Nµ = , = [0.4870, 0.8325]. W 2.30 2.00 Aggregating ν (nonmembership): Nν = [0.30, 0.50]·[0.10, 0.25] + [0.70, 0.80]·[0.10, 0.20] + [1, 1]·[0.20, 0.35] = [0.030, 0.125] + [0.070, 0.160] + [0.20, 0.35] = [0.300, 0.635], h 0.300 0.635 i Nν = , = [0.1304, 0.3175]. W 2.30 2.00 Thus, relative to the dominant “Permanent” context, the candidate’s intuitionistic IV assessment is ν (P ) (y) = (µ, ν)(P ) (y) ∈ [0.4870, 0.8325] × [0.1304, 0.3175]. Example 3.9.4 (Precision agriculture irrigation — Interval–Valued Neutrosophic Plithogenic (s=3, t=1)). Agriculture irrigation supplies controlled water to crops through canals, sprinklers, or drip systems, improving yields, reducing drought risk and erosion (cf. [617]). Let v = “irrigation strategy” with P v = {C = Conservative, B = Balanced, A = Aggressive}. For a plot z, take IV–neutrosophic triples (T, I, F ): pdfIV (z, C) = ( [0.60, 0.80], [0.10, 0.25], [0.10, 0.20] ), pdfIV (z, B) = ( [0.70, 0.90], [0.05, 0.15], [0.05, 0.10] ), pdfIV (z, A) = ( [0.40, 0.60], [0.20, 0.30], [0.25, 0.40] ). Assume DCF intervals to the target (balanced) policy: pCFIV (C, B) = [0.10, 0.20], pCFIV (A, B) = [0.30, 0.50], pCFIV (B, B) = [0, 0]. Dominant value δ = B. Compatibility intervals and total weight: w(C | B) = [0.80, 0.90], w(B | B) = [1, 1], w(A | B) = [0.50, 0.70], W = [2.30, 2.60]. Truth component: NT = [0.80, 0.90]·[0.60, 0.80] + [1, 1]·[0.70, 0.90] + [0.50, 0.70]·[0.40, 0.60] = [0.48, 0.72] + [0.70, 0.90] + [0.20, 0.42] = [1.38, 2.04], h 1.38 2.04 i NT T (B) (z) = = , = [0.5308, 0.8869]. W 2.60 2.30 Indeterminacy component: NI = [0.80, 0.90]·[0.10, 0.25] + [1, 1]·[0.05, 0.15] + [0.50, 0.70]·[0.20, 0.30] = [0.08, 0.225] + [0.05, 0.15] + [0.10, 0.21] = [0.23, 0.585], h 0.23 0.585 i NI I (B) (z) = = , = [0.0885, 0.2543]. W 2.60 2.30 Falsity component: NF = [0.80, 0.90]·[0.10, 0.20] + [1, 1]·[0.05, 0.10] + [0.50, 0.70]·[0.25, 0.40] = [0.08, 0.18] + [0.05, 0.10] + [0.125, 0.28] = [0.255, 0.56], h 0.255 0.56 i NF F (B) (z) = = , = [0.0981, 0.2435]. W 2.60 2.30 Therefore, relative to the dominant “Balanced” policy, the plot’s IV–neutrosophic irrigation assessment is (T, I, F )(B) (z) ∈ [0.5308, 0.8869] × [0.0885, 0.2543] × [0.0981, 0.2435]. # Page. 60 ![Page Image](https://bcdn.docswell.com/page/VJNYW3DV78.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.10 Plithogenic OffSet A plithogenic offset allows membership and contradiction degrees beyond [0, 1], modeling negative and over-membership under attribute-based contradictions in complex systems [618–621]. Definition 3.10.1 (Plithogenic Offset). [260,622] Let S be a universal set, and P ⊆ S. A Plithogenic Offset P Soffset is defined as: P Soffset = (P, v, P v, pdf, pCF ), where: • v is an attribute. • P v is the set of possible values for the attribute v. • pdf : P × P v → [Ψv , Ωv ]s is the Degree of Appurtenance Function (DAF), where Ψv < 0 and Ωv > 1. • pCF : P v × P v → [Ψv , Ωv ]t is the Degree of Contradiction Function (DCF). In a Plithogenic Offset, membership degrees pdf (x, a) range from below 0 (negative membership) to above 1 (over-membership), providing flexibility to model diverse membership states. Table 3.16 presents the comparison of plithogenic sets, oversets, undersets, and offsets based on the range of their degree of appurtenance functions. Concept Plithogenic Set Membership range of DAF pdf pdf : P × P v → [0, 1]s Plithogenic OverSet pdf : P × P v → (1, Ωv ]s with Ωv > 1 Plithogenic UnderSet pdf : P × P v → [Ψv , 0)s with Ψv < 0 Plithogenic OffSet pdf : P × P v → [Ψv , Ωv ]s with Ψv < 0 < 1 < Ωv Brief description Standard plithogenic model: truth / indeterminacy / falsity components in [0, 1], with a contradiction function pCF : P v × P v → [0, 1]t controlling attribute–based aggregation. Over–membership only: each component strictly larger than 1, up to an upper bound Ωv . Models redundancy, over–satisfaction, or “super–belonging” under plithogenic contradictions. Negative–membership only: each component strictly below 0, down to a lower bound Ψv . Represents active opposition, counter–evidence, or anti–belonging in a plithogenic way. Full offset scale: a single plithogenic structure that simultaneously admits under–membership (< 0), classical membership ([0, 1]), and over–membership (> 1), together with a (possibly extended) contradiction function pCF : P v ×P v → [Ψv , Ωv ]t . Table 3.16: Comparison of plithogenic set, overset, underset, and offset via the range of the degree of appurtenance function. # Page. 61 ![Page Image](https://bcdn.docswell.com/page/YE9PX9GVJ3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.17: Special cases of the plithogenic offset P Soff (s, t) for t = 1. s t Offset Type 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 Plithogenic fuzzy offset (cf. [623, 624]) (/ Plithogenic Grey sets (cf. [625–627]) ) Plithogenic intuitionistic fuzzy offset (cf. [623, 624]) Plithogenic neutrosophic offset (cf. [623, 628, 629]) Plithogenic quadripartitioned neutrosophic offset Plithogenic pentapartitioned neutrosophic offset Plithogenic hexapartitioned neutrosophic offset Plithogenic heptapartitioned neutrosophic offset Plithogenic octapartitioned neutrosophic offset Plithogenic nonapartitioned neutrosophic offset The following table 3.17 summarizes important special cases of the plithogenic offset P Soff (s, t) when t = 1. (For t = 0, each case reduces to the corresponding classical offset.) We now present concrete examples of the proposed concept. Example 3.10.2 (Bank credit limit upgrade with plithogenic fuzzy offsets (fuzzy case, s=1, t=1)). Bank credit limit is the maximum amount a customer may borrow on a credit facility, set by risk assessment policies (cf. [630]). Let P = {cust} be a customer under review. The attribute is v = “recommended credit limit tier” with P v = {Low, Medium, High}. Because we allow offsets, the DAF can take values below 0 and above 1: pdf (cust, Low) = −0.10, pdf (cust, Medium) = 0.80, pdf (cust, High) = 1.40. Use a contradiction map pCF : P v × P v → [0, 1] (we keep [0, 1] for simplicity): pCF (Low, High) = 0.85, pCF (Medium, High) = 0.30, pCF (Low, Medium) = 0.50, pCF (a, a) = 0. Fix the dominant context δ = High and set weights w(a | δ) = 1 − pCF (a, δ). Then w(Low | High) = 0.15, w(Medium | High) = 0.70, w(High | High) = 1. The plithogenic aggregate relative to δ is the weighted mean Γ(High) (cust) = 0.15 · (−0.10) + 0.70 · 0.80 + 1 · 1.40 −0.015 + 0.56 + 1.40 1.945 = = ≈ 1.0514. 0.15 + 0.70 + 1 1.85 1.85 Interpretation. Although there is weak negative evidence for “Low”, strong over-membership for “High” and moderate support for “Medium” yield an over-membership aggregate slightly above 1, supporting an upgrade beyond the standard “High” threshold. Example 3.10.3 (Hospital triage with plithogenic intuitionistic fuzzy offsets (IF-offset, s=2, t=1)). Let P = {pt} be a patient and v = “triage category” with P v = {Immediate, Urgent, Routine}. Use a two-component DAF pdf (·, ·) = (µ, ν) allowing offsets: pdf (pt, Immediate) = (−0.10, 1.05), pdf (pt, Urgent) = (1.10, 0.00), pdf (pt, Routine) = (0.20, 0.95). Here µ is the membership and ν the non-membership; both may be < 0 or > 1. Set contradictions pCF (Immediate, Urgent) = 0.20, pCF (Routine, Urgent) = 0.60, and choose δ = Urgent. Weights w = 1 − pCF give w(Immediate | Urgent) = 0.80, pCF (a, a) = 0, # Page. 62 ![Page Image](https://bcdn.docswell.com/page/GE8D29V4ED.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets w(Urgent | Urgent) = 1, w(Routine | Urgent) = 0.40, W = 0.80 + 1 + 0.40 = 2.20. Aggregate membership and non-membership separately: Γ(δ) µ (pt) = 0.80(−0.10) + 1(1.10) + 0.40(0.20) −0.08 + 1.10 + 0.08 1.10 = = = 0.50, 2.20 2.20 2.20 0.80(1.05) + 1(0.00) + 0.40(0.95) 0.84 + 0 + 0.38 1.22 = = ≈ 0.5545. 2.20 2.20 2.20 Interpretation. The raw offsets include an over-membership 1.10 for “Urgent” and a negative membership for “Immediate”. After contradiction-aware aggregation toward δ = Urgent, the final intuitionistic pair is (µ, ν) ≈ (0.50, 0.5545), reflecting balanced support with slightly higher aggregated hesitation/contradiction against urgency. Γ(δ) ν (pt) = Example 3.10.4 (Power-grid dispatch with plithogenic neutrosophic offsets (NS-offset, s=3, t=1)). Power-grid dispatch coordinates generation units in real time to balance supply and demand while respecting network constraints and reliability requirements (cf. [631]). Consider P = {hour? } (a specific operating hour) and v = “primary energy source” with P v = {Coal, Gas, Solar}. Use neutrosophic triples pdf = (T, I, F ) with offsets allowed: pdf (hour? , Coal) = (−0.20, 0.10, 1.15), pdf (hour? , Gas) = (0.80, 0.20, 0.20), pdf (hour? , Solar) = (1.30, −0.05, 0.00). Contradictions to a solar-dominant context δ = Solar are pCF (Coal, Solar) = 0.85, pCF (Gas, Solar) = 0.40, pCF (a, a) = 0. Weights w = 1 − pCF yield w(Coal | Solar) = 0.15, w(Gas | Solar) = 0.60, w(Solar | Solar) = 1, W = 1.75. Aggregate each neutrosophic component: (δ) ΓT = (δ) ΓI = (δ) 0.15(−0.20) + 0.60(0.80) + 1(1.30) −0.03 + 0.48 + 1.30 1.75 = = = 1.00, 1.75 1.75 1.75 0.15(0.10) + 0.60(0.20) + 1(−0.05) 0.015 + 0.12 − 0.05 0.085 = = ≈ 0.0486, 1.75 1.75 1.75 ΓF = 0.15(1.15) + 0.60(0.20) + 1(0.00) 0.1725 + 0.12 + 0 0.2925 = = ≈ 0.1671. 1.75 1.75 1.75 Interpretation. Offsets encode strong policy/subsidy preference for Solar (T =1.30) and counterevidence for Coal (T = − 0.20, F =1.15). The contradiction-aware neutrosophic aggregate under a solar-dominant context is (T, I, F ) ≈ (1.00, 0.0486, 0.1671), i.e., decisive endorsement of Solar with very low indeterminacy and modest falsity. 3.11 Plithogenic Cubic Set A Plithogenic Cubic Set assigns each element interval and crisp membership data, weighted by attribute-wise contradiction, unifying fuzzy, intuitionistic fuzzy, and neutrosophic cubic models [217, 484, 632–636]. # Page. 63 ![Page Image](https://bcdn.docswell.com/page/LELM2W5V7R.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.11.1 (Plithogenic Cubic Set). [217, 632, 633] Let A be a nonempty universe. Let an attribute v take values in a set R (“attribute value set”), equipped with a degree of contradiction function C : R × R → [0, 1] satisfying (i) C(r, r) = 0 and (ii) C(r1 , r2 ) = C(r2 , r1 ). Write I([0, 1]) = { [α, β] ⊆ [0, 1] | 0 ≤ α ≤ β ≤ 1 }. A Plithogenic Cubic Set (PCS) on A is a collection of triples  P = ha, I(a), c(a)i a∈A , where I(·) is an interval–valued plithogenic appurtenance object and c(·) is its point–valued counterpart of the same type, chosen from one of the following canonical instances. (1) Plithogenic Fuzzy Cubic Set (PFCS). Γ = hX, µi, X : A → I([0, 1]), X(a) = [X − (a), X + (a)], µ : A → [0, 1], with 0 ≤ X − (a) ≤ X + (a) ≤ 1 for all a ∈ A. Here X is an interval–valued plithogenic fuzzy set (its interval endpoints arise from plithogenic appurtenance aggregation driven by C and, optionally, a dominant value in R), while µ is an ordinary fuzzy membership. (2) Plithogenic Intuitionistic Fuzzy Cubic Set (PIFCS). Ψ = hY, δi, Y : A → I([0, 1]) × I([0, 1]),  Y (a) = [µ− (a), µ+ (a)], [ν − (a), ν + (a)] , δ(a) = (µ(a), ν(a)) ∈ [0, 1]2 , subject to 0 ≤ µ− ≤ µ+ ≤ 1, 0 ≤ ν − ≤ ν + ≤ 1, and the intuitionistic constraint µ+ (a) + ν + (a) ≤ 1 for all a ∈ A. Here Y is an interval–valued plithogenic intuitionistic fuzzy set and δ is an intuitionistic fuzzy membership pair. (3) Plithogenic Neutrosophic Cubic Set (PNCS). Ω = hZ, λi, Z : A → I([0, 1])3 ,  Z(a) = [T − (a), T + (a)], [I − (a), I + (a)], [F − (a), F + (a)] , λ(a) = (T (a), I(a), F (a)) ∈ [0, 1]3 , with 0 ≤ T − ≤ T + ≤ 1, 0 ≤ I − ≤ I + ≤ 1, 0 ≤ F − ≤ F + ≤ 1 for all a ∈ A. Here Z is an interval–valued plithogenic neutrosophic set and λ is a (single–valued) neutrosophic triple. We now present concrete examples of the proposed concept. Example 3.11.2 (E–commerce targeting with a Plithogenic Fuzzy Cubic Set (PFCS)). Let A = {item? } be a specific product. The attribute is v = “customer segment” with value set R = {Budget, Standard, Premium}. Let the contradiction C : R × R → [0, 1] be symmetric with C(Budget, Premium) = 0.80, C(Standard, Premium) = 0.30, C(Budget, Standard) = 0.40, C(r, r) = 0. Fix the dominant segment r = Premium and set compatibility weights w(r | r∗ ) = 1 − C(r, r∗ ), hence w(Budget | r∗ ) = 0.20, w(Standard | r∗ ) = 0.70, ∗ w(Premium | r∗ ) = 1.00, W = 0.20 + 0.70 + 1.00 = 1.90. # Page. 64 ![Page Image](https://bcdn.docswell.com/page/4JMY893NJW.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.18: Classical cubic set families as special cases of the Plithogenic Cubic Set Classical cubic set Interval (set–valued) component Point (crisp) component Generalization Plithogenic Cubic Fuzzy cubic set X(a) = [X − (a), X + (a)] ⊆ [0, 1] µ(a) ∈ [0, 1] Obtain PFCS by letting the interval endpoints arise from plithogenic aggregation over attribute values and contradiction C(·, ·); the fuzzy cubic set is the case with no (or trivial) contradiction and a single attribute value. Intuitionistic fuzzy cubic set Y (a) = ([µ− (a), µ+ (a)], [ν − (a), ν + (a)]) with µ+ (a) + ν + (a) ≤ 1 (µ(a), ν(a)) with µ(a) + ν(a) ≤ 1 Obtain PIFCS by generating both interval pairs through a plithogenic DAF driven by an attribute v and a DCF C; the classical intuitionistic fuzzy cubic set is recovered when C ≡ 0 and there is only one attribute value. Neutrosophic cubic set [637–642] Z(a) ([T − , T + ], [I − , I + ], [F − , F + ]) ⊆ [0, 1]3 (T (a), I(a), F (a)) ∈ [0, 1]3 Obtain PNCS by letting each neutrosophic interval come from plithogenic fusion against a dominant value and contradiction C; the ordinary neutrosophic cubic set is the specialization with C ≡ 0 (or a single value of v). = inside Experts supply segment–wise fuzzy intervals and crisp memberships for item? : r Budget Standard Premium [X − (r), X + (r)] µ(r) [0.30, 0.55] 0.50 [0.60, 0.80] 0.70 [0.85, 0.95] 0.90 Aggregate interval endpoints and crisp membership by a weighted mean: X − (item? ) = 0.20 · 0.30 + 0.70 · 0.60 + 1.00 · 0.85 0.06 + 0.42 + 0.85 1.33 = = = 0.7000, 1.90 1.90 1.90 X + (item? ) = 0.20 · 0.55 + 0.70 · 0.80 + 1.00 · 0.95 0.11 + 0.56 + 0.95 1.62 = = ≈ 0.8526, 1.90 1.90 1.90 µ(item? ) = 0.20 · 0.50 + 0.70 · 0.70 + 1.00 · 0.90 0.10 + 0.49 + 0.90 1.49 = = ≈ 0.7842. 1.90 1.90 1.90 Therefore the PFCS triple is hitem? , [0.7000, 0.8526], 0.7842i. Interpretation: contradiction–aware weighting toward the dominant “Premium” segment tightens the interval near high affinity and raises the crisp membership. # Page. 65 ![Page Image](https://bcdn.docswell.com/page/PJR95GQ479.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.11.3 (Scholarship selection with a Plithogenic Intuitionistic Fuzzy Cubic Set (PIFCS)). Scholarship selection evaluates applicants’ academic merit, research potential, financial need, and community impact to allocate limited educational available funding fairly (cf. [643]). Let A = {applicant? } and v = “evidence source” with R = {Grades, Research, Community}. Choose r∗ = Research. Let the symmetric contradiction be C(Grades, Research) = 0.25, C(Community, Research) = 0.40, C(r, r) = 0, so that w(Grades | r∗ ) = 0.75, w(Community | r∗ ) = 0.60, w(Research | r∗ ) = 1.00, and W = 2.35. Each source provides an intuitionistic interval pair and a crisp pair (µ, ν): r Grades Research Community [µ− (r), µ+ (r)] [ν − (r), ν + (r)] [0.65, 0.80] [0.10, 0.20] [0.75, 0.92] [0.03, 0.10] [0.50, 0.70] [0.10, 0.20] (µ(r), ν(r)) (0.72, 0.18) (0.85, 0.08) (0.60, 0.18) Contradiction–weighted aggregation (componentwise) yields µ− (app? ) = 0.75 · 0.65 + 1.00 · 0.75 + 0.60 · 0.50 0.4875 + 0.75 + 0.30 1.5375 = = ≈ 0.6543, 2.35 2.35 2.35 0.75 · 0.80 + 1.00 · 0.92 + 0.60 · 0.70 0.60 + 0.92 + 0.42 1.94 = = ≈ 0.8255, 2.35 2.35 2.35 0.75 · 0.10 + 1.00 · 0.03 + 0.60 · 0.10 0.075 + 0.03 + 0.06 0.165 ν − (app? ) = = = ≈ 0.0702, 2.35 2.35 2.35 0.75 · 0.20 + 1.00 · 0.10 + 0.60 · 0.20 0.15 + 0.10 + 0.12 0.37 ν + (app? ) = = = ≈ 0.1574, 2.35 2.35 2.35 which respects the intuitionistic bound µ+ + ν + ≤ 1. For the crisp pair, µ+ (app? ) = µ(app? ) = 0.75 · 0.72 + 1.00 · 0.85 + 0.60 · 0.60 0.54 + 0.85 + 0.36 1.75 = = ≈ 0.7447, 2.35 2.35 2.35 0.75 · 0.18 + 1.00 · 0.08 + 0.60 · 0.18 0.135 + 0.08 + 0.108 0.323 = = ≈ 0.1372. 2.35 2.35 2.35 Hence the PIFCS triple is D E applicant? , ([µ− , µ+ ], [ν − , ν + ]) = ([0.6543, 0.8255], [0.0702, 0.1574]), (µ, ν) = (0.7447, 0.1372) . ν(app? ) = Interpretation: emphasizing research reduces disagreement and concentrates the interval toward eligibility while keeping nonmembership low. Example 3.11.4 (Urban bike-station siting with a Plithogenic Neutrosophic Cubic Set (PNCS)). Let A = {site? } and v = “planning criterion” with R = {Demand, Safety, Cost}. Choose r∗ = Demand. Define a symmetric contradiction C(Safety, Demand) = 0.30, C(Cost, Demand) = 0.60, C(r, r) = 0, so that w(Safety | r∗ ) = 0.70, w(Cost | r∗ ) = 0.40, w(Demand | r∗ ) = 1.00, with W = 2.10. Criterion–wise neutrosophic intervals and crisp triples (T, I, F ) are: r Demand Safety Cost [T − (r), T + (r)] [I − (r), I + (r)] [F − (r), F + (r)] [0.80, 0.95] [0.03, 0.10] [0.02, 0.07] [0.70, 0.88] [0.05, 0.12] [0.07, 0.15] [0.40, 0.60] [0.10, 0.20] [0.30, 0.45] r Demand Safety Cost (T (r), I(r), F (r)) (0.90, 0.06, 0.04) (0.80, 0.09, 0.11) (0.50, 0.15, 0.35) Aggregate (componentwise) with the weights: T − (site? ) = 1 · 0.80 + 0.70 · 0.70 + 0.40 · 0.40 0.80 + 0.49 + 0.16 1.45 = = ≈ 0.6905, 2.10 2.10 2.10 # Page. 66 ![Page Image](https://bcdn.docswell.com/page/PEXQKX5ZJX.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 1 · 0.95 + 0.70 · 0.88 + 0.40 · 0.60 0.95 + 0.616 + 0.24 1.806 = = = 0.8600, 2.10 2.10 2.10 1 · 0.03 + 0.70 · 0.05 + 0.40 · 0.10 0.03 + 0.035 + 0.04 0.105 I − (site? ) = = = = 0.0500, 2.10 2.10 2.10 1 · 0.10 + 0.70 · 0.12 + 0.40 · 0.20 0.10 + 0.084 + 0.08 0.264 I + (site? ) = = = ≈ 0.1257, 2.10 2.10 2.10 1 · 0.02 + 0.70 · 0.07 + 0.40 · 0.30 0.02 + 0.049 + 0.12 0.189 F − (site? ) = = = = 0.0900, 2.10 2.10 2.10 1 · 0.07 + 0.70 · 0.15 + 0.40 · 0.45 0.07 + 0.105 + 0.18 0.355 F + (site? ) = = = ≈ 0.1690. 2.10 2.10 2.10 For the crisp neutrosophic triple: T + (site? ) = T (site? ) = 1 · 0.90 + 0.70 · 0.80 + 0.40 · 0.50 0.90 + 0.56 + 0.20 1.66 = = ≈ 0.7905, 2.10 2.10 2.10 1 · 0.06 + 0.70 · 0.09 + 0.40 · 0.15 0.06 + 0.063 + 0.06 0.183 = = ≈ 0.0871, 2.10 2.10 2.10 1 · 0.04 + 0.70 · 0.11 + 0.40 · 0.35 0.04 + 0.077 + 0.14 0.257 F (site? ) = = = ≈ 0.1224. 2.10 2.10 2.10 Thus the PNCS triple is I(site? ) = site? , ([0.6905, 0.8600], [0.0500, 0.1257], [0.0900, 0.1690]), (0.7905, 0.0871, 0.1224) . Interpretation: prioritizing demand yields high truth and low indeterminacy about suitability; modest falsity persists due to cost and safety trade–offs. 3.12 Plithogenic Soft, HyperSoft, and SuperHyperSoft Set Recall that soft set represents objects by parameterized approximate subsets, enabling flexible, simple modeling of uncertainty in decision-making contexts under vague information [307,644]. HyperSoft set extends soft sets to multi-attribute, multi-valued parameter configurations, capturing higher-order, context-dependent uncertainties in datasets with structured aggregation rules [311,312,645,646]. SuperHyperSoft set iterates powerset-based soft constructions, linking hyper-parameters across levels to encode hierarchical, multilayer, interacting uncertainties over complex object families [330, 647, 648]. Plithogenic SuperHyperSoft set augments superhypersoft structures with contradiction-aware membership functions, modeling conflicting multi-attribute information across hierarchical parameters in realistic systems [649]. Definition 3.12.1 (Plithogenic SuperHyperSoft Set). Let U be a universe and let a1 , . . . , an be n ≥ 1 distinct attributes. For each i, let Ai be a finite, pairwise–disjoint set of values (Ai ∩ Aj = ∅ for i 6= j), and write C = P(A1 ) × · · · × P(An ). Fix a nonempty Y ⊆ U . A Plithogenic SuperHyperSoft Set (PSHSS) over U is the data  PSHSS = U, Y, {ai }ni=1 , {Ai }ni=1 , C, d, c , together with a mapping F : C → P(U ), where:  • d : Y × C → P [0, 1]j is the (possibly multi–valued) appurtenance function, with j ∈ {1, 2, 3} indicating fuzzy (j=1), intuitionistic fuzzy (j=2), or neutrosophic (j=3) membership codomain;  • for each attribute ai , c : P(Ai ) × P(Ai ) → P [0, 1]j is a (possibly multi–valued) contradiction function satisfying reflexivity and symmetry: c(α, α) = {0}, c(α, β) = c(β, α) (α, β ⊆ Ai ); # Page. 67 ![Page Image](https://bcdn.docswell.com/page/3EK95W6VED.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets • the image F (γ) ⊆ U for γ = (α1 , . . . , αn ) ∈ C is determined plithogenically from {dy (γ)}y∈Y together with the contradictions {c(αi , αi ), c(αi , αi0 )} via the chosen plithogenic aggregation/composition rules. Table 3.20 presents the reductions from a PSHSS to Plithogenic HyperSoft and Plithogenic Soft Sets. In addition, Table 3.19 summarizes the relationships between a PSHSS and the Fuzzy SuperHyperSoft Set, Intuitionistic Fuzzy SuperHyperSoft Set, and Neutrosophic SuperHyperSoft Set. Table 3.19: Plithogenic SuperHyperSoft Set (PSHSS) as a common generalization of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and partitioned Neutrosophic SuperHyperSoft Sets. Model Codomain of d Plithogenic SuperHyperSoft Set (PSHSS) Fuzzy SuperHyperSoft Set [650] Intuitionistic Fuzzy SuperHyperSoft Set Neutrosophic SuperHyperSoft Set [651–654] Quadripartitioned Neutrosophic SuperHyperSoft Set Pentapartitioned Neutrosophic SuperHyperSoft Set c Specialization of PSHSS P([0, 1] ), j ∈ {1, . . . , 5} reflexive, symmetric c Base model P([0, 1]) c(α, β) ≡ {0} j = 1, c ≡ {0} P([0, 1]2 ) c(α, β) ≡ {0} j = 2, c ≡ {0} P([0, 1]3 ) c(α, β) ≡ {0} j = 3, c ≡ {0} P([0, 1]4 ) c(α, β) ≡ {0} j = 4, c ≡ {0} P([0, 1]5 ) c(α, β) ≡ {0} j = 5, c ≡ {0} j A concrete example of this concept is provided below. Example 3.12.2 (Hospital bed allocation (neutrosophic PSHSS, j = 3)). Hospital bed allocation is the process where hospitals assign limited beds to patients to optimize care, reduce delays, manage capacity, and ensure efficient treatment flow (cf. [691]). Universe U = {pA , pB , pC } (patients); focus subset Y = {pA , pB }. Attributes and value-sets (pairwise disjoint): a1 = Severity, A1 = {Mild, Moderate, Severe}; a2 = Resources, A2 = {Low, High}; a3 = InfectionRisk, A3 = {Low, High}. Superhyper–parameter domain C = P(A1 ) × P(A2 ) × P(A3 ). Choose configuration γ = ({Moderate, Severe}, {High}, {High}) and dominant configuration γ ∗ = ({Severe}, {High}, {High}). Element-level (symmetric, reflexive) contradictions c1 , c2 , c3 : Ai × Ai → [0, 1]: c1 (Mild, Severe) = 0.90, c1 (Moderate, Severe) = 0.30, c2 (Low, High) = 0.80, c2 (x, x) = 0; c3 (Low, High) = 0.90, c3 (x, x) = 0. c1 (x, x) = 0; Lift to subset–level via the max–min Hausdorff–type rule n o b ci (H, K) := max max min ci (h, k), max min ci (h, k) . h∈H k∈K k∈K h∈H Then b c1 ({Mod, Sev}, {Sev}) = max{ c1 (Mod, Sev), c1 (Sev, Sev) } = max{0.30, 0} = 0.30, # Page. 68 ![Page Image](https://bcdn.docswell.com/page/L73WK18Q75.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.20: PSHSS reductions to Plithogenic HyperSoft / Plithogenic Soft Sets and their fuzzy, intuitionistic fuzzy, and neutrosophic specializations. Target structure Plithogenic HyperSoft Set (PHS [655–659] Domain restriction Replace the power–set domain by singletons: C 0 = A1 × · · · × An (identify {ωi } with ωi ∈ Ai ) Plithogenic Soft Set (PSS) [206, 660, 661] Take a single attribute (n = 1) and singletons: C 00 = A1 (From PSS) fuzzy / IF / neutrosophic soft Same as above, single attribute, singletons (From PHS) fuzzy / IF / neutrosophic hypersoft Same as above, C 0 = A1 × · · · × An b c2 ({High}, {High}) = 0, Function restriction Restrict d to d|Y ×C 0 : Y × C 0 → P([0, 1]j ); restrict c from P(Ai )×P(Ai ) to Ai × Ai via c0 (ω, ω 0 ) := c({ω}, {ω 0 }) d|Y ×A1 : Y ×A1 → P([0, 1]j ); c|A1 ×A1 via c0 (ω, ω 0 ) := c({ω}, {ω 0 }) Choose j = 1 (fuzzy [662,663] and shadowed [664, 665]), j = 2 (intuitionistic [604,666] and vague [667, 668]), j = 3 (neutrosophic [669, 670], hesitant Fuzzy [671, 672], picture fuzzy [673, 674], spherical fuzzy [675, 676]), j = 4 (quadripartitioned neutrosophic [377, 677, 678] and Double-valued neutrosophic [679, 680]), j = 5 (pentapartitioned neutrosophic) [9, 681] and set pCF ≡ 0 Choose j = 1, 2, 3 respectively and set contradiction to zero or constant Outcome Yields the standard PHS on A1 × · · · × An Reduces to a plithogenic soft set over parameter set A1 Recovers fuzzy soft, IF soft, neutrosophic soft Recovers fuzzy [682, 683]/IF [684, 685]/neutrosophic [422, 686](/hesitant fuzzy [687]/picture fuzzy [688, 689]/spherical fuzzy [690]) hypersoft set b c3 ({High}, {High}) = 0. Compatibility weights per attribute wi := 1 − b ci (αi , αi∗ ) and global weight w := 3 Y wi = (1 − 0.30)(1 − 0)(1 − 0) = 0.70. i=1 Neutrosophic appurtenance (singleton image of d) for pA at γ: dpA (γ) = {(T, I, F )} = {(0.85, 0.10, 0.08)}. One admissible plithogenic selection score is score(pA | γ, γ ∗ ) := w (T −F )−I = 0.70 (0.85−0.08)−0.10 = 0.70·0.77−0.10 = 0.439−0.10 = 0.339. With threshold τ = 0, score > 0 so pA ∈ F (γ) (assign bed). The same pipeline applies to pB , possibly yielding exclusion if the score falls below τ due to higher contradictions or lower truth. Example 3.12.3 (Retail assortment planning (fuzzy PSHSS, j = 1)). Retail assortment planning is the process where retailers determine optimal product mixes to meet customer demand, maximize sales, reduce stockouts, and improve profitability (cf. [692]). # Page. 69 ![Page Image](https://bcdn.docswell.com/page/87DK3XMWJG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Universe U is the SKU catalog; focus item s? ∈ Y ⊆ U . Attributes and values: a1 = Season, A1 = {Spring, Summer, Autumn, Winter}; a2 = PriceBand, A2 = {Low, Mid, High}; a3 = BrandStyle, A3 = {Classic, Sport, Tech}. Choose γ = ({Summer, Autumn}, {Mid, High}, {Sport, Tech}), γ ∗ = ({Autumn}, {High}, {Sport}). Singleton contradictions (symmetric, reflexive), nonzero entries: c1 (Summer, Autumn) = 0.20, c2 (Mid, High) = 0.30, c3 (Tech, Sport) = 0.40. Consider four singleton triples inside γ: θ1 = (Autumn, High, Sport), θ2 = (Summer, High, Sport), θ3 = (Autumn, Mid, Sport), θ4 = (Autumn, High, Tech). Per-triple compatibility w(θ) = Q3 i=1 w(θ1 ) = 1,  1 − ci (θi , θi∗ ) with θ∗ = (Autumn, High, Sport) gives w(θ2 ) = (1 − 0.20) · 1 · 1 = 0.80, w(θ3 ) = 1 · (1 − 0.30) · 1 = 0.70, w(θ4 ) = 1 · 1 · (1 − 0.40) = 0.60. Let the fuzzy memberships (from ds? ) be µ(θ1 ) = 0.86, µ(θ2 ) = 0.70, µ(θ3 ) = 0.75, µ(θ4 ) = 0.68. Normalize and aggregate: W = 4 X w(θk ) = 1.00 + 0.80 + 0.70 + 0.60 = 3.10, k=1 ∗ µagg (s? | γ, γ ) = P4 k=1 w(θk ) µ(θk ) W 1 · 0.86 + 0.8 · 0.70 + 0.7 · 0.75 + 0.6 · 0.68 2.353 = = ≈ 0.7584. 3.10 3.10 Decision: include s? in the autumn–premium–sport capsule if µagg ≥ 0.75 (here borderline but acceptable under a 0.75 cap with managerial override). Example 3.12.4 (University timetabling (intuitionistic PSHSS, j = 2)). University timetabling is the process where institutions schedule courses, rooms, instructors, and times to avoid conflicts, optimize resources, and support students’ learning (cf. [693]). Universe U is the set of candidate time–room slots; focus y? ∈ Y ⊆ U . Attributes and values: a1 = Period, A1 = {Morning, Afternoon}; a2 = RoomType, A2 = {Lecture, Lab}; a3 = Day, A3 = {Mon, Tue}. Take γ = ({Morning, Afternoon}, {Lecture, Lab}, {Mon, Tue}), ∗ γ = ({Morning}, {Lecture}, {Tue}). # Page. 70 ![Page Image](https://bcdn.docswell.com/page/VJPK4PRXE8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Singleton contradictions (symmetric, reflexive): c1 (Afternoon, Morning) = 0.40, c2 (Lab, Lecture) = 0.50, c3 (Mon, Tue) = 0.30. Select four representative triples inside γ: θ1 = (Morning, Lecture, Tue), θ2 = (Afternoon, Lecture, Tue), θ3 = (Morning, Lecture, Mon), θ4 = (Morning, Lab, Tue). Weights: w(θ2 ) = (1 − 0.40) = 0.60, w(θ1 ) = 1, w(θ3 ) = (1 − 0.30) = 0.70, w(θ4 ) = (1 − 0.50) = 0.50; W = 2.80. From dy? obtain intuitionistic pairs (µ, ν): (µ, ν)(θ1 ) = (0.85, 0.10), (µ, ν)(θ2 ) = (0.62, 0.25), (µ, ν)(θ3 ) = (0.70, 0.20), (µ, ν)(θ4 ) = (0.60, 0.22). Weighted aggregation: µagg = 1 · 0.85 + 0.6 · 0.62 + 0.7 · 0.70 + 0.5 · 0.60 0.85 + 0.372 + 0.49 + 0.30 2.012 = = ≈ 0.7186, 2.80 2.80 2.80 1 · 0.10 + 0.6 · 0.25 + 0.7 · 0.20 + 0.5 · 0.22 0.10 + 0.15 + 0.14 + 0.11 0.50 = = ≈ 0.1786, 2.80 2.80 2.80 which satisfies µagg + νagg ≤ 1. Decision rule (example): schedule y? if µagg − νagg ≥ 0.50; here 0.7186 − 0.1786 = 0.5400 ≥ 0.50, so y? ∈ F (γ) (slot accepted). νagg = 3.13 Hesitant Plithogenic Set A Hesitant Plithogenic Set assigns to each object and attribute value a finite nonempty set [588] of plithogenic membership vectors in [0, 1]s , together with a contradiction function on the attribute value set. It unifies hesitant fuzzy [6] and hesitant neutrosophic sets as special cases by choosing s = 1 and s = 3, respectively, and by setting the contradiction degrees to zero. Notation 3.13.1 (Finite hesitation families). Let X be a nonempty set. Denote by  H(X) := H ⊆ X H 6= ∅, H is finite the family of all finite nonempty subsets of X. Elements of H(X) will be called hesitation sets over X. Definition 3.13.2 (Hesitant Plithogenic Set). Let S be a universal set and let P ⊆ S be a nonempty subset. Let v be an attribute with value set P v. Fix integers s ≥ 1 and t ≥ 1. A Hesitant Plithogenic Set (HPS) of dimension (s, t) is a tuple (s,t) P SH =  P, v, P v, hpdf, pCF , where 1. the hesitant plithogenic degree of appurtenance function (hesitant DAF)  hpdf : P × P v −→ H [0, 1]s assigns to each pair (x, a) ∈ P × P v a finite nonempty set  hpdf (x, a) = α(1) (x, a), . . . , α(mx,a ) (x, a) ⊆ [0, 1]s , where mx,a ∈ N and each vector α(r) (x, a) = (α1 (x, a), . . . , αs (x, a)) has components in [0, 1]; (r) (r) # Page. 71 ![Page Image](https://bcdn.docswell.com/page/2EVVX2R3EQ.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 2. the degree of contradiction function (DCF) pCF : P v × P v −→ [0, 1]t satisfies the usual plithogenic axioms: • (Reflexivity) for all a ∈ P v, pCF (a, a) = (0, . . . , 0) ∈ [0, 1]t ; • (Symmetry) for all a, b ∈ P v, pCF (a, b) = pCF (b, a). When every hesitation set hpdf (x, a) is a singleton, P SH plithogenic set of dimension (s, t). (s,t) reduces to the classical (non-hesitant) Table 3.21 presents a summary of the roles of the parameters s and t in hesitant plithogenic memberships. Table 3.21: Roles of the parameters s and t in hesitant plithogenic memberships Parameter s=1 s=2 s=3 s=3 s=3 s=4 s=5 t Interpretation Encodes hesitant fuzzy [6, 359]-type memberships (each hesitant element is a list of scalar membership degrees). Encodes hesitant intuitionistic fuzzy [694,695]-type pairs (each hesitant element is a list of membership/nonmembership pairs). Encodes hesitant neutrosophic triples (T, I, F ) [696,697] (each hesitant element is a list of truth, indeterminacy, and falsity triples). Encodes hesitant picture fuzzy triples [698, 699]. Encodes spherical hesitant fuzzy triples [700, 701]. Encodes hesitant Quadripartitioned neutrosophic quadruple. Encodes hesitant Pentapartitioned neutrosophic quintuple. Counts the number of contradiction components in pCF (a, b) ∈ [0, 1]t . The special case pCF (a, b) ≡ (0, . . . , 0), for all attribute values a, b, recovers the usual hesitant structures without contradiction-awareness. A concrete example of this concept is provided below. Example 3.13.3 (Hesitant Plithogenic Set in Medical Diagnosis Triage). Consider an emergency department that must triage incoming patients according to the severity of a suspected condition. Let the universe of discourse be the set of patients P = {x1 , x2 }, where x1 is a young adult patient and x2 is an elderly patient with comorbidities. Let v be the attribute “clinical severity”, with value set P v = {mild, moderate, severe}. We work with dimension (s, t) = (3, 1) so that each membership vector has the neutrosophic form (T, I, F ) ∈ [0, 1]3 , and there is a single contradiction component. # Page. 72 ![Page Image](https://bcdn.docswell.com/page/57GLVRMYEL.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets The degree of contradiction function pCF : P v × P v → [0, 1] is chosen as  0,      0.4, pCF (a, b) =  0.6,     0.9, a = b, {a, b} = {mild, moderate}, {a, b} = {moderate, severe}, {a, b} = {mild, severe}, and extended symmetrically, so that the values reflect how “far apart” the severity labels are. For each patient x ∈ P and each attribute value a ∈ P v, the hesitant plithogenic degree of appurtenance  hpdf (x, a) ∈ H [0, 1]3 collects several candidate neutrosophic assessments (T, I, F ) from different physicians or diagnostic tools. For the young adult patient x1 , suppose we have:  hpdf (x1 , mild) = (0.75, 0.15, 0.10), (0.68, 0.22, 0.10) ,  hpdf (x1 , moderate) = (0.40, 0.30, 0.30) ,  hpdf (x1 , severe) = (0.10, 0.20, 0.70) . Here, for instance, (0.75, 0.15, 0.10) means that one expert believes with truth 0.75 and falsity 0.10 (with indeterminacy 0.15) that the patient is “mild”, while another expert provides the slightly different assessment (0.68, 0.22, 0.10); the set of such vectors forms the hesitation set. For the elderly patient x2 , assume:  hpdf (x2 , mild) = (0.20, 0.25, 0.55) ,  hpdf (x2 , moderate) = (0.55, 0.20, 0.25), (0.48, 0.32, 0.20) ,  hpdf (x2 , severe) = (0.35, 0.25, 0.40) . The tuple (3,1) P SH = P, v, P v, hpdf, pCF  is then a concrete Hesitant Plithogenic Set. It simultaneously captures: • multiple competing neutrosophic opinions for each (patient, severity-label) pair; and • the degrees of contradiction between different labels (mild, moderate, severe), which can be used in plithogenic aggregation rules (for example, giving more weight to labels with smaller contradiction against a chosen dominant label). A triage policy can subsequently aggregate each hesitation set (e.g. via weighted averaging or ordered selection) while explicitly accounting for the contradiction values pCF in order to prioritize treatments or allocate limited resources. # Page. 73 ![Page Image](https://bcdn.docswell.com/page/4EQY6VR6JP.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.13.4 (Hesitant Plithogenic Set in Supplier Risk Assessment). Consider a manufacturing company that evaluates suppliers according to their reliability in delivering critical components on time. Let the universe be the set of candidate suppliers P = {s1 , s2 , s3 }. Let v be the attribute “reliability class”, with attribute value set P v = {high, medium, low}. Here we work with dimension (s, t) = (1, 1), so each membership vector is a single fuzzy-type grade α ∈ [0, 1], and there is a single contradiction component. The contradiction function pCF : P v × P v → [0, 1] is given by  0, a = b,      0.6, {a, b} = {high, medium}, pCF (a, b) =  0.7, {a, b} = {medium, low},     0.95, {a, b} = {high, low}, extended symmetrically. Thus “high” and “low” reliability classes are almost maximally contradictory. The hesitant plithogenic degree of appurtenance hpdf : P × P v −→ H([0, 1]) collects multiple fuzzy membership scores obtained from different departments (e.g. logistics, finance, and quality control). For supplier s1 (long-term partner with mostly on-time deliveries) we may have: hpdf (s1 , high) = {0.82, 0.88, 0.91}, hpdf (s1 , medium) = {0.10, 0.15}, hpdf (s1 , low) = {0.02}. Here, for example, 0.82 and 0.88 could come from historical on-time rates as assessed by two different analysts, while 0.91 comes from a recent predictive model. For supplier s2 (new supplier with limited track record), suppose: hpdf (s2 , high) = {0.40, 0.55}, hpdf (s2 , medium) = {0.45, 0.50, 0.60}, hpdf (s2 , low) = {0.15, 0.25}. The hesitation sets express the fact that different stakeholders hold somewhat conflicting views about s2 ’s reliability. For supplier s3 (supplier with recurring delays), assume: hpdf (s3 , high) = {0.05}, hpdf (s3 , medium) = {0.25, 0.30}, hpdf (s3 , low) = {0.70, 0.80}. Thus the tuple (1,1) P SH = P, v, P v, hpdf, pCF  is a Hesitant Plithogenic Set that models supplier reliability assessment. A decision maker can choose a dominant attribute value (for example, “high” reliability) and perform plithogenic aggregation of each hesitation set hpdf (si , a) by: # Page. 74 ![Page Image](https://bcdn.docswell.com/page/KJ4W4M8M71.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets • first combining the hesitant fuzzy memberships inside each hpdf (si , a) (e.g. by average, median, or optimistic/pessimistic selection); and • then contradiction-weighting the resulting aggregated scores using pCF (high, a) so that memberships coming from values more contradictory to “high” (such as “low”) are penalized more strongly. This provides a flexible, contradiction-aware mechanism to rank or screen suppliers under heterogeneous and hesitant expert opinions. For completeness, we recall the standard hesitant fuzzy and hesitant neutrosophic sets in a form compatible with Definition 3.13.2. Definition 3.13.5 (Hesitant Fuzzy Set). [359] Let P be a nonempty universe. A Hesitant Fuzzy Set (HFS) on P is a pair HF = (P, hF ), where hF : P −→ H([0, 1]) assigns to each x ∈ P a finite nonempty set hF (x) ⊆ [0, 1] of possible membership degrees of x in the set. Definition 3.13.6 (Hesitant Neutrosophic Set). [697] Let P be a nonempty universe. A Hesitant Neutrosophic Set (HNS) on P is a pair HN = (P, hN ), where hN : P −→ H [0, 1]3  assigns to each x ∈ P a finite nonempty set hN (x) ⊆ [0, 1]3 of neutrosophic triples (T, I, F ) ∈ [0, 1]3 , interpreted as hesitant degrees of truth, indeterminacy, and falsity for x. We now show formally that the Hesitant Plithogenic Set simultaneously generalizes Hesitant Fuzzy Sets and Hesitant Neutrosophic Sets by suitable choices of s, t, and the attribute value set. Theorem 3.13.7 (Hesitant Plithogenic Set generalizes Hesitant Fuzzy Set). Fix a nonempty universe P . Consider the class of all Hesitant Fuzzy Sets on P ,  HFS(P ) := HF = (P, hF ) hF : P → H([0, 1]) , and the subclass of Hesitant Plithogenic Sets n (1,1) HPSF (P ) := P SH = (P, v, P v, hpdf, pCF ) Then there exists a bijection o P v = {a0 }, pCF (a, b) ≡ 0 . ΦF : HFS(P ) −→ HPSF (P ), so every Hesitant Fuzzy Set can be identified with a unique Hesitant Plithogenic Set with s = 1, t = 1, a single attribute value, and zero contradiction. # Page. 75 ![Page Image](https://bcdn.docswell.com/page/LE1Y482Y7G.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Proof. Step 1. Construction of the map ΦF : HFS(P ) → HPSF (P ). Let HF = (P, hF ) ∈ HFS(P ) be arbitrary. Fix an attribute name v and a singleton value set P v = {a0 }. Define pCF (a, b) := 0 ∈ [0, 1] for all a, b ∈ P v. Clearly pCF is reflexive and symmetric. Define hpdf : P × P v → H([0, 1]1 ) by hpdf (x, a0 ) :=  (α) ∈ [0, 1]1 α ∈ hF (x) , x ∈ P. (3.2) Since hF (x) is finite and nonempty, the set on the right-hand side is also finite and nonempty, so hpdf (x, a0 ) ∈ H([0, 1]1 ). Thus (1,1) P SH (HF ) := P, v, P v, hpdf, pCF  is a well-defined Hesitant Plithogenic Set of dimension (s, t) = (1, 1) with P v = {a0 } and pCF ≡ 0. We set (1,1) ΦF (HF ) := P SH (HF ). Step 2. Construction of the inverse map ΨF : HPSF (P ) → HFS(P ). Let (1,1) P SH = (P, v, {a0 }, hpdf, pCF ) ∈ HPSF (P ) be arbitrary. Define hF : P → H([0, 1]) by  hF (x) := α ∈ [0, 1] (α) ∈ hpdf (x, a0 ) , x ∈ P. (3.3) Since each hpdf (x, a0 ) is finite and nonempty, the set of its first coordinates hF (x) is also finite and nonempty, so hF (x) ∈ H([0, 1]). Hence HF := (P, hF ) is a Hesitant Fuzzy Set on P . Define (1,1) ΨF (P SH ) := HF . Step 3. Verification that ΨF and ΦF are mutual inverses. (i) For HF = (P, hF ) ∈ HFS(P ) and x ∈ P , combine (3.2) and (3.3) to compute    (1,1) ΨF ◦ ΦF (HF )(x) = ΨF P SH (HF ) (x) = α ∈ [0, 1] (α) ∈ hpdf (x, a0 ) . By (3.2), hpdf (x, a0 ) = Therefore  (β) ∈ [0, 1]1 n  ΨF ◦ ΦF (HF )(x) = α ∈ [0, 1] β ∈ hF (x) . o  (α) ∈ (β) | β ∈ hF (x) . The condition (α) ∈ {(β) | β ∈ hF (x)} is equivalent to ∃β ∈ hF (x) with (α) = (β). Equality of 1-dimensional vectors (α) = (β) implies α = β. Thus   ΨF ◦ ΦF (HF )(x) = α ∈ [0, 1] α ∈ hF (x) = hF (x). Since this holds for all x ∈ P , we obtain ΨF ◦ ΦF = idHFS(P ) . # Page. 76 ![Page Image](https://bcdn.docswell.com/page/GEWGXZR1J2.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets (ii) Conversely, let P SH = (P, v, {a0 }, hpdf, pCF ) ∈ HPSF (P ) and x ∈ P . Then ΨF (P SH (P, hF ) with hF given by (3.3). Applying ΦF we obtain a new hesitant plithogenic DAF  hpdf 0 (x, a0 ) := (α) ∈ [0, 1]1 α ∈ hF (x) . (1,1) (1,1) )= Using (3.3), this becomes hpdf 0 (x, a0 ) = Thus n hpdf 0 (x, a0 ) = n (α) ∈ [0, 1]1 (α) ∈ [0, 1]1  α ∈ β ∈ [0, 1] (β) ∈ hpdf (x, a0 ) o . o ∃β ∈ [0, 1] with (β) ∈ hpdf (x, a0 ) and α = β . But the condition “there exists β with (β) ∈ hpdf (x, a0 ) and α = β” is equivalent to “(α) ∈ hpdf (x, a0 )”. Hence  hpdf 0 (x, a0 ) = (α) ∈ [0, 1]1 (α) ∈ hpdf (x, a0 ) = hpdf (x, a0 ). Therefore the DAFs of P SH and ΦF ΨF (P SH are unchanged by construction. Thus (1,1) ) coincide, and the other components (P, v, P v, pCF ) (1,1)  ΦF ◦ ΨF = idHPSF (P ) . Step 4. Since ΨF ◦ ΦF = idHFS(P ) and ΦF ◦ ΨF = idHPSF (P ) , the maps ΦF and ΨF are mutual inverses and hence ΦF is a bijection. This proves that every Hesitant Fuzzy Set is canonically realized as a Hesitant Plithogenic Set with (s, t) = (1, 1), a single attribute value, and zero contradiction. Theorem 3.13.8 (Hesitant Plithogenic Set generalizes Hesitant Neutrosophic Set). Fix a nonempty universe P . Consider the class of all Hesitant Neutrosophic Sets on P ,  HNS(P ) := HN = (P, hN ) hN : P → H([0, 1]3 ) , and the subclass of Hesitant Plithogenic Sets n (3,1) HPSN (P ) := P SH = (P, v, P v, hpdf, pCF ) Then there exists a bijection o P v = {a0 }, pCF (a, b) ≡ 0 . ΦN : HNS(P ) −→ HPSN (P ), so every Hesitant Neutrosophic Set can be identified with a unique Hesitant Plithogenic Set with s = 3, t = 1, a single attribute value, and zero contradiction. Proof. The proof is parallel to Theorem 3.13.7, but now s = 3 and no change of dimension is needed. Step 1. For HN = (P, hN ) ∈ HNS(P ), fix an attribute v, set P v = {a0 }, and define pCF (a, b) := 0 for all a, b ∈ P v. Define hpdf (x, a0 ) := hN (x) ⊆ [0, 1]3 , x ∈ P. Since hN (x) is finite and nonempty, this yields hpdf (x, a0 ) ∈ H([0, 1]3 ), so (3,1) P SH Set ΦN (HN ) := P SH (3,1) Step 2. For P SH (3,1) (HN ) :=  P, v, {a0 }, hpdf, pCF ∈ HPSN (P ). (HN ). = (P, v, {a0 }, hpdf, pCF ) ∈ HPSN (P ), define hN : P → H([0, 1]3 ) simply by hN (x) := hpdf (x, a0 ), x ∈ P, # Page. 77 ![Page Image](https://bcdn.docswell.com/page/47ZL61RXJ3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets and put ΨN (P SH (3,1) ) := (P, hN ) ∈ HNS(P ). Step 3. For any HN = (P, hN ) and x ∈ P we have   (3,1) ΨN ◦ ΦN (HN )(x) = ΨN P SH (HN ) (x) = hpdf (x, a0 ) = hN (x), so ΨN ◦ ΦN = idHNS(P ) . Conversely, for any P SH and x ∈ P we have  (3,1) ΦN ◦ ΨN (P SH )(x, a0 ) = hN (x) = hpdf (x, a0 ), (3,1) so the DAF is preserved and the remaining components (P, v, P v, pCF ) are unchanged. Thus ΦN ◦ ΨN = idHPSN (P ) . Step 4. Hence ΦN is a bijection between HNS(P ) and HPSN (P ). This shows that every Hesitant Neutrosophic Set is realized as a particular Hesitant Plithogenic Set with (s, t) = (3, 1), a single attribute value, and trivial contradiction. 3.14 Spherical Plithogenic Sets Spherical plithogenic sets model attribute-valued elements using neutrosophic triples constrained on a sphere, integrating contradiction degrees for complex uncertainty interactions. In this section we recall the standard notions of spherical fuzzy set and spherical neutrosophic set, then introduce the notion of a spherical plithogenic set, and finally prove that it generalizes both previous concepts. Definition 3.14.1 (Spherical Fuzzy Set). [366,367,702] Let U be a nonempty universe. A spherical fuzzy set A on U is determined by three membership functions TA , IA , FA : U −→ [0, 1] such that, for every x ∈ U , 0 ≤ TA (x)2 + IA (x)2 + FA (x)2 ≤ 1. (3.4) We usually write A =   x, TA (x), IA (x), FA (x) : x ∈ U . Definition 3.14.2 (Spherical Neutrosophic Set). [703–705] Let U be a nonempty universe. A spherical neutrosophic set A on U is described by three functions TA , IA , FA : U −→ [0, 3] such that, for every x ∈ U , 0 ≤ TA (x)2 + IA (x)2 + FA (x)2 ≤ 3. (3.5) As before we may write A =   x, TA (x), IA (x), FA (x) : x ∈ U . Example 3.14.3 (Spherical Neutrosophic Set). Let U = {x1 , x2 , x3 } be a universe, where x1 = “normal”, x2 = “warning”, x3 = “critical” state of a machine. Define a spherical neutrosophic set A on U by TA (x1 ) = 0.9, IA (x1 ) = 0.2, FA (x1 ) = 0.1, TA (x2 ) = 0.6, IA (x2 ) = 0.6, FA (x2 ) = 0.4, TA (x3 ) = 0.3, IA (x3 ) = 0.8, FA (x3 ) = 0.9. Then, for each x ∈ U we have TA (x1 )2 + IA (x1 )2 + FA (x1 )2 = 0.92 + 0.22 + 0.12 = 0.81 + 0.04 + 0.01 = 0.86 ≤ 3, TA (x2 )2 + IA (x2 )2 + FA (x2 )2 = 0.62 + 0.62 + 0.42 = 0.36 + 0.36 + 0.16 = 0.88 ≤ 3, TA (x3 )2 + IA (x3 )2 + FA (x3 )2 = 0.32 + 0.82 + 0.92 = 0.09 + 0.64 + 0.81 = 1.54 ≤ 3, so the constraint (3.5) is satisfied for all x ∈ U , and hence A is a spherical neutrosophic set on U . # Page. 78 ![Page Image](https://bcdn.docswell.com/page/YJ6W2LRPJV.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets We now impose a spherical constraint on the three components of d and thus obtain the notion of a spherical plithogenic set. Definition 3.14.4 (Spherical Plithogenic Set). Let U be a nonempty universe, let a be a fixed attribute, and let V be a nonempty set of attribute values. Fix a radius λ > 0 and a contradiction degree function c : V × V → [0, 1] with c(v, v) = 0, c(v1 , v2 ) = c(v2 , v1 ) for all v, v1 , v2 ∈ V. A spherical plithogenic set of radius λ on (U, a, V ) is a plithogenic set Asph = (U, a, V, dA , c) such that • the degree of appurtenance function dA : U × V −→ [0, λ]3 is given by dA (x, v) =  TA (x, v), IA (x, v), FA (x, v) , • and for every (x, v) ∈ U × V we have the spherical constraint 0 ≤ TA (x, v)2 + IA (x, v)2 + FA (x, v)2 ≤ λ2 . (3.6)  We call TA (x, v), IA (x, v), FA (x, v) the spherical plithogenic neutrosophic triple of x with respect to the attribute value v. Remark 3.14.5. Note that if |V | = 1 and c ≡ 0, a spherical plithogenic set reduces to a single spherical triple assigned to each element x ∈ U , which is precisely the situation of spherical fuzzy and spherical neutrosophic sets, once λ is chosen appropriately. The specialization relationships between Spherical Plithogenic Sets and previous spherical models are summarized in Table 3.22. Table 3.22: Spherical fuzzy–type models as special cases of a Spherical Plithogenic Set (single attribute a∗ , pCF ≡ 0). Model pdf (x, a∗ ) Spherical constraint Spherical Fuzzy Set [367, 702] Spherical Neutrosophic Set [703–705] Spherical Picture Fuzzy Set [706, 707] Spherical Hesitant Fuzzy Set [701] (µ(x), η(x), ν(x)) µ(x)2 + η(x)2 + ν(x)2 ≤ 1 (T (x), I(x), F (x)) T (x)2 + I(x)2 + F (x)2 ≤ 1 (ξ(x), ψ(x), ρ(x), γ(x)) ξ(x)2 + ψ(x)2 + ρ(x)2 + γ(x)2 ≤ 1  for all (µ, η, ν) ∈ Hµ (x) × Hη (x) × Hν (x): µ2 + η 2 + ν 2 ≤ 1 Hµ (x), Hη (x), Hν (x) A concrete example of this concept is provided below. # Page. 79 ![Page Image](https://bcdn.docswell.com/page/GJ5M218PJ4.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.14.6 (Spherical plithogenic air–quality risk assessment). Let U be the set of city districts and let a be the attribute “air–quality health risk.” Take the value set V = {low, medium, high}. We consider a spherical plithogenic set of radius λ = 1, Asph = (U, a, V, dA , c), with contradiction degrees c(low, low) = c(medium, medium) = c(high, high) = 0, c(low, medium) = c(medium, low) = 0.5, c(medium, high) = c(high, medium) = 0.6, c(low, high) = c(high, low) = 1. For a specific district x0 ∈ U , suppose the environmental authority assigns the following spherical plithogenic neutrosophic triples:  dA (x0 , low) = TA , IA , FA (x0 , low) = (0.10, 0.20, 0.95),  dA (x0 , medium) = TA , IA , FA (x0 , medium) = (0.50, 0.30, 0.40),  dA (x0 , high) = TA , IA , FA (x0 , high) = (0.75, 0.40, 0.10). Each triple satisfies the spherical constraint T 2 + I 2 + F 2 ≤ 1: 0.102 + 0.202 + 0.952 = 0.01 + 0.04 + 0.9025 = 0.9525 ≤ 1, 0.502 + 0.302 + 0.402 = 0.25 + 0.09 + 0.16 = 0.50 ≤ 1, 0.752 + 0.402 + 0.102 = 0.5625 + 0.16 + 0.01 = 0.7325 ≤ 1. Here TA (x0 , v) is the degree to which x0 truly has risk level v, IA (x0 , v) is the degree of indeterminacy (due to sensor noise, seasonal variation, or incomplete data), and FA (x0 , v) is the degree to which x0 fails to have risk level v. The contradiction function c expresses that “low” and “high” risk are maximally contradictory, while “medium” is partially compatible with both. When aggregating expert opinions or monitoring data, the plithogenic machinery weights the triples using c, so that evidence supporting “high” is discounted when combined with evidence supporting “low,” and vice versa. Thus Asph provides a spherical, contradiction–aware representation of uncertain air–quality risk for all districts x ∈ U . Example 3.14.7 (Spherical plithogenic evaluation of diabetes severity). Diabetes severity quantifies how advanced the disease is, combining glycemic control, complications, comorbidities, treatment intensity, and long-term risk of outcomes (cf. [708]). Let U be the set of patients in a clinic and let a be the attribute “diabetes severity.” Consider the set V = {mild, moderate, severe}, and a spherical plithogenic set of radius λ = 1, B sph = (U, a, V, dB , c), # Page. 80 ![Page Image](https://bcdn.docswell.com/page/9E2941KZ7R.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets with contradiction degrees c(mild, mild) = c(moderate, moderate) = c(severe, severe) = 0, c(mild, moderate) = c(moderate, mild) = 0.3, c(moderate, severe) = c(severe, moderate) = 0.4, c(mild, severe) = c(severe, mild) = 1. For a particular patient y ∈ U , suppose laboratory tests and physician assessments yield:  dB (y, mild) = TB , IB , FB (y, mild) = (0.25, 0.40, 0.80),  dB (y, moderate) = TB , IB , FB (y, moderate) = (0.60, 0.35, 0.30),  dB (y, severe) = TB , IB , FB (y, severe) = (0.75, 0.30, 0.15). Again, each triple lies in the unit sphere: 0.252 + 0.402 + 0.802 = 0.0625 + 0.16 + 0.64 = 0.8625 ≤ 1, 0.602 + 0.352 + 0.302 = 0.36 + 0.1225 + 0.09 = 0.5725 ≤ 1, 0.752 + 0.302 + 0.152 = 0.5625 + 0.09 + 0.0225 = 0.675 ≤ 1. Here TB (y, v) represents the degree to which the current medical evidence supports severity level v for patient y, IB (y, v) captures uncertainty (e.g., missing tests, conflicting indicators), and FB (y, v) encodes the degree of rejection of severity level v. The contradiction function c encodes that “mild” and “severe” are highly incompatible, whereas “moderate” is closer to each. In multi–expert decision support, spherical plithogenic aggregation uses c to reconcile conflicting evaluations (e.g., one expert supporting “mild” and another supporting “severe”) while preserving the spherical neutrosophic constraint for each (y, v) ∈ U × V . This yields a medically interpretable, contradiction–aware assessment of diabetes severity for every patient. We now formalize and prove the generalization property. Theorem 3.14.8. Let U be a nonempty universe. (i) ( Spherical fuzzy case) Consider the class SFS(U ) of all spherical fuzzy sets on U . Then SFS(U ) is in bijection with the subclass SPS 1 (U ) of spherical plithogenic sets of radius λ = 1 such that V = {v0 } is a singleton and c(v0 , v0 ) = 0. (ii) ( Spherical neutrosophic case) Consider the class SN S(U ) of all spherical neutrosophic sets on U . Then SN S(U ) is in bijection with the subclass SPS √3 (U ) of spherical plithogenic sets of √ radius λ = 3 satisfying V = {v0 } and c(v0 , v0 ) = 0. Consequently, the notion of spherical plithogenic set strictly generalizes both spherical fuzzy sets and spherical neutrosophic sets. # Page. 81 ![Page Image](https://bcdn.docswell.com/page/D7Y4MZNNEM.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Proof. We prove (i) and (ii) separately. Proof of (i). Fix λ = 1, a singleton value set V = {v0 }, and let c(v0 , v0 ) = 0. Step 1: From spherical fuzzy to spherical plithogenic. Let A ∈ SFS(U ) be a spherical fuzzy set. By definition we have three maps TA , IA , FA : U → [0, 1] such that for every x ∈ U , 0 ≤ TA (x)2 + IA (x)2 + FA (x)2 ≤ 1. Define a degree of appurtenance function dA : U × V → [0, 1]3 by dA (x, v0 ) :=  TA (x), IA (x), FA (x) . (3.7) Since TA (x), IA (x), FA (x) ∈ [0, 1] for all x, the codomain condition dA (x, v0 ) ∈ [0, 1]3 is satisfied. Moreover, the spherical inequality TA (x)2 + IA (x)2 + FA (x)2 ≤ 1 is precisely (3.6) with λ = 1 and v = v0 . Therefore, Asph := (U, a, V, dA , c) is a spherical plithogenic set of radius 1 with singleton value set and trivial contradiction function. This defines a map Φ : SFS(U ) −→ SPS 1 (U ), A 7−→ Asph . Step 2: From spherical plithogenic (radius 1, singleton V ) to spherical fuzzy. Conversely, let B sph = (U, a, V, dB , c) ∈ SPS 1 (U ) with V = {v0 } and c(v0 , v0 ) = 0. By Definition 3.14.4, for each x ∈ U there exist real numbers TB (x, v0 ), IB (x, v0 ), FB (x, v0 ) ∈ [0, 1] such that dB (x, v0 ) =  TB (x, v0 ), IB (x, v0 ), FB (x, v0 ) and 0 ≤ TB (x, v0 )2 + IB (x, v0 )2 + FB (x, v0 )2 ≤ 1 for every x ∈ U . Define TB∗ (x) := TB (x, v0 ), ∗ IB (x) := IB (x, v0 ), FB∗ (x) := FB (x, v0 ). ∗ Then TB∗ , IB , FB∗ : U → [0, 1] and for each x ∈ U , ∗ 0 ≤ TB∗ (x)2 + IB (x)2 + FB∗ (x)2 = TB (x, v0 )2 + IB (x, v0 )2 + FB (x, v0 )2 ≤ 1. Therefore  ∗ B := (x, TB∗ (x), IB (x), FB∗ (x)) : x ∈ U is a spherical fuzzy set, i.e. B ∈ SFS(U ). This defines a map Ψ : SPS 1 (U ) −→ SFS(U ), B sph 7−→ B. # Page. 82 ![Page Image](https://bcdn.docswell.com/page/VENYW3XVJ8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Step 3: Φ and Ψ are inverse to each other. Take A ∈ SFS(U ) and compute Ψ(Φ(A)). By construction, Φ(A) is the spherical plithogenic set Asph with  dA (x, v0 ) = TA (x), IA (x), FA (x) . Applying Ψ recovers the spherical fuzzy triple TA∗ (x) = TA (x), ∗ IA (x) = IA (x), FA∗ (x) = FA (x) for all x ∈ U . Hence Ψ(Φ(A)) = A. Conversely, take B sph ∈ SPS 1 (U ) and compute Φ(Ψ(B sph )). The spherical fuzzy set Ψ(B sph ) has components ∗ TB∗ (x) = TB (x, v0 ), IB (x) = IB (x, v0 ), FB∗ (x) = FB (x, v0 ). e sph with degree function Then Φ(Ψ(B sph )) is the spherical plithogenic set B   ∗ d˜B (x, v0 ) = TB∗ (x), IB (x), FB∗ (x) = TB (x, v0 ), IB (x, v0 ), FB (x, v0 ) = dB (x, v0 ). Thus d˜B = dB and hence Φ(Ψ(B sph )) = B sph . Therefore Φ and Ψ are mutually inverse bijections between SFS(U ) and SPS 1 (U ), proving (i). Proof of (ii). Now fix λ = √ 3, V = {v0 }, and c(v0 , v0 ) = 0. Step 1: From spherical neutrosophic to spherical plithogenic. Let A ∈ SN S(U ) be a spherical neutrosophic set. Then TA , IA , FA : U → [0, 3] and for every x ∈ U , 0 ≤ TA (x)2 + IA (x)2 + FA (x)2 ≤ 3. Define (3.8) √ dA : U × V → [0, 3]3 by dA (x, v0 ) :=  TA (x), IA (x), FA (x) . First we check the codomain condition. Fix x ∈ U . From (3.8) we have TA (x)2 ≤ TA (x)2 + IA (x)2 + FA (x)2 ≤ 3, hence TA (x)2 ≤ 3 and therefore 0 ≤ TA (x) ≤ √ 3. The same argument applies to IA (x) and FA (x), so 0 ≤ TA (x), IA (x), FA (x) ≤ √ for every x ∈ U . Thus dA (x, v0 ) ∈ [0, 3]3 . Next, the spherical constraint (3.6) with λ = √ √ 3 3 becomes √ 0 ≤ TA (x)2 + IA (x)2 + FA (x)2 ≤ ( 3)2 = 3, which holds by (3.8). Therefore Asph := (U, a, V, dA , c) (3.9) # Page. 83 ![Page Image](https://bcdn.docswell.com/page/Y79PX98VE3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets √ is a spherical plithogenic set of radius 3 with singleton value set and trivial contradiction function. This defines a map Φ0 : SN S(U ) −→ SPS √3 (U ), A 7−→ Asph . √ Step 2: From spherical plithogenic (radius 3, singleton V ) to spherical neutrosophic. Conversely, let B sph = (U, a, V, dB , c) ∈ SPS √3 (U ) with V = {v0 } and c(v0 , v0 ) = 0. Again, by Definition 3.14.4, we have for each x ∈ U √  dB (x, v0 ) = TB (x, v0 ), IB (x, v0 ), FB (x, v0 ) ∈ [0, 3]3 and Define √ 0 ≤ TB (x, v0 )2 + IB (x, v0 )2 + FB (x, v0 )2 ≤ ( 3)2 = 3. TB† (x) := TB (x, v0 ), Then and for every x ∈ U , Thus † IB (x) := IB (x, v0 ), FB† (x) := FB (x, v0 ). √ † TB† , IB , FB† : U → [0, 3] ⊆ [0, 3], † 0 ≤ TB† (x)2 + IB (x)2 + FB† (x)2 ≤ 3.  † B := (x, TB† (x), IB (x), FB† (x)) : x ∈ U is a spherical neutrosophic set, i.e. B ∈ SN S(U ). This gives a map Ψ0 : SPS √3 (U ) −→ SN S(U ), B sph 7−→ B. Step 3: Φ0 and Ψ0 are inverse to each other. The verification is identical in structure to the spherical fuzzy case. For any A ∈ SN S(U ), the composite Ψ0 (Φ0 (A)) recovers the original triple (TA (x), IA (x), FA (x)) for each x ∈ U , so Ψ0 (Φ0 (A)) = A. Conversely, for any B sph ∈ SPS √3 (U ), the composite Φ0 (Ψ0 (B sph )) preserves dB pointwise (via the same coordinate equalities as in the proof of (i)), hence Φ0 (Ψ0 (B sph )) = B sph . Therefore Φ0 and Ψ0 are mutually inverse bijections between SN S(U ) and SPS √3 (U ), establishing (ii). Combining (i) and (ii), we conclude that: • every spherical fuzzy set can be realized as a spherical plithogenic set of radius 1 with a single attribute value and zero contradiction; √ • every spherical neutrosophic set can be realized as a spherical plithogenic set of radius 3 with a single attribute value and zero contradiction; • spherical plithogenic sets in general allow multiple attribute values and a nontrivial contradiction degree c, and hence strictly extend both frameworks. This completes the proof. # Page. 84 ![Page Image](https://bcdn.docswell.com/page/G78D29847D.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.15 T-Spherical Plithogenic Set A T -Spherical Plithogenic Set models uncertainty by assigning triples constrained by a t-spherical radius, incorporating attribute-based contradictions to capture richer multi-valued information. Definition 3.15.1 (T-Spherical Plithogenic Set). Let U be a nonempty universe, let a be a fixed attribute, and let V be a nonempty set of attribute values. Fix a radius λ > 0, a real parameter t ≥ 1, and a contradiction degree function c : V × V → [0, 1] with c(v, v) = 0, c(v1 , v2 ) = c(v2 , v1 ) for all v, v1 , v2 ∈ V. A T-spherical plithogenic set of order t and radius λ on (U, a, V ) is a plithogenic set ATSph = (U, a, V, dA , c) such that • the degree of appurtenance function dA : U × V −→ [0, λ]3 is given by dA (x, v) =  TA (x, v), IA (x, v), FA (x, v) , • and for every (x, v) ∈ U × V we have the t–spherical constraint 0 ≤ TA (x, v)t + IA (x, v)t + FA (x, v)t ≤ λt . (3.10)  The triple TA (x, v), IA (x, v), FA (x, v) is called the t–spherical plithogenic neutrosophic triple of x with respect to the attribute value v. Table 3.23 provides a brief overview showing that various T-spherical fuzzy–type models arise as special cases of a T-Spherical Plithogenic Set. Each model is obtained by fixing a single attribute value, setting the contradiction function to zero, and applying the t-spherical constraint to its corresponding membership structure. Table 3.23: T-spherical fuzzy–type models as special cases of a T-Spherical Plithogenic Set (single attribute a∗ , pCF ≡ 0, order t ≥ 1, radius 1). Model pdf (x, a∗ ) t-spherical constraint T-Spherical Fuzzy Set [709, 710] T-Spherical Neutrosophic Set [711, 712] T-Spherical Picture Fuzzy Set T-Spherical Hesitant Fuzzy Set [701, 713] (µ(x), η(x), ν(x)) µ(x)t + η(x)t + ν(x)t ≤ 1t (T (x), I(x), F (x)) T (x)t + I(x)t + F (x)t ≤ 1t (ξ(x), ψ(x), ρ(x), γ(x)) ξ(x)t + ψ(x)t + ρ(x)t + γ(x)t ≤ 1t  for all (µ, η, ν) ∈ Hµ (x) × Hη (x) × Hν (x): µt + η t + ν t ≤ 1t Hµ (x), Hη (x), Hν (x) A concrete example of this concept is provided below. # Page. 85 ![Page Image](https://bcdn.docswell.com/page/L7LM2WRVJR.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.15.2 (A simple T-Spherical Plithogenic Set). Let U = {x1 , x2 } be a set of two alternatives (e.g. two candidate suppliers), and let V = {v1 , v2 } be a set of two attribute values (e.g. v1 = “cost-efficient”, v2 = “high quality”). Fix λ = 1 and t = 3. Define the contradiction function c : V × V → [0, 1] by c(v1 , v1 ) = c(v2 , v2 ) = 0, c(v1 , v2 ) = c(v2 , v1 ) = 0.4. Define dA : U × V → [0, 1]3 by: dA (x1 , v1 ) = (0.8, 0.3, 0.2), dA (x1 , v2 ) = (0.5, 0.5, 0.2), dA (x2 , v1 ) = (0.4, 0.7, 0.2), dA (x2 , v2 ) = (0.6, 0.2, 0.6). We check the constraint (3.10) for t = 3. For (x1 , v1 ): 0.83 + 0.33 + 0.23 = 0.512 + 0.027 + 0.008 = 0.547 ≤ 13 . For (x1 , v2 ): 0.53 + 0.53 + 0.23 = 0.125 + 0.125 + 0.008 = 0.258 ≤ 13 . For (x2 , v1 ): 0.43 + 0.73 + 0.23 = 0.064 + 0.343 + 0.008 = 0.415 ≤ 13 . For (x2 , v2 ): 0.63 + 0.23 + 0.63 = 0.216 + 0.008 + 0.216 = 0.440 ≤ 13 . Thus all triples satisfy TA (x, v)3 + IA (x, v)3 + FA (x, v)3 ≤ 13 , so ATSph = (U, a, V, dA , c) is a Tspherical plithogenic set of order t = 3 and radius λ = 1. In this framework, the following theorems hold. Theorem 3.15.3 (T-Spherical Plithogenic Set generalizes Spherical Plithogenic Set). Every spherical plithogenic set of Definition 3.14.4 is a T-spherical plithogenic set of order t = 2 in the sense of Definition 3.15.1. Hence the class of T-spherical plithogenic sets strictly generalizes the class of spherical plithogenic sets. # Page. 86 ![Page Image](https://bcdn.docswell.com/page/4EMY89RNEW.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Proof. Let Asph = (U, a, V, dA , c) be a spherical plithogenic set of radius λ as in Definition 3.14.4. By definition we have  dA (x, v) = TA (x, v), IA (x, v), FA (x, v) ∈ [0, λ]3 and, for all (x, v) ∈ U × V , (3.11) 0 ≤ TA (x, v)2 + IA (x, v)2 + FA (x, v)2 ≤ λ2 . Now fix t = 2. Compare (3.11) with the t–spherical constraint (3.10) in Definition 3.15.1: 0 ≤ TA (x, v)t + IA (x, v)t + FA (x, v)t ≤ λt . For t = 2 these two inequalities coincide exactly, and the remaining data (U, a, V, dA , c) are unchanged. Therefore Asph satisfies all requirements of a T-spherical plithogenic set of order t = 2 and radius λ. Consequently, the mapping Φ : {spherical plithogenic sets} −→ {T-spherical plithogenic sets}, Φ(Asph ) = Asph with t = 2 embeds the class of spherical plithogenic sets into the class of T-spherical plithogenic sets. Since for t > 2 there exist T-spherical plithogenic sets which do not satisfy the quadratic constraint of Definition 3.14.4, the inclusion is proper, so T-spherical plithogenic sets strictly generalize spherical plithogenic sets. 3.16 Plithogenic Rough Set A plithogenic rough set approximates a subset using plithogenic relations combining membership and contradiction, generalizing fuzzy and neutrosophic rough models [714]. Definition 3.16.1 (Plithogenic rough approximation space). [714] Let U 6= ∅ be a universe and let R be a plithogenic relation on U . A plithogenic relation is specified by pdfR : U × U −→ [0, 1]s , pCFR : U × U −→ [0, 1]t , where pdfR (x, y) is the (vector) plithogenic degree of appurtenance of y to x, and pCFR (x, y) is the (vector) degree of contradiction between the attribute values of x and y. Assume there is a fixed aggregation Φ : [0, 1]s × [0, 1]t −→ [0, 1] that is monotone in each argument and satisfies Φ(a, 0) = Φ(0, b) = 0. For convenience write  R̃(x, y) := Φ pdfR (x, y), pCFR (x, y) ∈ [0, 1]. Then (U, R) is called a plithogenic rough approximation space. Definition 3.16.2 (Plithogenic lower approximation). [714] Let A ⊆ U and let (U, R) be as above. The plithogenic lower approximation of A with respect to R is the fuzzy set  P LR (A) : U → [0, 1], P LR (A)(x) := inf max 1 − R̃(x, y), 1 − R̃(y, x) , x ∈ U. y∈A In the scalar case s = t = 1 and Φ(a, b) = max{a, b}, this becomes  P LR (A)(x) = inf max 1 − pdfR (x, y), 1 − pCFR (x, y) , y∈A x ∈ U. # Page. 87 ![Page Image](https://bcdn.docswell.com/page/PER95GR4J9.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.16.3 (Plithogenic upper approximation). [714] Under the same assumptions, the plithogenic upper approximation of A with respect to R is the fuzzy set  P LR (A) : U → [0, 1], P LR (A)(x) := sup min R̃(x, y), 1 − R̃(y, x) , x ∈ U. y∈U In the scalar case s = t = 1 and Φ(a, b) = max{a, b}, this reduces to  P LR (A)(x) = sup min pdfR (x, y), 1 − pCFR (x, y) , x ∈ U. y∈U Definition 3.16.4 (Plithogenic Rough Set). Let A ⊆ U . The plithogenic rough set of A (with respect to R) is the ordered pair  P LR (A) := P LR (A), P LR (A) . Table 3.24 presents the relationships between plithogenic rough sets and their related concepts. Table 3.24: Examples of rough-type models that are special cases of the plithogenic rough set P LR (A) = (P LR (A), P LR (A)) by choosing the DAF dimension s and (optionally) the DCF dimension t. Target rough model s t Generalization by plithogenic rough set Fuzzy Rough Set [715–717] Membership type single fuzzy degree µ(x) ∈ [0, 1] 1 0 Intuitionistic Fuzzy Rough Set [718–720] intuitionistic pair (µ, ν) 2 0 Vague Rough Set [721–723] Neutrosophic Rough Set [724–727] Vague pair (µ, ν) 2 0 neutrosophic triple (T, I, F ) 3 0 Picture Fuzzy Rough Set [728–730] Hesitant Fuzzy Rough Set [731–733] Spherical Fuzzy Rough Set [734–736] Quadripartitioned Neutrosophic Rough Set (cf. [737]) Pentapartitioned Neutrosophic Rough Set picture fuzzy triple Hesitant fuzzy triple Spherical fuzzy triple 4-part neutrosophic/rough components 5-part neutrosophic/rough components 3 0 3 0 3 0 4 0 5 0 obtained by taking pdf (x, a) ∈ [0, 1] and pCF ≡ 0; lower/upper are fuzzygraded approximations recovered by letting pdf (x, a) ∈ [0, 1]2 and pCF ≡ 0; lower/upper use both membership and nonmembership recovered by letting pdf (x, a) ∈ [0, 1]2 and pCF ≡ 0; recovered by letting pdf (x, a) ∈ [0, 1]3 and pCF ≡ 0; lower/upper defined componentwise on (T, I, F ) recovered by letting pdf (x, a) ∈ [0, 1]3 and pCF ≡ 0; recovered by letting pdf (x, a) ∈ [0, 1]3 and pCF ≡ 0; recovered by letting pdf (x, a) ∈ [0, 1]3 and pCF ≡ 0; obtained by pdf (x, a) ∈ [0, 1]4 ; plithogenic framework stores four uncertainty channels per element obtained by pdf (x, a) ∈ [0, 1]5 ; plithogenic rough set directly supports higher partitioned rough information A concrete example of this concept is provided below. Example 3.16.5 (Hospital triage: “needs ICU” under plithogenic clinical similarity/contradiction). Let the universe be patients U = {p1 , p2 , p3 }. Target concept A = {p2 } (clinically judged “needs ICU”). For a plithogenic relation R, take scalar degrees s = t = 1 and adopt the aggregation   R̃(x, y) := Φ pdfR (x, y), pCFR (x, y) = pdfR (x, y) · 1 − pCFR (x, y) , so that high clinical similarity and low contradiction increase R̃ ∈ [0, 1]. Assume (diagonal values omitted): pdfR (x, y) x = p1 p2 p3 y = p1 − 0.70 0.50 p2 0.80 − 0.60 p3 0.30 0.40 − # Page. 88 ![Page Image](https://bcdn.docswell.com/page/P7XQKXRZEX.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets pCFR (x, y) x = p1 p2 p3 y = p1 − 0.30 0.50 p2 0.20 − 0.40 p3 0.60 0.40 − Then (again off–diagonal shown)  R̃(x, y) = pdfR (x, y) 1 − pCFR (x, y) : R̃(x, y) x = p1 p2 p3 y = p1 p2 p3 − 0.80 × 0.80 = 0.64 0.30 × 0.40 = 0.12 0.70 × 0.70 = 0.49 − 0.40 × 0.60 = 0.24 0.50 × 0.50 = 0.25 0.60 × 0.60 = 0.36 − Plithogenic lower/upper approximations (finite U ⇒ inf = min, sup = max):  P LR (A)(x) = min max 1 − R̃(x, y), 1 − R̃(y, x) , y∈A  P LR (A)(x) = max min R̃(x, y), 1 − R̃(y, x) . y∈U For x = p1 (unlabeled):  P LR (A)(p1 ) = max 1 − R̃(p1 , p2 ), 1 − R̃(p2 , p1 ) = max(1 − 0.64, 1 − 0.49) = max(0.36, 0.51) = 0.51.  P LR (A)(p1 ) = max min(R̃(p1 , p1 ), 1 − R̃(p1 , p1 )), min(R̃(p1 , p2 ), 1 − R̃(p2 , p1 )), min(R̃(p1 , p3 ), 1 − R̃(p3 , p1 )) . Taking R̃(p1 , p1 ) = 1 (self–consistency), the three mins are min(1, 0) = 0, min(0.64, 1−0.49) = min(0.64, 0.51) = 0.51, min(0.12, 1−0.25) = min(0.12, 0.75) = 0.12, so P LR (A)(p1 ) = max(0, 0.51, 0.12) = 0.51. Hence p1 lies on the boundary: P L = 0.51, P L = 0.51. For x = p3 (unlabeled): P LR (A)(p3 ) = max(1 − R̃(p3 , p2 ), 1 − R̃(p2 , p3 )) = max(1 − 0.36, 1 − 0.24) = max(0.64, 0.76) = 0.76,  P LR (A)(p3 ) = max min(1, 0), min(0.36, 1 − 0.49) = min(0.36, 0.51) = 0.36, min(0.25, 1 − 0.25) = 0.25 = 0.36. Thus p3 is weakly covered in the upper region (0.36) with a comparatively large lower penalty (0.76), reflecting moderate similarity to the ICU case but notable contradictions. Example 3.16.6 (Credit risk screening: “high–risk borrowers”). Universe U = {a, b, c} (loan applicants); target A = {b} (“high–risk” profile from historical labels). Use the same scalar model with R̃(x, y) = pdfR (x, y) 1 − pCFR (x, y) . Assume pdfR x=a b c y=a b c − 0.65 0.30 0.55 − 0.35 0.25 0.50 − pCFR x=a b c y=a b − 0.25 0.40 − 0.30 0.45 c 0.50 0.35 − Hence R̃(a, b) = 0.65(1 − 0.25) = 0.4875, R̃(b, a) = 0.55(1 − 0.40) = 0.33, R̃(c, b) = 0.50(1 − 0.45) = 0.275, R̃(b, c) = 0.35(1 − 0.35) = 0.2275, # Page. 89 ![Page Image](https://bcdn.docswell.com/page/37K95WRV7D.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets and R̃(a, c) = 0.30(1 − 0.50) = 0.15, R̃(c, a) = 0.25(1 − 0.30) = 0.175. Lower/upper for x = a: P LR (A)(a) = max(1 − R̃(a, b), 1 − R̃(b, a)) = max(1 − 0.4875, 1 − 0.33) = max(0.5125, 0.67) = 0.67,  P LR (A)(a) = max min(1, 0), min(0.4875, 1 − 0.33) = min(0.4875, 0.67) = 0.4875, min(0.15, 1 − 0.175) = 0.15 = 0.4875. Applicant a sits close to the boundary (upper ≈ 0.49) but with a sizable lower penalty (0.67), i.e., some similarity to high–risk patterns yet notable contradictions. For x = c: P LR (A)(c) = max(1−R̃(c, b), 1−R̃(b, c)) = max(1−0.275, 1−0.2275) = max(0.725, 0.7725) = 0.7725,  P LR (A)(c) = max min(1, 0), min(0.275, 1 − 0.33) = min(0.275, 0.67) = 0.275, min(0.175, 1 − 0.15) = 0.175 = 0.275. Thus c is loosely covered by the upper region (0.275) and far from the lower region, indicating predominantly non–high–risk behavior with mild resemblance to b. Example 3.16.7 (Manufacturing quality control: “defective batches”). Manufacturing quality control monitors production processes, inspects products, and corrects defects to ensure consistent standards, safety, compliance, and customer satisfaction (cf. [738]). Universe U = {B1 , B2 , B3 , B4 } (batches). Target A = {B3 , B4 } (confirmed defective). Let R̃ = pdf · (1 − pCF ) as before, with s = t = 1. Suppose the (off–diagonal) entries summarizing similarity in fault signatures (PDF) and contradictions due to differing process settings (DCF) are pdfR B1 B2 B3 B4 B1 B2 B3 B4 − 0.55 0.70 0.40 0.60 − 0.45 0.35 0.50 0.40 − 0.80 0.45 0.30 0.75 − pCFR B1 B2 B3 B4 B1 B2 B3 B4 − 0.30 0.20 0.40 0.35 − 0.25 0.30 0.25 0.35 − 0.15 0.40 0.30 0.10 − Compute the needed R̃ values: R̃(B1 , B3 ) = 0.70(1 − 0.20) = 0.56, R̃(B3 , B1 ) = 0.50(1 − 0.25) = 0.375, R̃(B1 , B4 ) = 0.40(1 − 0.40) = 0.24, R̃(B4 , B1 ) = 0.45(1 − 0.40) = 0.27. Lower/upper for x = B1 w.r.t. A = {B3 , B4 }: P LR (A)(B1 ) = min y∈{B3 ,B4 }  max 1 − R̃(B1 , y), 1 − R̃(y, B1 ) . For y = B3 : max(1 − 0.56, 1 − 0.375) = max(0.44, 0.625) = 0.625. For y = B4 : max(1 − 0.24, 1 − 0.27) = max(0.76, 0.73) = 0.76. Thus P LR (A)(B1 ) = min{0.625, 0.76} = 0.625.  P LR (A)(B1 ) = max min R̃(B1 , y), 1 − R̃(y, B1 ) . y∈U Evaluate four mins (incl. self): min(1, 0) = 0, min(0.55(1 − 0.30) = 0.385, 1 − 0.60(1 − 0.35) = 1 − 0.39 = 0.61) = 0.385, min(0.56, 1 − 0.375 = 0.625) = 0.56, min(0.24, 1 − 0.27 = 0.73) = 0.24. Therefore P LR (A)(B1 ) = max{0, 0.385, 0.56, 0.24} = 0.56. Interpretation: B1 is moderately similar to defective batches (upper 0.56) but fails certainty (lower 0.625), i.e., flagged for re–inspection. # Page. 90 ![Page Image](https://bcdn.docswell.com/page/LJ3WK1GQJ5.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets For x = B2 (quick check against A): R̃(B2 , B3 ) = 0.45(1 − 0.25) = 0.3375, R̃(B3 , B2 ) = 0.40(1 − 0.35) = 0.26, R̃(B2 , B4 ) = 0.35(1 − 0.30) = 0.245, R̃(B4 , B2 ) = 0.30(1 − 0.30) = 0.21. P LR (A)(B2 ) = min{max(1 − 0.3375, 1 − 0.26), max(1 − 0.245, 1 − 0.21)} = min{max(0.6625, 0.74), max(0.755, 0.79)} = min{0.74, 0.79} = 0.74, P LR (A)(B2 ) ≥ max{min(0.3375, 1 − 0.26) = 0.3375, min(0.245, 1 − 0.21) = 0.245} = 0.3375, hence B2 is weakly in the upper region, suggesting a lower likelihood of defect yet non–negligible resemblance to faulty signatures. 3.17 Plithogenic soft rough set A plithogenic soft rough set combines parameterized soft memberships with plithogenic rough approximations, modeling contradiction-aware, granular uncertainty in decision-making contexts (cf. [739]). In this section we couple the plithogenic rough approximation machinery with plithogenic soft information. The resulting structure will be called a Plithogenic soft rough set; it will be shown to specialize both to a plithogenic rough set and to a plithogenic soft set. Definition 3.17.1 (Plithogenic soft approximation space). Let U 6= ∅ be a universe and let E be a nonempty set of parameters. A plithogenic soft approximation space is a tuple S := (U, E, µS , pCFE , {Re }e∈E , Φ, Ψ), where: • µS : U × E → [0, 1]j is a plithogenic degree of appurtenance (DAF) assigning to each pair (x, e) ∈ U × E a (possibly vector‐valued) membership µS (x, e), with j ∈ {1, 2, 3, 4, 5}; • pCFE : E × E → [0, 1]t is a degree of contradiction function between parameters, satisfying pCFE (e, e) = 0, pCFE (e, e0 ) = pCFE (e0 , e) (e, e0 ∈ E); • for each e ∈ E, Re is a plithogenic relation on U given by pdfRe : U × U → [0, 1]s , pCFRe : U × U → [0, 1]t , together with a fixed aggregation Φ : [0, 1]s × [0, 1]t −→ [0, 1] which is monotone in each argument and satisfies Φ(a, 0) = Φ(0, b) = 0 (a ∈ [0, 1]s , b ∈ [0, 1]t ). As in the plithogenic rough case, we abbreviate  R̃e (x, y) := Φ pdfRe (x, y), pCFRe (x, y) ∈ [0, 1]. • Ψ : [0, 1]j × [0, 1] → [0, 1]j is a fixed plithogenic aggregation operator which is monotone in each argument, i.e., µ1 ≤ µ2 , λ1 ≤ λ2 =⇒ Ψ(µ1 , λ1 ) ≤ Ψ(µ2 , λ2 ), and satisfies the boundary conditions Ψ(0, λ) = 0, Ψ(µ, 0) = 0 (µ ∈ [0, 1]j , λ ∈ [0, 1]). # Page. 91 ![Page Image](https://bcdn.docswell.com/page/8JDK3X1WEG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets For each e ∈ E, the pair (U, Re ) is a plithogenic rough approximation space in the sense of the previous subsection, and hence admits plithogenic lower and upper approximations. Definition 3.17.2 (Plithogenic soft rough lower and upper approximations). Let S be a plithogenic soft approximation space as above, and let A ⊆ U . For each fixed parameter e ∈ E and element x ∈ U , define the local plithogenic rough lower and upper approximations of A with respect to Re by  P LRe (A)(x) := inf max 1 − R̃e (x, y), 1 − R̃e (y, x) , y∈A  P LRe (A)(x) := sup min R̃e (x, y), 1 − R̃e (y, x) , y∈U exactly as in the plithogenic rough case, but with R replaced by Re . The plithogenic soft rough lower approximation of A is the mapping P SR(A) : U × E → [0, 1]j , defined by  P SR(A)(x, e) := Ψ µS (x, e), P LRe (A)(x) , (x, e) ∈ U × E, and the plithogenic soft rough upper approximation of A is the mapping P SR(A) : U × E → [0, 1]j , given by  P SR(A)(x, e) := Ψ µS (x, e), P LRe (A)(x) , (x, e) ∈ U × E. Thus, for each fixed e ∈ E, the functions x 7→ P SR(A)(x, e) and x 7→ P SR(A)(x, e) describe how strongly x belongs to the plithogenic rough lower/upper approximation of A under parameter e, after combining the soft membership µS (x, e) with the rough information derived from Re . Definition 3.17.3 (Plithogenic soft rough set). Let S and A be as above. The ordered pair P SRS (A) := P SR(A), P SR(A)  is called the plithogenic soft rough set of A over the plithogenic soft approximation space S. Equivalently, for each e ∈ E we may regard FL (e) : U → [0, 1]j , FL (e)(x) := P SR(A)(x, e), FU (e) : U → [0, 1]j , FU (e)(x) := P SR(A)(x, e), so that P SRS (A) naturally corresponds to the pair of plithogenic soft sets (FL , FU ) on (U, E). The following Table 3.25 presents how the plithogenic soft rough set serves as a common generalization of several existing soft rough models. A concrete example is given below. # Page. 92 ![Page Image](https://bcdn.docswell.com/page/VEPK4PQX78.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.25: Plithogenic soft rough set as a common generalization of several soft rough models. Target soft rough model Codomain FL , F U of j Plithogenic soft rough [0, 1]j set Fuzzy soft rough set [0, 1] [740, 741] Intuitionistic fuzzy soft [0, 1]2 rough set [742–744] Neutrosophic soft rough [0, 1]3 set [745–748] Hesitant fuzzy soft [0, 1]3 rough set [749–751] Picture fuzzy soft rough [0, 1]3 set [752] Spherical fuzzy soft [0, 1]3 rough set [709, 710, 753] Quadripartitioned Neu- [0, 1]4 trosophic soft rough set Pentapartitioned Neu- [0, 1]5 trosophic soft rough set Specialization of P SRS (A) j ∈ Base model with general pCF on attribute values. {1, 2, 3} j=1 Take FL , FU : U → [0, 1] and set pCF ≡ 0. j=2 Take FL , FU : U → [0, 1]2 and set pCF ≡ 0. j=3 Take FL , FU : U → [0, 1]3 and set pCF ≡ 0. j=3 j=4 Represent hesitant information in [0, 1]3 and set pCF ≡ 0. Use picture fuzzy acceptance/neutrality/rejection in [0, 1]3 with pCF ≡ 0. Use spherical fuzzy components in [0, 1]3 and set pCF ≡ 0. Take FL , FU : U → [0, 1]4 and set pCF ≡ 0. j=5 Take FL , FU : U → [0, 1]5 and set pCF ≡ 0. j=3 j=3 Example 3.17.4 (Supplier selection under two parameters). Consider a purchasing department that must decide whether to treat a supplier as “strategic partner.” Universe and parameters. U = {S1 , S2 }, E = {e1 , e2 }, where S1 = local supplier, S2 = global supplier, e1 = “on–time delivery,” e2 = “eco–compliance.” The target concept is A = {S2 }, meaning “strategic partner candidates.” Soft (plithogenic) parameter–based view. For each pair (x, e) ∈ U × E, the decision maker gives a fuzzy soft membership µS (x, e) ∈ [0, 1]: µS (x, e) S1 S2 e1 (on–time) 0.70 0.90 e2 (eco) 0.60 0.80 Plithogenic rough relations. For each parameter e ∈ E we have a plithogenic relation Re on U with scalar degree of appurtenance pdfRe and contradiction pCFRe . We work in the scalar case s = t = 1 and define  R̃e (x, y) := pdfRe (x, y) 1 − pCFRe (x, y) ∈ [0, 1], setting R̃e (x, x) = 1 by convention. For on–time delivery e1 : pdfRe1 (x, y) S1 S2 S1 − 0.75 S2 0.80 − pCFRe1 (x, y) S1 S2 S1 − 0.25 S2 0.20 − so R̃e1 (S1 , S2 ) = 0.80(1 − 0.20) = 0.64, R̃e1 (S2 , S1 ) = 0.75(1 − 0.25) = 0.5625. # Page. 93 ![Page Image](https://bcdn.docswell.com/page/27VVX2537Q.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets For eco–compliance e2 : pdfRe2 (x, y) S1 S2 S1 S2 − 0.60 0.70 − pCFRe2 (x, y) S1 S2 S1 − 0.20 S2 0.30 − thus R̃e2 (S1 , S2 ) = 0.60(1 − 0.30) = 0.42, R̃e2 (S2 , S1 ) = 0.70(1 − 0.20) = 0.56. Plithogenic rough lower/upper approximations (per parameter). For A = {S2 } and any x ∈ U , the local plithogenic rough lower and upper approximations are  P LRe (A)(x) = max 1 − R̃e (x, S2 ), 1 − R̃e (S2 , x) ,  P LRe (A)(x) = max min R̃e (x, y), 1 − R̃e (y, x) . y∈U For x = S1 and e1 :  P LRe1 (A)(S1 ) = max 1 − 0.64, 1 − 0.5625 = max(0.36, 0.4375) = 0.4375. For the upper approximation, n  o P LRe1 (A)(S1 ) = max min R̃e1 (S1 , S1 ), 1 − R̃e1 (S1 , S1 ) , min R̃e1 (S1 , S2 ), 1 − R̃e1 (S2 , S1 )   = max min(1, 0), min(0.64, 1 − 0.5625) = max 0, min(0.64, 0.4375) = max{0, 0.4375} = 0.4375. For x = S1 and e2 :  P LRe2 (A)(S1 ) = max 1 − 0.42, 1 − 0.56 = max(0.58, 0.44) = 0.58,  P LRe2 (A)(S1 ) = max min(1, 0), min(0.42, 1 − 0.56) = max{0, min(0.42, 0.44)} = 0.42. For the “prototypical” strategic supplier x = S2 we have R̃e (S2 , S2 ) = 1 and thus P LRe (A)(S2 ) = max(1 − 1, 1 − 1) = 0 for both e1 , e2 , while the upper penalties remain moderate, e.g.  P LRe1 (A)(S2 ) = max min(1, 0), min(0.5625, 1 − 0.64) = max{0, min(0.5625, 0.36)} = 0.36,  P LRe2 (A)(S2 ) = max min(1, 0), min(0.56, 1 − 0.42) = max{0, min(0.56, 0.58)} = 0.56. Plithogenic soft rough approximations. Take j = 1 and choose the scalarization Ψ(a, b) := a (1 − b), a, b ∈ [0, 1], so that the rough “penalty” b is converted into a support factor (1 − b) and combined with the soft membership a. The plithogenic soft rough lower and upper approximations of A are   P SR(A)(x, e) = Ψ µS (x, e), P LRe (A)(x) = µS (x, e) 1 − P LRe (A)(x) , # Page. 94 ![Page Image](https://bcdn.docswell.com/page/5JGLVR6Y7L.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets   P SR(A)(x, e) = Ψ µS (x, e), P LRe (A)(x) = µS (x, e) 1 − P LRe (A)(x) . For x = S1 :  P SR(A)(S1 , e1 ) = 0.70 1 − 0.4375 = 0.70 × 0.5625 = 0.39375,  P SR(A)(S1 , e1 ) = 0.70 1 − 0.4375 = 0.39375,  P SR(A)(S1 , e2 ) = 0.60 1 − 0.58 = 0.60 × 0.42 = 0.252,  P SR(A)(S1 , e2 ) = 0.60 1 − 0.42 = 0.60 × 0.58 = 0.348. For x = S2 :  P SR(A)(S2 , e1 ) = 0.90 1 − 0 = 0.90,  P SR(A)(S2 , e1 ) = 0.90 1 − 0.36 = 0.90 × 0.64 = 0.576,  P SR(A)(S2 , e2 ) = 0.80 1 − 0 = 0.80,  P SR(A)(S2 , e2 ) = 0.80 1 − 0.56 = 0.80 × 0.44 = 0.352. Interpretation. For the “strategic partner” concept A = {S2 }, the plithogenic soft rough set  P SRS (A) = P SR(A), P SR(A) yields, parameter by parameter, a pair of graded soft approximations that jointly reflect: • the soft, parameterized preferences of the purchasing team (µS on on–time delivery and eco–compliance), • the plithogenic rough neighborhood of the strategic supplier S2 under each parameter (via Re1 , Re2 , including contradiction), • and a combined assessment through Ψ. Here S2 has large lower values (0.90 for e1 , 0.80 for e2 ), meaning it is strongly inside the plithogenic soft rough lower approximation of “strategic partner.” Supplier S1 has moderate values, especially under eco–compliance, indicating a borderline candidate that may require contractual improvements or further audits before being upgraded to strategic status. Next we show that the above notion simultaneously extends the plithogenic rough set and the plithogenic soft set. Theorem 3.17.5 (Generalization of plithogenic rough set). Let (U, R) be a plithogenic rough approximation space as in Definition 2.XX (Plithogenic rough approximation space), and let A ⊆ U . Consider the plithogenic soft approximation space SR := (U, E, µS , pCFE , {Re }e∈E , Φ, Ψ) obtained as follows: • take a singleton parameter set E := {e0 }; # Page. 95 ![Page Image](https://bcdn.docswell.com/page/47QY6V46EP.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets • define the plithogenic soft membership to be constant µS (x, e0 ) := 1 ∈ [0, 1]j , x ∈ U, and the parameter contradiction to be trivial pCFE (e0 , e0 ) := 0; • set Re0 := R and use the same aggregation Φ as in the definition of R̃(x, y); • choose the plithogenic aggregation Ψ : [0, 1]j × [0, 1] → [0, 1] in the scalar case j = 1 by Ψ(a, λ) := λ, a, λ ∈ [0, 1], i.e. Ψ simply projects to the second argument. Then, for every x ∈ U , P SR(A)(x, e0 ) = P LR (A)(x), P SR(A)(x, e0 ) = P LR (A)(x), so that P SRSR (A) coincides with the plithogenic rough set P LR (A) = (P LR (A), P LR (A)). Proof. Fix x ∈ U . Since E = {e0 }, all expressions are evaluated at e0 . By definition of P SR(A) we have  P SR(A)(x, e0 ) = Ψ µS (x, e0 ), P LRe0 (A)(x) . In SR we imposed µS (x, e0 ) = 1, Re0 = R, hence  P SR(A)(x, e0 ) = Ψ 1, P LR (A)(x) . The chosen Ψ(a, λ) = λ yields P SR(A)(x, e0 ) = P LR (A)(x). The computation for the upper approximation is analogous:   P SR(A)(x, e0 ) = Ψ µS (x, e0 ), P LRe0 (A)(x) = Ψ 1, P LR (A)(x) = P LR (A)(x). Therefore the pair  P SRSR (A) = P SR(A)(·, e0 ), P SR(A)(·, e0 ) is exactly the plithogenic rough set P LR (A), up to the natural identification of the singleton parameter e0 . This shows that every plithogenic rough set arises as a special case of a plithogenic soft rough set. Theorem 3.17.6 (Generalization of plithogenic soft set). Let (U, E, µS , pCFE ) be a plithogenic soft set (cf. the reduction to PSS in Table 3.20), where µS : U × E → [0, 1]j , pCFE : E × E → [0, 1]t , and let A ⊆ U be arbitrary. Consider any family of plithogenic relations {Re }e∈E on U and define SS by SS := (U, E, µS , pCFE , {Re }e∈E , Φ, Ψ), where Φ is as before and Ψ is now chosen as the projection to the first argument: Ψ(µ, λ) := µ, µ ∈ [0, 1]j , λ ∈ [0, 1]. Then, for all (x, e) ∈ U × E, P SR(A)(x, e) = µS (x, e), P SR(A)(x, e) = µS (x, e), so that P SRSS (A) coincides with the original plithogenic soft set (U, E, µS , pCFE ) (viewed as a pair of identical lower/upper plithogenic soft sets). # Page. 96 ![Page Image](https://bcdn.docswell.com/page/KE4W4MQMJ1.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Proof. Fix (x, e) ∈ U × E. By definition of P SR(A),  P SR(A)(x, e) = Ψ µS (x, e), P LRe (A)(x) . With Ψ(µ, λ) = µ we obtain P SR(A)(x, e) = µS (x, e), independently of the value of P LRe (A)(x). The same argument applies to the upper approximation: P SR(A)(x, e) = Ψ µS (x, e), P LRe (A)(x)  = µS (x, e). Therefore, for each e ∈ E the maps x 7−→ P SR(A)(x, e), x 7−→ P SR(A)(x, e) both coincide with the original plithogenic soft membership x 7→ µS (x, e). Equivalently, the pair of plithogenic soft sets (FL , FU ) encoded by P SRSS (A) satisfies (e ∈ E), FL (e) = FU (e) = µS (·, e) which is exactly the given plithogenic soft set (U, E, µS , pCFE ). Hence every plithogenic soft set is realized as a specialization of a plithogenic soft rough set. 3.18 Linear Diophantine Plithogenic set A linear Diophantine plithogenic set uses multi-component memberships and integer coefficients, constrained by linear Diophantine equations, inside a contradiction-aware plithogenic attribute structure for complex uncertainty. Definition 3.18.1 (Linear Diophantine plithogenic number). Fix an integer s ≥ 1 and a constant C > 0. An s–dimensional linear Diophantine plithogenic number is a pair  Λ := µ, α ∈ [0, 1]s × [0, 1]s , where µ = (µ1 , . . . , µs ), α = (α1 , . . . , αs ), satisfy the linear Diophantine–type constraints 0 ≤ s X αi µi ≤ C, 0 ≤ i=1 s X αi ≤ C. (3.12) i=1 The quantity π(Λ) := C − s X α i µi i=1 plays the role of a (reference–dependent) hesitancy or residual degree. Definition 3.18.2 (Linear Diophantine plithogenic set). Let X be a nonempty universe and let v be a plithogenic attribute with set of attribute values Pv . Let pCF : Pv × Pv −→ [0, 1] be the plithogenic degree–of–contradiction function. Fix an integer s ≥ 1 and a constant C > 0. A linear Diophantine plithogenic set (LDPS) on X with respect to (v, Pv , pCF ) is a structure  PSLD := X, v, Pv , pCF, LDpdf, C , # Page. 97 ![Page Image](https://bcdn.docswell.com/page/L71Y48PYJG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets where LDpdf : X × Pv −→ [0, 1]s × [0, 1]s ,  (x, a) 7−→ µ(x, a), α(a) , such that, for all x ∈ X and a ∈ Pv , 0 ≤ s X αi (a) µi (x, a) ≤ C, 0 ≤ i=1 s X (3.13) αi (a) ≤ C. i=1 Here  µ(x, a) = µ1 (x, a), . . . , µs (x, a) ,  α(a) = α1 (a), . . . , αs (a) , are, respectively, plithogenic component degrees of x with respect to the value a and the associated reference parameters for a. For each (x, a) the pair Λ(x, a) := µ(x, a), α(a)  is an s–dimensional linear Diophantine plithogenic number in the sense of the previous definition. The residual degree at (x, a) is π(x, a) := C − s X αi (a) µi (x, a). i=1 Table 3.26 presents the description of the Linear Diophantine plithogenic set as a common generalization of several existing Linear Diophantine models. Table 3.26: Linear Diophantine plithogenic set as a common generalization Model Components Linear Dio- (µD , νD ) phantine fuzzy set [754–756] Linear Diophan- (TD , ID , FD ) tine neutrosophic set [757] Linear Diophan- (TD , UD , KD ) tine spherical fuzzy set [758–760] + Linear Diophan- (h− D , hD ) tine hesitant fuzzy set [761] Linear Diophan- (µ1 , . . . , µs ) tine plithogenic set Ref. params Diophantine constraint (α, β) 0 ≤ αµD + βνD ≤ 1, (α, δ, β) 0 ≤ αTD + δID + βFD ≤ 2, δ+β ≤2 (α, β, η) 0 ≤ αTD + βUD + ηKD ≤ 1, α+β+η ≤1 (α, β) + 0 ≤ αh− D + βhD ≤ 1, (α1 , . . . , αs ) 0≤ C Ps i=1 αi µi ≤ C, Plithogenic part 0 ≤ α + β ≤ 1 none 0 ≤ α + none 0 ≤ none 0 ≤ α + β ≤ 1 none 0≤ Ps i=1 αi ≤ (v, Pv , pCF ) A concrete example is provided below. Example 3.18.3 (Linear Diophantine plithogenic set for smartphone selection). Consider a household choosing among smartphones X = {x1 , x2 , x3 } = {EcoPhone, GamePhone, BudgetPhone}. Let v be the plithogenic attribute “evaluation criterion’’ with value set Pv = {price, battery, camera}. The plithogenic degree of contradiction pCF : Pv × Pv → [0, 1] is set as pCF (price, battery) = 0.30, pCF (price, camera) = 0.20, pCF (battery, camera) = 0.40, # Page. 98 ![Page Image](https://bcdn.docswell.com/page/G7WGXZ21E2.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets with pCF (a, a) = 0 and symmetry pCF (a, b) = pCF (b, a). Take s = 3 components and C = 2. For each value a ∈ Pv let α(a) = (α1 (a), α2 (a), α3 (a)) = (1, 0.5, 0.5), so that 0≤ 3 X αi (a) = 1 + 0.5 + 0.5 = 2 ≤ C. i=1 Interpret the three membership components for (x, a) as a neutrosophic-type triple µ(x, a) = (µ1 (x, a), µ2 (x, a), µ3 (x, a)) = (T, I, F ) ∈ [0, 1]3 , where T measures support that x is good on criterion a, I expresses indeterminacy, and F expresses counter-evidence. For the EcoPhone x1 define µ(x1 , price) = (0.70, 0.10, 0.20), µ(x1 , battery) = (0.90, 0.05, 0.10), µ(x1 , camera) = (0.75, 0.15, 0.20). The Linear Diophantine plithogenic constraints at (x1 , a) read 0≤ 3 X αi (a)µi (x1 , a) ≤ C, i=1 0≤ 3 X αi (a) ≤ C. i=1 Check explicitly for a = battery: 3 X αi (battery)µi (x1 , battery) = 1 · 0.90 + 0.5 · 0.05 + 0.5 · 0.10 = 0.90 + 0.025 + 0.05 = 0.975. i=1 Thus 0 ≤ 0.975 ≤ C = 2, 0 ≤ 1 + 0.5 + 0.5 = 2 ≤ 2, so the Linear Diophantine conditions are satisfied at (x1 , battery). The corresponding residual degree is 3 X π(x1 , battery) := C − αi (battery)µi (x1 , battery) = 2 − 0.975 = 1.025. i=1 Similarly, one defines µ(x2 , a) and µ(x3 , a) for a ∈ Pv (e.g. GamePhone better on camera, BudgetPhone better on price), always ensuring 0≤ 3 X αi (a)µi (x, a) ≤ 2. i=1 The resulting structure with  PSLD = X, v, Pv , pCF, LDpdf, C ,  LDpdf(x, a) = µ(x, a), α(a) , (x, a) ∈ X × Pv , is a Linear Diophantine plithogenic set. It combines: (i) Diophantine-type linear constraints on the weighted memberships, (ii) plithogenic contradictions between criteria via pCF , to model a realistic multi-criteria smartphone selection problem. # Page. 99 ![Page Image](https://bcdn.docswell.com/page/4JZL61DXE3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.19 TreePlithogenic Set A TreePlithogenic set organizes attributes in a rooted tree and aggregates their plithogenic memberships using contradiction-aware hierarchical weighting across levels [762]. Related concepts such as TreeSoft Set [319–321, 763, 764] and TreeRough Set [762, 765] are also known in the literature. Definition 3.19.1 (TreePlithogenic Set). [762] Let S be a universal set and P ⊆ S a nonempty subset. Let Tree(A) be a finite rooted attribute–tree whose nodes are attributes ai arranged in levels 1, . . . , m. For every node (attribute) ai ∈ Tree(A) let P vi be the set of admissible values of ai , and let pdfi : P × P vi −→ [0, 1]si be the (possibly multi–component) plithogenic degree of appurtenance attached to ai . Assume further a global degree of contradiction function [  [  pCF : P vi × P vi −→ [0, 1]t i i satisfying (i) pCF (a, a) = 0 for all a (reflexivity), and (ii) pCF (a, b) = pCF (b, a) for all a, b (symmetry). A TreePlithogenic Set is the tuple TPS = P, Tree(A), {P vi }ai ∈Tree(A) ,  {pdfi }ai ∈Tree(A) , pCF , which, for every subset of nodes X ⊆ Tree(A) and every element x ∈ P , collects all plithogenic memberships pdfi (x, ·) of x with respect to the attributes present in X, while modulating/weighting them by pCF according to the mutual contradiction of their value–labels. In particular, if the tree has only one level and one attribute, this reduces to the ordinary plithogenic set. Table 3.27 presents the description of a TreePlithogenic Set as a unifying model for tree–based fuzzy, intuitionistic fuzzy, and neutrosophic sets. Table 3.27: TreePlithogenic Set as a unifying model for tree–based fuzzy / intuitionistic fuzzy / neutrosophic sets. Target (tree) TreeFuzzy Set [762] Attribute/tree restriction One rooted attribute–tree; each node has a single scalar membership in [0, 1] Tree-Intuitionistic Fuzzy Set [762] (TreeVague Set) TreeNeutrosophic Set [762] (TreePicture Fuzzy, TreeHesitant Fuzzy, and Tree-Spherical Fuzzy Set) TreeQuadripartitioned Neutrosophic Set TreePentapartitioned Neutrosophic Set One rooted attribute–tree; each node stores (µ, ν) with µ + ν ≤ 1 Outcome in TPS TreePlithogenic Set with si = 1 for all nodes and pCF ≡ 0; fuzzy case recovered TreePlithogenic Set with si = 2 and pCF ≡ 0; intuitionistic tree recovered One rooted attribute–tree; each node stores (T, I, F ) ∈ [0, 1]3 TreePlithogenic Set with si = 3 and pCF ≡ 0; neutrosophic tree recovered One rooted attribute–tree; TreePlithogenic Set with si = 4 and pCF ≡ 0; neutrosophic tree recovered One rooted attribute–tree; TreePlithogenic Set with si = 5 and pCF ≡ 0; neutrosophic tree recovered A concrete example of this concept is provided below. # Page. 100 ![Page Image](https://bcdn.docswell.com/page/YE6W2L1PEV.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.19.2 (Hiring a data engineer: contradiction-aware hierarchical fit). Let the attribute-tree be root Fit with three children: Technical, Experience, Culture. Each child has two leaf attributes: Technical = {Coding, DataModeling}, Experience = {Projects, Domain}, Culture = {Communication, Teamwork}. Each node uses a fuzzy plithogenic membership (scalar in [0, 1]), with dominant value set to High at every level. Let the linguistic labels be {Low, Med, High} and the contradiction to High be pCF (High, High) = 0, pCF (Med, High) = 0.3, pCF (Low, High) = 0.8. The compatibility weight is w(` | High) = 1 − pCF (`, High). Leaf evaluations for candidate c (observed label; base membership µ): Leaf Coding DataModeling Projects Domain Communication Teamwork Label High Med High Med Med High µ 0.82 0.68 0.75 0.55 0.62 0.78 w(· | High) 1.0 0.7 1.0 0.7 0.7 1.0 Aggregate each internal node by weighted mean µ b = Pwµ w : P µTechnical = 0.82 · 1 + 0.68 · 0.7 0.82 + 0.476 1.296 = = = 0.76235. 1 + 0.7 1.7 1.7 0.75 · 1 + 0.55 · 0.7 0.75 + 0.385 1.135 = = = 0.66765. 1 + 0.7 1.7 1.7 0.62 · 0.7 + 0.78 · 1 0.434 + 0.78 1.214 µCulture = = = = 0.71412. 0.7 + 1 1.7 1.7 µExperience = Intermediate node labels (for root-level contradiction) via thresholding: High if µ ≥ 0.70, Med if 0.50 ≤ µ < 0.70, Low otherwise. Hence Technical = High, Experience = Med, Culture = High, so the root weights are w(High | High) = 1, w(Med | High) = 0.7. Root aggregation (Fit): µFit = 0.76235 · 1 + 0.66765 · 0.7 + 0.71412 · 1 0.76235 + 0.46736 + 0.71412 1.94383 = = = 0.72068. 1 + 0.7 + 1 2.7 2.7 Thus, candidate c attains a contradiction-aware hierarchical Fit score ≈ 0.721. Example 3.19.3 (Emergency triage: pneumonia severity under a clinical tree). Tree: root Severity with children Vitals, Imaging, Labs; leaves Vitals={SpO2 , RespRate}, Imaging={CXR, CT}, Labs={CRP, WBC}. Dominant label at all levels is Severe. Labels {Mild, Moderate, Severe} with contradictions pCF (Severe, Severe) = 0, pCF (Moderate, Severe) = 0.35, and weights w(` | Severe) = 1 − pCF (`, Severe). pCF (Mild, Severe) = 0.85, # Page. 101 ![Page Image](https://bcdn.docswell.com/page/GE5M214PE4.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Leaf assessments for patient p: Leaf SpO2 RespRate CXR CT CRP WBC Label Severe Moderate Moderate Mild Moderate Severe µ 0.88 0.65 0.70 0.40 0.60 0.75 w(· | Severe) 1.0 0.65 0.65 0.15 0.65 1.0 Node aggregations: µVitals = 0.88 · 1 + 0.65 · 0.65 0.88 + 0.4225 1.3025 = = = 0.78939. 1 + 0.65 1.65 1.65 0.70 · 0.65 + 0.40 · 0.15 0.455 + 0.06 0.515 = = = 0.64375. 0.65 + 0.15 0.80 0.80 0.60 · 0.65 + 0.75 · 1 0.39 + 0.75 1.14 µLabs = = = = 0.69091. 0.65 + 1 1.65 1.65 µImaging = Intermediate labels (High/Severe if µ ≥ 0.70): Vitals=Severe, Imaging=Moderate, Labs=Moderate. Root weights: w(Severe | Severe) = 1, w(Moderate | Severe) = 0.65. Root aggregation (Severity): µSeverity = 0.78939 · 1 + 0.64375 · 0.65 + 0.69091 · 0.65 1 + 0.65 + 0.65 0.78939 + 0.41844 + 0.44909 1.65692 = = 0.72040. 2.30 2.30 The patient’s contradiction-aware severity score is ≈ 0.720. = Example 3.19.4 (Supplier selection: suitability under cost–quality–delivery tree). Tree: root Suitability with children Cost, Quality, Delivery. Dominant label at all levels is Preferred. Labels {Risky, Acceptable, Preferred} with pCF (Preferred, Preferred) = 0, pCF (Acceptable, Preferred) = 0.4, pCF (Risky, Preferred) = 0.85, and w(` | Preferred) = 1 − pCF (`, Preferred) ∈ {1, 0.6, 0.15}. Leaves for supplier s: Leaf UnitPrice TotalCost DefectRate Certifications OnTime LeadTime Label Acceptable Preferred Preferred Acceptable Acceptable Risky µ 0.62 0.78 0.83 0.58 0.65 0.35 w(· | Preferred) 0.6 1.0 1.0 0.6 0.6 0.15 Node aggregations: µCost = 0.62 · 0.6 + 0.78 · 1 0.372 + 0.78 1.152 = = = 0.72. 0.6 + 1 1.6 1.6 # Page. 102 ![Page Image](https://bcdn.docswell.com/page/972941VZJR.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 0.83 · 1 + 0.58 · 0.6 0.83 + 0.348 1.178 = = = 0.73625. 1 + 0.6 1.6 1.6 0.65 · 0.6 + 0.35 · 0.15 0.39 + 0.0525 0.4425 µDelivery = = = = 0.59. 0.6 + 0.15 0.75 0.75 µQuality = Intermediate labels: Cost=Preferred, Quality=Preferred, Delivery=Acceptable. Root weights: w(Preferred | Preferred) = 1, w(Acceptable | Preferred) = 0.6. Root aggregation (Suitability): µSuitability = 0.72 · 1 + 0.73625 · 1 + 0.59 · 0.6 0.72 + 0.73625 + 0.354 1.81025 = = = 0.69625. 1 + 1 + 0.6 2.6 2.6 Hence the supplier receives a contradiction-aware suitability of ≈ 0.696, i.e., borderline “preferred” under the specified hierarchy and contradictions. 3.20 ForestPlithogenic Set A ForestPlithogenic set spans multiple attribute trees and fuses their plithogenic memberships while weighting inter-tree contradictions for consistent evaluation results [762]. Definition 3.20.1 (ForestPlithogenic Set). [762] Let { TPSt }t∈T be a family of TreePlithogenic Sets, (t) (t) where the t–th tree has attribute–tree Tree(A(t) ), value–sets {P vi }, appurtenance maps {pdfi }, (t) and (possibly) its own contradiction map pCF . Form the disjoint union (forest) of attribute–trees G  Forest {A(t) }t∈T := Tree(A(t) ). t∈T A ForestPlithogenic Set on the common universe P is the tuple FPS =  (t) (t) ] , P, Forest({A(t) }), {P vi }, {pdfi }, pCF ] extends all pCF (t) ’s to the union of value–labels coming from every tree. For any subset where pCF of forest–nodes X and any element x ∈ P , the forest–level membership of x is obtained by aggregating the plithogenic memberships provided by every tree whose nodes appear in X, under the ] . If the forest consists of exactly one tree, the above reduces to a common contradiction control pCF TreePlithogenic Set. Table 3.28 provides an explanation of the ForestPlithogenic Set as a unifying model for forest–based fuzzy, intuitionistic fuzzy, and neutrosophic sets. A concrete example of this concept is provided below. Example 3.20.2 (Smart-city site selection: forest of environmental, infrastructure, and socioeconomic trees). City site selection evaluates potential urban locations using criteria like transport, resources, risk, environment, and growth to choose optimal development (cf. [766]). Consider three TreePlithogenic Sets over parcels P : TPSEnv , TPSInfra , and TPSSocio . Each node uses a fuzzy membership in [0, 1]. Linguistic labels are {Low, Med, High} with contradiction to the dominant label High: pCF (High, High) = 0, pCF (Med, High) = 0.3, # Page. 103 ![Page Image](https://bcdn.docswell.com/page/DJY4MZ3N7M.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.28: ForestPlithogenic Set as a unifying model for forest–based fuzzy / intuitionistic fuzzy / neutrosophic sets. Target (forest) ForestFuzzy Set [762] ForestIntuitionistic Fuzzy Set [762] Forest restriction Finite disjoint family of attribute–trees; every node fuzzy in [0, 1]; no cross–tree contradiction Forest of trees; each node has (µ, ν); no cross–tree contradiction ForestNeutrosophic Set [762] Forest of trees; each node has (T, I, F ) ForestQuadripartitioned Neutrosophic Set ForestPentapartitioned Neutrosophic Set Forest of trees Forest of trees Outcome in FPS ForestPlithogenic Set with si = 1 for ] ≡0 all nodes and pCF ForestPlithogenic Set with si = 2 and ] ≡0 pCF ForestPlithogenic Set with si = 3 and ] ≡0 pCF ForestPlithogenic Set with si = 4 and ] ≡0 pCF ForestPlithogenic Set with si = 5 and ] ≡0 pCF pCF (Low, High) = 0.8, and weights w(` | High) = 1 − pCF (`, High) ∈ {1, 0.7, 0.2}. For a parcel x ∈ P , leaf assessments (label; base membership µ) are: (Env) AirQuality=(High; 0.80), GreenCover=(Med; 0.65), Noise=(High; 0.75). 0.80 · 1 + 0.65 · 0.7 + 0.75 · 1 0.80 + 0.455 + 0.75 2.005 = = = 0.74259. 1 + 0.7 + 1 2.7 2.7 µEnv (x) = (Infra) TransitAccess=(Med; 0.68), RoadConnectivity=(High; 0.77), Utilities=(Med; 0.60). µInfra (x) = 0.68 · 0.7 + 0.77 · 1 + 0.60 · 0.7 0.476 + 0.77 + 0.42 1.666 = = = 0.69417. 0.7 + 1 + 0.7 2.4 2.4 (Socio) Safety=(Med; 0.70), CommunitySupport=(High; 0.74), Rent=(Low; 0.45). µSocio (x) = 0.70 · 0.7 + 0.74 · 1 + 0.45 · 0.2 0.49 + 0.74 + 0.09 1.32 = = = 0.69474. 0.7 + 1 + 0.2 1.9 1.9 Thresholding (High if µ ≥ 0.70, Med if 0.50 ≤ µ < 0.70) gives Env=High, Infra=Med, Socio=Med. The forest–level (dominant High) aggregate is µForest (x) = = 0.74259 · 1 + 0.69417 · 0.7 + 0.69474 · 0.7 1 + 0.7 + 0.7 0.74259 + 0.48592 + 0.48632 1.71483 = = 0.71451. 2.4 2.4 Thus x attains a contradiction-aware forest score ≈ 0.715. # Page. 104 ![Page Image](https://bcdn.docswell.com/page/V7NYW3VVE8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.20.3 (Hospital readmission risk: forest of ClinicalHistory, CurrentStatus, SocialDeterminants). Hospital readmission risk estimates the probability a discharged patient returns soon, guiding care coordination, follow up planning, and resource allocation (cf. [767]). Trees: TPSCH , TPSCS , TPSSD with labels {Low, Moderate, High} and dominant High risk. Contradictions to High: pCF (High, High) = 0, pCF (Moderate, High) = 0.35, pCF (Low, High) = 0.85, so w(· | High) ∈ {1, 0.65, 0.15}. Patient p leaf evaluations (label; µ): (ClinicalHistory) Comorbidity=(High; 0.82), PriorAdmissions=(Moderate; 0.66). µCH (p) = 0.82 · 1 + 0.66 · 0.65 0.82 + 0.429 1.249 = = = 0.75697. 1 + 0.65 1.65 1.65 (CurrentStatus) BPControl=(Moderate; 0.55), Adherence=(Low; 0.40), Mobility=(Moderate; 0.60). µCS (p) = 0.55 · 0.65 + 0.40 · 0.15 + 0.60 · 0.65 0.3575 + 0.06 + 0.39 0.8075 = = = 0.55690. 0.65 + 0.15 + 0.65 1.45 1.45 (SocialDeterminants) Support=(Low; 0.35), Housing=(Moderate; 0.58), Access=(Moderate; 0.62). µSD (p) = 0.35 · 0.15 + 0.58 · 0.65 + 0.62 · 0.65 0.0525 + 0.377 + 0.403 0.8325 = = = 0.57310. 0.15 + 0.65 + 0.65 1.45 1.45 Tree labels: CH=High, CS=Moderate, SD=Moderate. Forest aggregation (dominant High): µForest (p) = 0.75697 · 1 + 0.55690 · 0.65 + 0.57310 · 0.65 1 + 0.65 + 0.65 0.75697 + 0.36199 + 0.37252 1.49148 = = 0.64847. 2.30 2.30 Hence p’s contradiction-aware forest readmission score is ≈ 0.648 (moderate–high). = Example 3.20.4 (Portfolio selection: forest of Risk, Return, and Liquidity). Portfolio selection chooses an optimal mix of assets balancing expected return, risk tolerance, diversification, and constraints like liquidity or regulations (cf. [768]). Trees: TPSRisk , TPSReturn , TPSLiq . Labels {Undesirable, Acceptable, Attractive} with dominant Attractive. Contradictions: , , , so w ∈ {1, 0.6, 0.15}. pCF (Attractive, Attractive) = 0 pCF (Acceptable, Attractive) = 0.4 pCF (Undesirable, Attractive) = 0.85 # Page. 105 ![Page Image](https://bcdn.docswell.com/page/YJ9PX96V73.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Asset a leaf evaluations (label; µ): (Risk) Volatility=(Acceptable; 0.68), Drawdown=(Acceptable; 0.70), Diversification=(Attractive; 0.74). µRisk (a) = 0.68 · 0.6 + 0.70 · 0.6 + 0.74 · 1 0.408 + 0.42 + 0.74 1.568 = = = 0.71273. 0.6 + 0.6 + 1 2.2 2.2 (Return) CAGR=(Attractive; 0.80), Sharpe=(Acceptable; 0.66), Alpha=(Attractive; 0.75). µReturn (a) = 0.80 · 1 + 0.66 · 0.6 + 0.75 · 1 0.80 + 0.396 + 0.75 1.946 = = = 0.74846. 1 + 0.6 + 1 2.6 2.6 (Liquidity) Turnover=(Acceptable; 0.64), BidAsk=(Undesirable; 0.40), Depth=(Acceptable; 0.67). µLiq (a) = 0.64 · 0.6 + 0.40 · 0.15 + 0.67 · 0.6 0.384 + 0.06 + 0.402 0.846 = = = 0.62667. 0.6 + 0.15 + 0.6 1.35 1.35 Tree labels: Risk=Attractive, Return=Attractive, Liquidity=Acceptable. Forest aggregation (dominant Attractive): µForest (a) = 0.71273 · 1 + 0.74846 · 1 + 0.62667 · 0.6 0.71273 + 0.74846 + 0.37600 1.83719 = = = 0.70738. 1 + 1 + 0.6 2.6 2.6 Thus asset a achieves a contradiction-aware forest score ≈ 0.707, i.e., marginally attractive overall. 3.21 Plithogenic Soft Expert Set Soft expert set extends soft sets by incorporating multiple experts’ parameterized opinions for decisionmaking under uncertainty and differing viewpoints simultaneously [769–771]. A plithogenic soft expert set enriches the classical (fuzzy / intuitionistic / neutrosophic) soft expert set by attaching to each expert–parameter–opinion triple a plithogenic degree of appurtenance together with a contradiction-aware fusion [772]. This allows modeling heterogeneous multi-component memberships while explicitly penalizing conflicting labels. Definition 3.21.1 (Plithogenic Soft Expert Set (PSES)). [772] Let U be a universe of discourse; E a set of parameters; X a set of experts; and O = {1 = agree, 0 = disagree} a set of opinions. Put Z = E × X × O and fix a finite index set A ⊆ Z of activated triples. For each parameter e ∈ E, let P ve be its set of admissible value-labels. Define the label universe G  L := P ve t O e∈E as a disjoint union of all parameter value-sets and opinions. For every e ∈ E fix a (possibly vectorvalued) plithogenic degree-of-appurtenance (DAF) pdfe : U × P ve −→ [0, 1]se , and fix a global degree-of-contradiction function (DCF) pCF : L × L −→ [0, 1]t , pCF (`, `) = 0, pCF (`1 , `2 ) = pCF (`2 , `1 ). # Page. 106 ![Page Image](https://bcdn.docswell.com/page/GJ8D29Z4JD.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Let Φ : [0, 1]s × [0, 1]t → [0, 1] be an aggregation (s := maxe se ) which is monotone nondecreasing in its first argument and nonincreasing in its second, and satisfies the neutral/annihilation conditions Φ(0, b) = 0, Φ(a, 0) = Ψ(a) ∈ [0, 1]. A Plithogenic Soft Expert Set (PSES) on U is a tuple  PSES = U, E, X, O, A, {P ve }e∈E , {pdfe }e∈E , pCF, Φ, val , F where val : A → e∈E P ve assigns to each α = (e, x, o) ∈ A the value-label val(α) ∈ P ve used by expert x under opinion o. Its plithogenic soft expert mapping is FPL : A −→ [0, 1]U ,    FPL (α)(u) := Φ ιe pdfe (u, val(α)) , pCF val(α), o , where α = (e, x, o) and ιe : [0, 1]se ,→ [0, 1]s is the natural padding into the common s-dimensional cube. Remark 3.21.2 (Well-posedness and reductions). (i) If t = 0 (no contradiction channel) then FPL (α)(u) = Ψ ιe (pdfe (u, val(α))) , recovering multi-component memberships without contradiction penalization. (ii) Choosing s = 1 and t = 0 recovers fuzzy soft expert sets; choosing s = 2 and t = 0 recovers intuitionistic fuzzy soft expert sets; choosing s = 3 and t = 0 recovers neutrosophic soft expert sets. See Table 3.29. Describe the content in Table 3.29 as Reductions showing that a Plithogenic Soft Expert Set (PSES) subsumes fuzzy, intuitionistic fuzzy, and neutrosophic soft expert sets. Table 3.29: Reductions of a Plithogenic Soft Expert Set (PSES) to several soft expert models. Target model Image of F (α) Contradiction Reduction from PSES Fuzzy Soft Expert Set [771, 773, 774] Intuitionistic Fuzzy Soft Expert Set [769, 770, 775] Neutrosophic Soft Expert Set [776–778] [0, 1] t=0 [0, 1]2 t=0 [0, 1]3 t=0 Hesitant Fuzzy Soft Expert Set [779, 780] [0, 1]3 t=0 Picture Fuzzy Soft Expert Set [781] [0, 1]3 t=0 Spherical Fuzzy Soft Expert Set [782–784] [0, 1]3 t=0 Take s = 1; each pdfe (u, v) ∈ [0, 1]; set pCF ≡ 0 and Φ(a, 0) = a. Take s = 2; pdfe (u, v) = (µ, ν) with usual IF constraint; set pCF ≡ 0 and use standard IF-operations. Take s = 3; pdfe (u, v) = (T, I, F ); set pCF ≡ 0 and apply neutrosophic soft expert operations componentwise. Take s = 3; encode hesitant information in (T, I, F )-type components; set pCF ≡ 0 and use hesitant soft expert aggregation. Take s = 3; interpret (T, I, F ) as picture fuzzy acceptance/neutrality/rejection; set pCF ≡ 0 and use picture fuzzy rules. Take s = 3; impose spherical fuzzy constraint on (T, I, F ); set pCF ≡ 0 and adopt spherical soft expert operators. A concrete example of this concept is provided below. # Page. 107 ![Page Image](https://bcdn.docswell.com/page/LJLM2WDVER.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.21.3 (Sustainable supplier selection as a PSES). Systematic evaluation of suppliers using environmental, social, and economic criteria to reduce impacts and ensure reliable, ethical, long-term sourcing choices (cf. [785]). Let U = {u1 , u2 , u3 } be a set of candidate suppliers. Let E = {e1 = “cost”, e2 = “sustainability”}, X = {x1 = procurement manager, x2 = sustainability officer}, and O = {1 = agree, 0 = disagree}. For the parameters we take P ve1 = {low, medium, high}, P ve2 = {green, neutral, risky}, and let L be the disjoint union of all P ve together with O. Consider the activated triples  A = (e1 , x1 , 1), (e2 , x1 , 1), (e2 , x2 , 1) . F The value-assignment map val : A → e∈E P ve is given by val(e1 , x1 , 1) = low, val(e2 , x1 , 1) = neutral, val(e2 , x2 , 1) = green. Let se1 = se2 = 1 and define fuzzy DAFs pdfe1 , pdfe2 : U × P vei → [0, 1] by, for example, pdfe1 (u1 , low) = 0.9, pdfe1 (u2 , low) = 0.6, pdfe1 (u3 , low) = 0.2, pdfe2 (u1 , green) = 0.7, pdfe2 (u2 , green) = 0.4, pdfe2 (u3 , green) = 0.1, pdfe2 (u1 , neutral) = 0.5, pdfe2 (u2 , neutral) = 0.7, pdfe2 (u3 , neutral) = 0.6. To model plithogenic contradiction, let t = 1 and define pCF : L × L → [0, 1] by pCF (low, 1) = 0.1, pCF (green, 1) = 0.1, pCF (neutral, 1) = 0.3, and set pCF (`, `) = 0 and pCF (`1 , `2 ) = 0 for all other pairs, so that “neutral” sustainability is slightly more contradictory to a strong positive opinion than “green”. Take the aggregation Φ : [0, 1] × [0, 1] → [0, 1], Φ(a, b) = a(1 − b), which decreases the membership when the contradiction degree b is large. Then the plithogenic soft expert mapping FPL : A → [0, 1]U is given, for α = (e, x, o) ∈ A and u ∈ U , by  FPL (α)(u) = Φ pdfe (u, val(α)), pCF (val(α), o) . # Page. 108 ![Page Image](https://bcdn.docswell.com/page/47MY89PN7W.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets For instance, for α1 = (e1 , x1 , 1) (cost, manager, agree) we obtain  FPL (α1 )(u1 ) = Φ 0.9, 0.1 = 0.9 × (1 − 0.1) = 0.81, so supplier u1 is strongly supported as “low cost” by the procurement manager. Similarly, for α2 = (e2 , x1 , 1) and supplier u2 ,  FPL (α2 )(u2 ) = Φ pdfe2 (u2 , neutral), pCF (neutral, 1) = 0.7 × (1 − 0.3) = 0.49, reflecting that a “neutral” sustainability assessment is partially penalized by the contradiction degree 0.3. Thus this decision scenario forms a Plithogenic Soft Expert Set on the supplier universe U . Example 3.21.4 (Course recommendation as a PSES). Let U = {c1 , c2 , c3 } be a set of elective courses at a university. Let E = {e1 = “difficulty”, e2 = “job-market relevance”}, X = {x1 = advisor, x2 = industry mentor}, and again O = {1 = agree, 0 = disagree}. For the parameters take P ve1 = {easy, moderate, hard}, P ve2 = {low, medium, high}, and consider the activated triples  A = (e1 , x1 , 1), (e2 , x1 , 1), (e2 , x2 , 1) . Suppose val(e1 , x1 , 1) = moderate, val(e2 , x1 , 1) = high, val(e2 , x2 , 1) = high, so both experts emphasize high job-market relevance, while the advisor describes difficulty as moderate. Let se1 = se2 = 1, and define fuzzy DAFs by pdfe1 (c1 , moderate) = 0.8, pdfe1 (c2 , moderate) = 0.5, pdfe1 (c3 , moderate) = 0.3, pdfe2 (c1 , high) = 0.6, pdfe2 (c2 , high) = 0.9, pdfe2 (c3 , high) = 0.4. To keep the contradiction structure simple, let t = 1 and define pCF (moderate, 1) = 0.2, pCF (high, 1) = 0.1, and pCF (`, `) = 0, pCF (`1 , `2 ) = 0 for all other pairs. Thus “high job-market relevance” is almost fully compatible with a positive opinion, while “moderate difficulty” carries a slightly higher contradiction penalty. Using the same aggregation Φ(a, b) = a(1 − b) as above, we obtain for α = (e2 , x1 , 1) (advisor, high relevance) and course c2 :  FPL (α)(c2 ) = Φ pdfe2 (c2 , high), pCF (high, 1) = 0.9 × (1 − 0.1) = 0.81, indicating strong plithogenic support for recommending c2 due to high job-market relevance with low contradiction. In contrast, for α0 = (e1 , x1 , 1) (advisor, moderate difficulty) and course c3 :  FPL (α0 )(c3 ) = Φ pdfe1 (c3 , moderate), pCF (moderate, 1) = 0.3 × (1 − 0.2) = 0.24, showing weaker support for c3 under the same expert opinion. This university course recommendation scenario thus provides another concrete real-life instance of a Plithogenic Soft Expert Set on U . # Page. 109 ![Page Image](https://bcdn.docswell.com/page/P7R95G14E9.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.22 Dynamic Plithogenic Set A Dynamic Plithogenic Set (DPS) makes the plithogenic membership and contradiction depend on time, so each instant yields a (static) plithogenic set snapshot [786]. Definition 3.22.1 (Dynamic Plithogenic Set). Let P 6= ∅ be a universe, let v be an attribute with value–set Pv , and let T ⊆ R be a nonempty time domain. Fix integers s ≥ 1 and t ≥ 0. A Dynamic Plithogenic Set (of dimension (s, t)) is a tuple DPS = (P, v, Pv , pdf, pCF, T ), where pdf : T × P × Pv −→ [0, 1]s , pCF : T × Pv × Pv −→ [0, 1]t are, respectively, the time–dependent Degree of Appurtenance Function (DAF) and Degree of Contradiction Function (DCF). For each fixed t0 ∈ T , the snapshot at time t0 is the (static) plithogenic set  PS(t0 ) = P, v, Pv , pdf (t0 ) , pCF (t0 ) , with pdf (t0 ) (x, a) := pdf (t0 , x, a) and pCF (t0 ) (a, b) := pCF (t0 , a, b). We require, for all t ∈ T and all a, b ∈ Pv , pCF (t, a, a) = 0 ∈ [0, 1]t (reflexivity), pCF (t, a, b) = pCF (t, b, a) (symmetry). We provide in Table 3.30 a concise summary of how dynamic fuzzy, dynamic intuitionistic fuzzy, and dynamic neutrosophic models arise as special cases of the Dynamic Plithogenic Set (DPS) framework. Table 3.30: Dynamic fuzzy/intuitionistic/neutrosophic families as special cases of the Dynamic Plithogenic Set (DPS). Dynamic model Dynamic Fuzzy (DFS) [344, 787] Set Dynamic Intuitionistic Fuzzy Set (DIFS) [788, 789] Dynamic Neutrosophic Set (DNS) (cf. [786, 790]) Dynamic Hesitant Fuzzy Set (cf. [791, 792]) s 1 t 0 2 0 3 0 3 0 How it is obtained from DPS Take pdf : T × P × Pv → [0, 1] and pCF absent (or pCF ≡ 0). Each snapshot is a fuzzy set; membership varies with t. Take pdf (t, x, a) = (µ, ν) ∈ [0, 1]2 with µ+ν ≤ 1; pCF absent (or ≡ 0). Take pdf (t, x, a) = (T, I, F ) ∈ [0, 1]3 ; pCF absent (or ≡ 0). Take pdf (t, x, a) ∈ [0, 1]3 ; pCF absent (or ≡ 0). A concrete example of this concept is provided below. Example 3.22.2 (Dynamic plithogenic customer satisfaction in an online platform). Customer satisfaction measures how well products and services meet or exceed customer expectations, positively influencing loyalty, repeat purchases, and reputation (cf. [793]). Consider an online subscription platform that tracks how customers feel about its Basic and Premium plans before and after a major interface redesign. Let P = {Basic, Premium} be the universe of plans, and let the attribute be v = “overall satisfaction” with value set Pv = {low, medium, high}. # Page. 110 ![Page Image](https://bcdn.docswell.com/page/PJXQKXYZ7X.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets We take a time domain T = {t1 , t2 } ⊂ R, where t1 is “before redesign” and t2 is “after redesign”. Choose s = 3 and t = 1. For each (t, x, a) ∈ T × P × Pv the time–dependent DAF pdf : T × P × Pv −→ [0, 1]3 returns a triple pdf (t, x, a) = (µT , µI , µF ), interpreted as degrees of approval (µT ), hesitation (µI ), and rejection (µF ) of value a for plan x at time t. For instance, for the Premium plan we may set pdf (t, Premium, high) pdf (t, Premium, medium) pdf (t, Premium, low) t1 (before redesign) (0.45, 0.30, 0.25) (0.35, 0.25, 0.40) (0.10, 0.20, 0.70) t2 (after redesign) (0.80, 0.10, 0.10) (0.15, 0.15, 0.70) (0.05, 0.10, 0.85) and similarly for the Basic plan (with smaller improvements after t2 ). The contradiction degree function pCF : T × Pv × Pv −→ [0, 1] is taken time–independent for simplicity: pCF (·, ·, ·) low medium high low 0 0.3 0.9 medium 0.3 0 0.5 high 0.9 0.5 0 and symmetric by definition. Then DPScust = (P, v, Pv , pdf, pCF, T ) is a Dynamic Plithogenic Set. The dynamic aspect is captured by the change in the appurtenance triples from t1 to t2 : after the redesign, the degree of “high satisfaction” for Premium shifts from (0.45, 0.30, 0.25) to (0.80, 0.10, 0.10), while the contradiction structure between satisfaction levels is kept fixed. This allows the analyst to compare customer sentiment snapshots over time within a plithogenic framework. Example 3.22.3 (Dynamic plithogenic air–quality risk assessment in a city). Air-quality risk assessment evaluates pollutant exposure, health impacts, and uncertainty to prioritize mitigation policies, regulations, and urban planning decisions effectively (cf. [794]). A city monitors daily air quality in two districts, A and B, and classifies each district into qualitative health risk levels. Let the universe be P = {District A, District B}, and let the attribute be v = “air–quality health risk” with value set Pv = {low, moderate, high}. Take a discrete time domain of three consecutive days, T = {d1 , d2 , d3 } ⊂ R, # Page. 111 ![Page Image](https://bcdn.docswell.com/page/3JK95W3VJD.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets with d1 = “Monday”, d2 = “Tuesday”, d3 = “Wednesday”. We again fix s = 3, t = 1, and interpret pdf (d, x, a) = (µT , µI , µF ) ∈ [0, 1]3 as degrees of acceptable risk, uncertainty, and unacceptable risk associated with label a for district x on day d. Suppose that for District A we obtain the following DAF values: pdf (d,A, low) pdf (d,A, moderate) pdf (d,A, high) d1 (clear) (0.80, 0.10, 0.10) (0.15, 0.20, 0.65) (0.05, 0.10, 0.85) d2 (haze) (0.40, 0.20, 0.40) (0.40, 0.25, 0.35) (0.20, 0.20, 0.60) d3 (pollution episode) (0.10, 0.10, 0.80) (0.30, 0.20, 0.50) (0.60, 0.15, 0.25) On clear day d1 the city sees District A mostly as “low risk”, while during the pollution episode d3 the weight moves toward “high risk”. Similar (possibly different) evolutions can be specified for District B. The time–dependent contradiction degree pCF : T × Pv × Pv −→ [0, 1] can encode changing public–health emphasis. For example, the authorities may become more sensitive to differences between “moderate” and “high” after the pollution episode. A simple specification is pCF (d1 , ·, ·) low moderate high low 0 0.2 0.8 moderate 0.2 0 0.4 high 0.8 0.4 0 pCF (d3 , ·, ·) low moderate high low 0 0.3 0.9 moderate 0.3 0 0.6 high 0.9 0.6 0 with pCF (d2 , ·, ·) chosen in between and symmetry enforced. Here, the contradiction between “moderate” and “high” risk increases from 0.4 on d1 to 0.6 on d3 , reflecting stricter health–policy thresholds after observing a serious episode. Then DPSair = (P, v, Pv , pdf, pCF, T ) is a Dynamic Plithogenic Set whose snapshots PS(d1 ) , PS(d2 ) , and PS(d3 ) represent, respectively, the city’s plithogenic risk assessments on each day, taking into account both time-varying memberships and time-varying contradiction between risk labels. 3.23 Probabilistic Plithogenic Set A probabilistic plithogenic set attaches to each element a random (possibly multi–component) plithogenic degree, and fuses such degrees across attributes by a contradiction–aware aggregator. This subsumes probabilistic fuzzy, probabilistic intuitionistic fuzzy, and probabilistic neutrosophic sets. Definition 3.23.1 (Probabilistic Plithogenic Degree (PPD)). Let (Ω, A, P) be a probability space. Fix a finite attribute system Att = {a1 , . . . , am }, P vi (admissible values of ai ). For an underlying universe P , an si –component probabilistic plithogenic degree at node ai is a measurable map µi : P × P vi × Ω −→ [0, 1] si , (x, α, ω) 7−→ µi (x, α; ω), meaning that, for each fixed (x, α), the random vector µi (x, α; ·) describes the stochastic membership (possibly multi–valued such as (µ, ν) or (T, I, F )). # Page. 112 ![Page Image](https://bcdn.docswell.com/page/LE3WK1LQE5.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.23.2 (Probabilistic Plithogenic Set (PPS)). Let pCF : S i P vi  × S i P vi  → [0, 1] be a symmetric contradiction degree with pCF (a, a) = 0. Let AggpCF be a measurable contradiction–aware aggregator that, for any finite family of components {zj }j in [0, 1] and their pairwise contradictions {cjk }, returns a value in [0, 1]; a typical choice is the contradiction–weighted t–norm/t–conorm blend AggpCF (u, v; c) := (1 − c) T (u, v) + c S(u, v), c ∈ [0, 1], extended iteratively to n ≥ 2 inputs, where T is a t–norm and S a t–conorm. A Probabilistic Plithogenic Set is the tuple   PPS = P, {(ai , P vi , µi )}m , pCF, Agg i=1 pCF . Q For any selection of attribute–values γ = (α1 , . . . , αm ) ∈ i P vi and x ∈ P , define the aggregated random degree  µ(x, γ; ω) := AggpCF µi (x, αi ; ω) ∈ [0, 1] s , 1≤i≤m where the aggregation is taken componentwise when s := maxi si > 1 and the pairwise contradictions used inside AggpCF are pCF (αi , αj ). Representative crisp summaries can be extracted, e.g. E[µ(x, γ; ω)], or quantile profiles qp (x, γ) defined componentwise by   qp (x, γ) := inf t ∈ [0, 1] : P µ(x, γ; ω) ≤ t ≥ p . Remark 3.23.3 (Set operations (defined ω–wise)). For two PPSs on the same (P, {P vi }, pCF, AggpCF ) with degrees µA , µB : Union: Intersection: Complement:  (∪) µA∪B (x, γ; ω) := AggpCF µA (x, γ; ω), µB (x, γ; ω) ,  (∩) µA∩B (x, γ; ω) := AggpCF µA (x, γ; ω), µB (x, γ; ω) ,  µAc (x, γ; ω) := Comp µA (x, γ; ω) , where AggpCF (resp. AggpCF ) is typically obtained by biasing toward S (resp. T ) via the local contradiction degree(s), and Comp is the plithogenic complement (for s = 1, Comp(u) = 1 − u; for (µ, ν) or (T, I, F ), apply the standard dualities componentwise). A crisp output can be taken as E[·] of the above random outputs when needed. (∪) (∩) Remark 3.23.4 (Connection to probabilistic fuzzy sets). In a probabilistic fuzzy set (PFS), the membership grade is a random variable on a probability space, often formalized as a measurable map µA : T × Ω → [0, 1] (measurable in ω for fixed t, and a fuzzy membership in t for fixed ω). This yields a “3D” view where the grade has a probability distribution. The PPS reduces to a PFS when m = 1, s = 1, and pCF ≡ 0 (so AggpCF is the identity). See the measurable definition and random–grade viewpoint of PFS in the literature. Table 3.31 summarizes the reductions of the Probabilistic Plithogenic Set into its corresponding fuzzy, intuitionistic fuzzy, neutrosophic, and classical plithogenic special cases. A concrete example of this concept is provided below. Example 3.23.5 (Probabilistic plithogenic supplier evaluation in sustainable procurement). Consider a manufacturing company that needs to select a long–term raw–material supplier under uncertain future demand and regulatory conditions. # Page. 113 ![Page Image](https://bcdn.docswell.com/page/8EDK3XNW7G.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.31: Reductions of the Probabilistic Plithogenic Set (PPS). A PPS collapses to PFS/PIFS/PNS by fixing one attribute, taking pCF ≡ 0, and choosing s = 1, 2, 3 respectively. Target model Probabilistic Fuzzy Set (PFS) [795–798] Components (codomain) scalar grade u ∈ [0, 1] (random) Constraints none beyond u ∈ [0, 1] Probabilistic Intuitionistic Fuzzy Set (PIFS) [180, 799, 800] Probabilistic Neutrosophic Set (PNS) [801–803] Probabilistic Hesitant Fuzzy Set [804, 805, 805] Probabilistic Quadripartitioned Neutrosophic Set Probabilistic Pentapartitioned Neutrosophic Set pair (µ, ν) ∈ [0, 1]2 (random) µ + ν ≤ 1 a.s. triple (T, I, F ) ∈ [0, 1]3 (random) typically 0 ≤ T, I, F ≤ 1 hesitant fuzzy triple (random) Quadripartitioned Neutrosophic Quadruple (random) − Pentapartitioned Neutrosophic QuinTuple (random) − Reduction from PPS m = 1, s = 1, pCF ≡ 0; AggpCF identity; µ(x, α; ·) ∈ [0, 1] m = 1, s = 2, pCF ≡ 0; µ(x, α; ·) ∈ [0, 1]2 with a.s. µ + ν ≤ 1 m = 1, s = 3, pCF ≡ 0; µ(x, α; ·) ∈ [0, 1]3 m = 1, s = 3, pCF ≡ 0; µ(x, α; ·) ∈ [0, 1]3 m = 1, s = 4, pCF ≡ 0; µ(x, α; ·) ∈ [0, 1]4 − m = 1, s = 5, pCF ≡ 0; µ(x, α; ·) ∈ [0, 1]5 Universe and attributes. Let the universe of alternatives be P = {S1 , S2 , S3 }, where S1 is the incumbent supplier and S2 , S3 are new candidates. The company evaluates suppliers according to the finite attribute system Att = {a1 , a2 , a3 }, where a1 = “unit cost”, a2 = “quality stability”, a3 = “environmental performance”. For each ai we use the same linguistic value set P vi = {low, medium, high} interpreted as low/medium/high favorability for the company. Uncertain scenarios and probabilistic degrees. Future conditions are uncertain, so the company models three equally likely scenarios: Ω = {ω1 , ω2 , ω3 }, P({ωk }) = 13 (k = 1, 2, 3), representing ω1 = “demand boom”, ω2 = “baseline”, ω3 = “strict carbon regulation”. We choose si = 1 for all i, so each probabilistic plithogenic degree µi (x, α; ω) ∈ [0, 1] is an ordinary random fuzzy membership. For illustration, consider attribute a3 (environmental performance) and supplier S1 . The random membership µ3 (S1 , α; ω) for α ∈ P v3 is specified as follows: µ3 (S1 , α; ω) α = low α = medium α = high ω1 0.20 0.50 0.30 ω2 0.10 0.40 0.50 ω3 0.05 0.30 0.65 # Page. 114 ![Page Image](https://bcdn.docswell.com/page/V7PK4P5XJ8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Before strict regulation (ω1 , ω2 ), S1 is mostly “medium” to “high” in environmental performance, but under strict regulation ω3 , the company expects S1 to invest more in green technology, increasing the membership of “high” to 0.65. Similarly, we define µ1 and µ2 for a1 (cost) and a2 (quality) for each supplier, attribute–value, and scenario. For example, for S2 on cost: µ1 (S2 , α; ω) α = low α = medium α = high ω1 0.80 0.15 0.05 ω2 0.75 0.20 0.05 ω3 0.70 0.25 0.05 indicating that S2 is expected to be consistently low–cost, even with regulatory changes. Contradiction degrees and aggregation. We define a simple contradiction degree on the union of all value sets by pCF (low, high) = pCF (high, low) = 0.9, pCF (low, medium) = pCF (medium, low) = 0.4, pCF (medium, high) = pCF (high, medium) = 0.5, pCF (α, α) = 0, and extend pCF trivially across attributes (values from different attributes are considered moderately contradictory, if needed). As contradiction–aware aggregator, the company chooses AggpCF (u, v; c) = (1 − c) min{u, v} + c max{u, v}, which biases toward the t–norm min when c is small (compatible values) and toward the t–conorm max when c is large (strong contradiction). For a fixed supplier S2 and a particular value profile γ = (α1 , α2 , α3 ) = (low cost, high quality, high environment), the aggregated random degree  µ(S2 , γ; ω) = AggpCF µ1 (S2 , low; ω), µ2 (S2 , high; ω), µ3 (S2 , high; ω) is a scenario–dependent random membership in [0, 1]. A crisp summary such as 3   X E µ(S2 , γ; ω) = µ(S2 , γ; ωk ) P({ωk }) k=1 gives the expected plithogenic suitability of S2 under profile γ, while quantiles capture risk–averse views. Altogether,   PPSsupply = P, {(ai , P vi , µi )}3i=1 , pCF, AggpCF is a Probabilistic Plithogenic Set modeling sustainable supplier evaluation under uncertain future conditions. Example 3.23.6 (Probabilistic plithogenic disease–risk assessment in preventive medicine). Disease–risk assessment is systematic evaluation of likelihood and impact of developing specific diseases using medical data, biomarkers, lifestyle, genetics, and environment over time (cf. [806]). A public–health agency wants to assess an individual’s risk of developing a chronic disease (e.g. type–2 diabetes) using several uncertain indicators. # Page. 115 ![Page Image](https://bcdn.docswell.com/page/2JVVX2P3JQ.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Universe and attributes. Let P be the population of patients enrolled in a screening program. For a fixed patient x ∈ P we evaluate the following attribute system: Att = {a1 , a2 , a3 }, with a1 = “body–mass index (BMI)”, a2 = “fasting blood glucose”, a3 = “physical activity level”. Each attribute is described by a qualitative value set: P v1 = {normal, overweight, obese}, P v2 = {normal, impaired, high}, P v3 = {low, moderate, high}. Measurement uncertainty as probability space. Measurements are noisy and may vary across visits, so the agency considers three equiprobable “data states”: Ω = {ω1 , ω2 , ω3 }, P({ωk }) = 13 , where, for a given patient x, ω1 = “optimistic lab readings”, ω2 = “typical readings”, ω3 = “pessimistic readings”. We choose s1 = s2 = s3 = 3 and regard  µi (x, α; ω) = Ti (x, α; ω), Ii (x, α; ω), Fi (x, α; ω) ∈ [0, 1]3 as a neutrosophic–type triple of degrees of support (T ), indeterminacy (I), and counter–evidence (F ) for value α of attribute ai under data state ω. For example, for patient x with moderately high BMI, the random degrees for a1 may be µ1 (x, α; ω) α = normal α = overweight α = obese ω1 ω2 (0.30, 0.20, 0.50) (0.20, 0.20, 0.60) (0.50, 0.20, 0.30) (0.55, 0.20, 0.25) (0.20, 0.20, 0.60) (0.25, 0.25, 0.50) ω3 (0.10, 0.15, 0.75) (0.45, 0.20, 0.35) (0.45, 0.20, 0.35) indicating that, under pessimistic readings ω3 , the evidence for α = obese increases. Similarly, for fasting glucose a2 we might have, for the same patient x, µ2 (x, α; ω) α = normal α = impaired α = high ω1 ω2 (0.70, 0.10, 0.20) (0.55, 0.15, 0.30) (0.20, 0.15, 0.65) (0.30, 0.20, 0.50) (0.10, 0.20, 0.70) (0.15, 0.20, 0.65) ω3 (0.40, 0.15, 0.45) (0.35, 0.20, 0.45) (0.25, 0.20, 0.55) and analogous tables for a3 describing physical activity. Contradiction structure and risk aggregation. Some attribute values jointly increase disease risk (e.g. “obese” and “high glucose”), while others are protective or partially compensating (e.g. # Page. 116 ![Page Image](https://bcdn.docswell.com/page/5EGLVRPYJL.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets “obese” but “high activity”). This is captured by a plithogenic contradiction degree pCF on the union of all P vi . For instance, pCF (normal BMI, normal glucose) = 0.1, pCF (obese, high glucose) = 0.95, pCF (obese, high activity) = 0.6, and pCF (α, α) = 0 with symmetry. The contradiction–aware aggregator AggpCF : ([0, 1]3 )n × [0, 1]n×n −→ [0, 1]3 combines the neutrosophic triples across attributes. For a chosen risk–oriented profile γ = (α1 , α2 , α3 ) = (obese, high glucose, low activity), the aggregated random triple µ(x, γ; ω) ∈ [0, 1]3 represents the probabilistic plithogenic degree of being a “high–risk patient” for x under data state ω. Taking the expectation   E µ(x, γ; ω) yields an average neutrosophic risk triple, while quantiles of µ(x, γ; ω) provide conservative risk bounds. Thus   PPSmed = P, {(ai , P vi , µi )}3i=1 , pCF, AggpCF is a Probabilistic Plithogenic Set that models disease–risk assessment under measurement noise and uncertain future health states. 3.24 Triangular Plithogenic Set A triangular plithogenic set models each element’s appurtenance by one or more triangular fuzzy numbers on [0, 1], while modulating these degrees through an attribute–value contradiction function. Definition 3.24.1 (Triangular Fuzzy Number (TFN)). (cf. [807–809]) A TFN on [0, 1] is a triple τ = (`, m, u) with 0 ≤ ` ≤ m ≤ u ≤ 1. Its membership function µτ : [0, 1] → [0, 1] is   0, z ≤ `,     z−`   , ` < z ≤ m,  m−` µτ (z) = u−z    , m < z < u,   u −m    0, z ≥ u. Definition 3.24.2 (Triangular Plithogenic Set (TPS)). Let U be a universe and v an attribute with value set Pv . Fix integers p, q, r ≥ 0 (a triangular refinement signature) and set s := p+q+r ∈ {1, 2, 3}. Define the triangular plithogenic appurtenance map  p  q  r s tPDF : U × Pv −→ [0, 1]3 , (x, a) 7−→ Ti (x, a) Ij (x, a) Fk (x, a) , i=1 j=1 k=1 where each component Ti (x, a), Ij (x, a), Fk (x, a) is a TFN Ti (x, a) = (`Ti , mTi , uTi ), etc. Let the plithogenic contradiction be pCF : Pv × Pv → [0, 1]t , pCF (a, b) = pCF (b, a), pCF (a, a) = 0. # Page. 117 ![Page Image](https://bcdn.docswell.com/page/4JQY6V267P.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets A Triangular Plithogenic Set is the tuple  TPS = U, v, Pv , (p, q, r), tPDF, pCF, Φ , together with a fixed monotone aggregator Φ: [0, 1]3 s × [0, 1]t −→ [0, 1]3 that returns a TFN and satisfies Φ(·, 0) = componentwise TFN aggregation without contradiction. For any x ∈ U and dominant value a∗ ∈ Pv , the effective triangular degree of x is the TFN   τTPS (x | a∗ ) := Φ tPDF(x, a∗ ), pCF a∗ , · ∈ [0, 1]3 . Decision scores (if needed) can be scalarized by a TFN defuzzifier, e.g. the centroid cen(`, m, u) = `+m+u , applied to τTPS (x | a∗ ). 3 Table 3.32 explains that the classical triangular fuzzy, intuitionistic, and neutrosophic sets arise as exact special cases of the Triangular Plithogenic Set. Table 3.32: Classical triangular fuzzy/intuitionistic/neutrosophic sets as exact special cases of the Triangular Plithogenic Set by signature choice and pCF ≡ 0. Target triangular model Triangular Fuzzy Set (TFS) [810–812] Triangular Intuitionistic Fuzzy Set (TIFS) [813–815] Signature (p, q, r) (1, 0, 0) Triangular Neutrosophic Set (TNS) [816–818] (1, 1, 1) Triangular Hesitant Fuzzy Sets [819–821] (1, 1, 1) (1, 1, 0) Recovery inside TPS (set pCF ≡ 0) One TFN T1 (x, a) per x and a; the effective TFN is the aggregated T (no contradiction). A pair of TFNs (T1 , I1 ) per (x, a) with a consistency condition (e.g. peak values mT1 + mI1 ≤ 1); operations act componentwise on (T1 , I1 ). A triple of TFNs (T1 , I1 , F1 ) per (x, a); no coupling required beyond range constraints (optional bounds may be imposed per application). A triple per (x, a); no coupling required beyond range constraints (optional bounds may be imposed per application). A brief concrete example of this concept is provided below. Example 3.24.3 (Triangular plithogenic assessment of laptop options in online shopping). A customer wants to choose a new laptop from three alternatives U = {L1 , L2 , L3 }. They evaluate each laptop with respect to the attribute v = “overall perceived utility” and a set of linguistic attribute values Pv = {low, medium, high}. We consider a purely triangular fuzzy–type refinement with (p, q, r) = (1, 0, 0), s = p + q + r = 1, # Page. 118 ![Page Image](https://bcdn.docswell.com/page/K74W4MLME1.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets so that for every (x, a) ∈ U × Pv there is a single triangular fuzzy number T1 (x, a) describing the truth–membership of “laptop x has utility level a”. Thus  tPDF(x, a) = T1 (x, a) , T1 (x, a) = (`T1 , mT1 , uT1 ) ∈ [0, 1]3 . Suppose the customer’s approximate judgements (already normalized to [0, 1]) are as follows. Triangular degrees for L1 . Laptop L1 is a budget option: mostly between “medium” and “high”, but with some uncertainty. The triangular truth–memberships are T1 (L1 , low) = (0.0, 0.1, 0.3), T1 (L1 , medium) = (0.2, 0.5, 0.8), T1 (L1 , high) = (0.4, 0.7, 1.0). Triangular degrees for L2 . Laptop L2 is a mid–range balanced model: T1 (L2 , low) = (0.0, 0.0, 0.2), T1 (L2 , medium) = (0.3, 0.6, 0.9), T1 (L2 , high) = (0.5, 0.8, 1.0). Triangular degrees for L3 . Laptop L3 is a premium model; the user believes it is rarely “low” utility but sometimes only “medium”, depending on personal taste: T1 (L3 , low) = (0.0, 0.0, 0.1), T1 (L3 , medium) = (0.2, 0.4, 0.7), T1 (L3 , high) = (0.6, 0.9, 1.0). Plithogenic contradiction on utility levels. We impose a contradiction degree on Pv : pCF (low, high) = pCF (high, low) = 0.9, pCF (low, medium) = pCF (medium, low) = 0.4, pCF (medium, high) = pCF (high, medium) = 0.5, pCF (a, a) = 0 for a ∈ Pv . This reflects that “low” and “high” perceived utility are strongly contradictory, while “medium” is moderately contradictory to them. Contradiction–aware aggregation toward a dominant value. Assume the customer desires high utility as the dominant value a∗ = high. Then the triangular plithogenic degree   τTPS (x | a∗ ) = Φ tPDF(x, high), pCF (high, ·) is obtained via a chosen TFN aggregator Φ which biases the triangle according to how strongly “high” contradicts the other utility levels. For example, if Φ puts more weight on optimistic parts of the triangle whenever pCF (high, low) is large, then laptops which rarely have “low” utility (such as L3 ) will obtain a more favorable τTPS (L3 | high) than those with sizable “low” components. Defuzzifying via the centroid `+m+u 3 yields a scalar plithogenic score for each Li , thus providing a concrete Triangular Plithogenic Set  TPSlaptop = U, v, Pv , (1, 0, 0), tPDF, pCF, Φ cen(`, m, u) = for real–life online laptop selection under contradictory impressions of utility levels. # Page. 119 ![Page Image](https://bcdn.docswell.com/page/LJ1Y48LYEG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.24.4 (Triangular plithogenic evaluation of renewable–energy projects). Renewable energy refers to power from naturally replenished sources like sun, wind, water, and biomass, reducing emissions and dependence globally (cf. [822]). A city council considers three renewable–energy projects U = {P1 , P2 , P3 }, where P1 is a solar farm, P2 an onshore wind park, and P3 a biomass plant. The main attribute is v = “overall sustainability impact”, with attribute values Pv = {poor, acceptable, good, excellent}. Because sustainability must balance environmental, social, and economic dimensions, the council wants to model not only truth but also indeterminacy and falsity degrees, each in triangular fuzzy form. We therefore choose (p, q, r) = (1, 1, 1), so that s = p + q + r = 3,   tPDF(x, a) = T1 (x, a) I1 (x, a) F1 (x, a) , where each of T1 (x, a), I1 (x, a), F1 (x, a) is a triangular fuzzy number in [0, 1]3 . Sustainability profile for the solar farm P1 . For P1 , suppose the council’s expert panel provides the following triangular neutrosophic–type assessments for the value “good”: T1 (P1 , good) = (0.5, 0.7, 0.9), indicating that the truth of “P1 is good” is most plausible around 0.7, with minimum support 0.5 and maximum 0.9. Due to uncertainty in long–term lifecycle data, the indeterminacy is I1 (P1 , good) = (0.1, 0.2, 0.4), while falsity (counter–evidence, e.g. land–use concerns) is F1 (P1 , good) = (0.0, 0.1, 0.3). For a more ambitious value “excellent”, they specify T1 (P1 , excellent) = (0.2, 0.4, 0.7), I1 (P1 , excellent) = (0.2, 0.3, 0.5), F1 (P1 , excellent) = (0.1, 0.2, 0.5). Profiles for wind park P2 and biomass plant P3 . For P2 (wind park), visual impact and noise cause more conflict with stakeholders, but carbon savings are high. For the value “good” one may have T1 (P2 , good) = (0.4, 0.6, 0.8), I1 (P2 , good) = (0.2, 0.3, 0.4), F1 (P2 , good) = (0.1, 0.2, 0.4). # Page. 120 ![Page Image](https://bcdn.docswell.com/page/GJWGXZV172.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets For P3 (biomass), fuel supply chain and emissions create further uncertainty: T1 (P3 , good) = (0.3, 0.5, 0.7), I1 (P3 , good) = (0.3, 0.4, 0.6), F1 (P3 , good) = (0.1, 0.3, 0.5). Plithogenic contradiction between sustainability levels. We define a contradiction degree on Pv by viewing it as an ordered scale: poor ≺ acceptable ≺ good ≺ excellent. Set |ι(a) − ι(b)| , a, b ∈ Pv , 3 where ι(poor) = 0, ι(acceptable) = 1, ι(good) = 2, ι(excellent) = 3. Thus pCF (a, b) := pCF (poor, excellent) = 1, pCF (poor, good) = 23 , pCF (acceptable, good) = 13 , and pCF (a, a) = 0. Contradiction–aware triangular sustainability degree. Assume the council aims at excellent sustainability as dominant value a∗ = excellent. For each project x ∈ {P1 , P2 , P3 } the Triangular Plithogenic Set returns a neutrosophic–type TFN   τTPS (x | a∗ ) = Φ tPDF(x, excellent), pCF (excellent, ·) ∈ [0, 1]3 , where Φ aggregates the (T, I, F ) triangles while taking into account how much “excellent” contradicts lower levels such as “poor” or “acceptable”. Defuzzifying τTPS (x | a∗ ) by the centroid componentwise, cen(T ) = `T + mT + uT , 3 cen(I) = `I + mI + uI , 3 cen(F ) = `F + mF + uF , 3 gives a scalar summary of truth, indeterminacy, and falsity for the statement “project x is excellent in sustainability”. In this way  TPSsust = U, v, Pv , (1, 1, 1), tPDF, pCF, Φ is a concrete Triangular Plithogenic Set for real–life evaluation of renewable–energy projects with contradictory sustainability judgements. 3.25 Trapezoidal Plithogenic Sets A Trapezoidal Plithogenic Set models elements using trapezoidal truth, indeterminacy, falsity memberships, weighted by attribute-wise contradiction degrees for uncertain information. Definition 3.25.1 (Trapezoidal fuzzy number). (cf. [823–825]) Let (a, b, c, d) with a ≤ b ≤ c ≤ d. The trapezoidal membership profile Trap abcd : [0, 1] → [0, 1] is  0, x ≤ a,     x − a   , a < x < b,     b−a b ≤ x ≤ c, Trap abcd = 1,    d − x   , c < x < d,   d−c    0, x ≥ d. We abbreviate a trapezoidal fuzzy number by (a, b, c, d). # Page. 121 ![Page Image](https://bcdn.docswell.com/page/4EZL614X73.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.25.2 (Trapezoidal Plithogenic Set (TPS)). Let U be a universe, v an attribute with value-set Pv , and pCF : Pv × Pv → [0, 1] a plithogenic contradiction function with pCF (a, a) = 0 and pCF (a, b) = pCF (b, a). For each a ∈ Pv , a TPS on U assigns to every x ∈ U three trapezoidal fuzzy numbers on [0, 1]: T (x, a) = (t1 , t2 , t3 , t4 ), I(x, a) = (i1 , i2 , i3 , i4 ), F (x, a) = (f1 , f2 , f3 , f4 ), interpreted as truth-, indeterminacy-, and falsity-membership profiles, respectively. The data  TPS = U, v, Pv ; T, I, F ; pCF is called a Trapezoidal Plithogenic Set. A scalar plithogenic inclusion grade at a chosen dominant value a∗ ∈ Pv is obtained by first defuzzifying each trapezoid via the arithmetic mean S(a, b, c, d) := a+b+c+d and then contradiction-weighting: 4 λ(a∗ ) := 1 X pCF (a∗ , b), |Pv |     µpl (x | a∗ ) := 1−λ(a∗ ) S T (x, a∗ ) +λ(a∗ ) 1−S(F (x, a∗ )) −β S I(x, a∗ ) , b∈Pv with a design parameter β ∈ [0, 1]. One finally clips µpl to [0, 1] if needed. The set-operations (union/intersection/complement) are defined at the (T, I, F )-level by a chosen t-conorm/t-norm/negator and transported through the defuzzification. Table 3.33 explains how the classical trapezoidal fuzzy, intuitionistic fuzzy, and neutrosophic families arise as specializations of the Trapezoidal Plithogenic Set. Table 3.33: Classical trapezoidal fuzzy/intuitionistic/neutrosophic families as specializations of the Trapezoidal Plithogenic Set. Target model (recovered) Trapezoidal Fuzzy Set (TFS) [826–828] Trapezoidal Intuitionistic Fuzzy Set (TIFS) [829–831] Trapezoidal Vague Set Trapezoidal Neutrosophic Set (TrNS) [832–834] Trapezoidal Hesitant Fuzzy Sets [835, 836] Trapezoidal Picture Fuzzy Sets [837, 838] Trapezoidal Spherical Fuzzy Sets [839] Trapezoidal Quadripartitioned Neutrosophic Sets Trapezoidal Pentapartitioned Neutrosophic Sets Constraints inside TPS Recovery of the classical grade Set I(x, a) ≡ (0, 0, 0, 0), F (x, a) ≡ (0, 0, 0, 0) and pCF ≡ 0. Use (T, F ) only with the intuitionistic constraint t4 (x, a)+f4 (x, a) ≤ 1, and pCF ≡ 0. Use two value only with the vague constraint t4 (x, a) + f4 (x, a) ≤ 1, and pCF ≡ 0. Keep (T, I, F ) with t4 (x, a) + i4 (x, a)+f4 (x, a) ≤ 3 and pCF ≡ 0.  µ(x) = S T (x, a∗ ) gives the usual trapezoidal membership. µ(x) = S(T (x, a∗ )), ν(x) = S(F (x, a∗ )), with µ + ν ≤ 1. pCF ≡ 0 and Hesitant Fuzzy Constraints pCF ≡ 0. Picture Fuzzy Constraints pCF ≡ 0. Spherical Fuzzy Constraints pCF ≡ 0. Quadripartitioned Neutrosophic Constraints pCF ≡ 0. Pentapartitioned Neutrosophic Constraints A concrete example of this concept is provided below. Please refer to the references. T (x), I(x), F (x) are trapezoids; classical TrNS operations apply componentwise. Triple Values are trapezoids; classical operations apply componentwise. Triple Values are trapezoids; classical operations apply componentwise. Triple Values are trapezoids; classical operations apply componentwise. Quadruple Values are trapezoids; classical operations apply componentwise. Quinruple Values are trapezoids; classical operations apply componentwise. # Page. 122 ![Page Image](https://bcdn.docswell.com/page/Y76W2L3P7V.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.25.3 (Supplier sustainability evaluation as a Trapezoidal Plithogenic Set). Consider a procurement department that must evaluate two candidate suppliers U = {SA , SB } with respect to an attribute v = “sustainability level” whose plithogenic value-set is Pv = {low, medium, high}. The contradiction degrees between attribute values are chosen as pCF (·, ·) low medium high low 0 0.2 0.7 medium 0.2 0 0.3 high 0.7 0.3 0 so that pCF (a, a) = 0 and pCF (a, b) = pCF (b, a) for all a, b ∈ Pv . Suppose that the decision maker takes the dominant value a∗ = high. For each supplier x ∈ U and the dominant value a∗ , the Trapezoidal Plithogenic Set specifies trapezoidal truth-, indeterminacy-, and falsity-membership profiles (T, I, F ) on [0, 1] as follows: SA SB T (x, high) (0.6, 0.7, 0.9, 1.0) (0.3, 0.4, 0.5, 0.6) I(x, high) (0.1, 0.2, 0.2, 0.3) (0.2, 0.3, 0.3, 0.4) F (x, high) (0.0, 0.0, 0.1, 0.2) (0.3, 0.4, 0.5, 0.6) Interpreted linguistically, SA has a trapezoidal profile that is mostly in the “highly sustainable” region with small indeterminacy and low falsity, while SB is only moderately sustainable with larger falsity and indeterminacy. Using the arithmetic mean S(a, b, c, d) := a+b+c+d , 4 we obtain for SA at a∗ = high:  0.6 + 0.7 + 0.9 + 1.0 3.2 4 S T (SA , high) = = = , 4 4 5  0.1 + 0.2 + 0.2 + 0.3 0.8 1 S I(SA , high) = = = , 4 4 5  0 + 0 + 0.1 + 0.2 0.3 3 S F (SA , high) = = = . 4 4 40 The plithogenic global contradiction factor at the dominant value a∗ = high is λ(high) =  11 1 X 1 pCF (high, b) = 0 + 0.3 + 0.7 = . |Pv | 3 30 b∈Pv For a design parameter β ∈ [0, 1], the plithogenic inclusion grade is     µpl (x | a∗ ) := 1 − λ(a∗ ) S T (x, a∗ ) + λ(a∗ ) 1 − S(F (x, a∗ )) − β S I(x, a∗ ) , so, taking for instance β = 12 , we get  11  4 11  3 1 1 µpl (SA | high) = 1 − · + 1− − · . 30 5 30 40 2 5 We now simplify each term: 1− 11 19 = , 30 30 1− 3 37 = , 40 40 # Page. 123 ![Page Image](https://bcdn.docswell.com/page/G75M21LP74.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets  11  4 19 4 76 38 · = · = = , 30 5 30 5 150 75 11  3  11 37 407 1− = · = , 30 40 30 40 1200 1 1 1 · = . 2 5 10 1− Hence µpl (SA | high) = 38 407 1 608 407 120 895 179 + − = + − = = ≈ 0.746. 75 1200 10 1200 1200 1200 1200 240 Thus, under the trapezoidal plithogenic model, supplier SA attains a high inclusion grade (≈ 0.75) in the plithogenic sustainability set, quantitatively reflecting high truth, low falsity, limited indeterminacy, and the contradiction of the chosen attribute value “high” with other possible values. The data  TPS1 = U, v, Pv ; T, I, F ; pCF constitute a concrete Trapezoidal Plithogenic Set used for sustainable supplier evaluation. Example 3.25.4 (Chronic disease risk assessment as a Trapezoidal Plithogenic Set). Chronic disease risk assessment estimates an individual’s long-term likelihood of developing conditions using medical history, lifestyle, biomarkers, and demographics data [840]. Consider a hospital that stratifies patients with a chronic condition into risk categories based on multiple clinical indicators. Let U = {P1 , P2 } represent two patients, and let v = “cardiovascular risk level” with plithogenic value-set Pv = {low, moderate, high}. The contradiction degrees capture how far each category is from the desired “low risk” state: pCF (·, ·) low moderate high low 0 0.5 0.9 moderate 0.5 0 0.4 high 0.9 0.4 0 and the dominant value is chosen as a∗ = low, reflecting the clinical goal of keeping patients at low risk. For each patient x ∈ U and a∗ = low, the Trapezoidal Plithogenic Set prescribes trapezoidal truth-, indeterminacy-, and falsity-membership profiles for the statement “x is of low cardiovascular risk”: P1 P2 T (x, low) (0.5, 0.6, 0.8, 0.9) (0.1, 0.2, 0.3, 0.4) I(x, low) (0.1, 0.2, 0.2, 0.3) (0.2, 0.3, 0.4, 0.5) F (x, low) (0.1, 0.2, 0.3, 0.4) (0.5, 0.6, 0.7, 0.8) Patient P1 has relatively high truth-membership in “low risk” and moderate falsity-membership, whereas P2 has low truth-membership and high falsity-membership, reflecting a clinically higher risk. Using the same mean operator S(a, b, c, d) = a+b+c+d , we compute for P1 : 4  0.5 + 0.6 + 0.8 + 0.9 2.8 7 S T (P1 , low) = = = , 4 4 10  0.1 + 0.2 + 0.2 + 0.3 0.8 1 S I(P1 , low) = = = , 4 4 5 # Page. 124 ![Page Image](https://bcdn.docswell.com/page/9J2941LZER.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets  0.1 + 0.2 + 0.3 + 0.4 1.0 1 S F (P1 , low) = = = . 4 4 4 The contradiction factor at a∗ = low is λ(low) =  1.4 1 X 1 14 7 pCF (low, b) = 0 + 0.5 + 0.9 = = = . |Pv | 3 3 30 15 b∈Pv Fixing again β = 12 , the plithogenic inclusion grade of P1 in the low-risk plithogenic set is  7 7 7 1 1 1 µpl (P1 | low) = 1 − · + 1− − · . 15 10 15 4 2 5 We simplify term by term: 7 8 1 3 = , 1− = , 15 15 4 4   7 7 8 7 56 28 1− · = · = = , 15 10 15 10 150 75 7 1 7 3 21 7 1− = · = = , 15 4 15 4 60 20 1− 1 1 1 · = . 2 5 10 Therefore µpl (P1 | low) = 28 7 1 112 105 30 187 + − = + − = ≈ 0.623. 75 20 10 300 300 300 300 Thus patient P1 belongs to the plithogenic low-risk set with a moderate inclusion degree (about 0.62), determined jointly by the trapezoidal truth/indeterminacy/falsity profiles and the contradiction between the desired state “low” and the other risk categories. Collecting the ingredients  TPS2 = U, v, Pv ; T, I, F ; pCF , the hospital obtains a concrete Trapezoidal Plithogenic Set that can be used to rank patients, design follow-up intervals, or allocate monitoring resources under cardiovascular risk uncertainty. 3.26 Nonstandard Plithogenic Sets A nonstandard fuzzy set assigns each element a hyperreal membership near [0, 1], allowing infinitesimal underset/overset deviations and analysis via monads. A nonstandard neutrosophic set assigns hyperreal truth, indeterminacy, falsity degrees near [0, 1], permitting infinitesimal inconsistencies and refined uncertainty modeling variability [841]. A nonstandard plithogenic set maps items and attribute values to hyperreal membership vectors, aggregating evidence using contradiction-weighted t-norm/t-conorm operators internally. Definition 3.26.1 (Nonstandard primitives). [841] Let ∗R be a hyperreal field extending R. An element ε ∈ ∗R is infinitesimal if |ε| < n1 for all n ∈ N. Write the halo of [0, 1] as [0, 1]ns := { x ∈ ∗R | ∃y ∈ [0, 1] with x ≈ y }, and, more liberally, for a fixed positive infinitesimal δ, set the nonstandard band [0, 1](δ) := [−δ, 1 + δ] ⊂ ∗R. The standard part map st sends near–standard x ∈ ∗R to the unique y ∈ R with x ≈ y. # Page. 125 ![Page Image](https://bcdn.docswell.com/page/DEY4MZQNJM.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.26.2 (Nonstandard Neutrosophic Set (NSN)). [841] A nonstandard neutrosophic set on U is a triple of maps TA , IA , FA : U −→ [0, 1](δ) ⊂ ∗R, optionally subject to the near–standard neutrosophic bound TA (x) + IA (x) + FA (x) ≤ 3 + δ (x ∈ U ). Set operations act componentwise via an internal t-norm T∗ and t-conorm S∗ :  (T, I, F )A∪B (x) = S∗ (TA , TB ), S∗ (IA , IB ), T∗ (FA , FB ) ,  (T, I, F )A∩B (x) = T∗ (TA , TB ), T∗ (IA , IB ), S∗ (FA , FB ) ,  (T, I, F )A{ (x) = FA (x), 1 − IA (x), TA (x) . Definition 3.26.3 (Nonstandard Plithogenic Set (NSP)). Fix a plithogenic context  P S = P, v, P v, pdf, pCF , where P is a universe, v is an attribute with value set P v, the degree of appurtenance is s pdf : P × P v −→ [0, 1](δ) ⊂ (∗R)s , and the degree of contradiction is pCF : P v × P v −→ [0, 1](δ) ⊂ ∗R with pCF (a, a) = 0, pCF (a, b) = pCF (b, a). A nonstandard plithogenic set is a selection of (P, v, P v, pdf, pCF ) together with fixed internal tnorm/t-conorm (T∗ , S∗ ) used to aggregate across contradictory values. For a, b ∈ [0, 1](δ) and c ∈ [0, 1](δ) , define the DCF–weighted binary aggregator e c b := (1 − c) T∗ (a, b) + c S∗ (a, b), a∧ s and extend componentwise to vectors in [0, 1](δ) . Given a finite multiset of attribute values {u1 , . . . , um } ⊂ P v, the aggregated membership of x ∈ P is e c12 pdf (x; u2 ) ∧ e c13 · · · ∧ e c1m pdf (x; um ), µNSP (x; {uj }) := pdf (x; u1 ) ∧ c1j := pCF (u1 , uj ). When only one value u is considered, µNSP (x; {u}) = pdf (x; u). Table 3.34 presents the description of the nonstandard plithogenic set as a common generalization. Table 3.34: Nonstandard plithogenic set as a common generalization Model Nonstandard Fuzzy Set Membership form µ : U → [0, 1](δ) Nonstandard Intuitionistic Fuzzy Set (µ, ν) : 2 [0, 1](δ) U → Nonstandard Neutrosophic Set [842–844] (T, I, F ) : 3 [0, 1](δ) U → Nonstandard partitioned sophic Set Nonstandard partitioned sophic Set QuadriNeutro- q : U → [0, 1](δ) 4 PetaNeutro- p : U → [0, 1](δ) 5 A concrete example of this concept is provided below. Realization as NSP Take P v = {u0 }, s = 1, define pdf (x; u0 ) := µ(x), pCF (u0 , u0 ) = 0. Take P v = {u0 }, s = 2, set pdf (x; u0 ) := (µ(x), ν(x)), pCF (u0 , u0 ) = 0. Take P v = {u0 }, s = 3, set pdf (x; u0 ) := (T (x), I(x), F (x)), pCF (u0 , u0 ) = 0. Take P v = {u0 }, s = 4, set pdf (x; u0 ) := q(x), pCF (u0 , u0 ) = 0. Take P v = {u0 }, s = 5, set pdf (x; u0 ) := p(x), pCF (u0 , u0 ) = 0. # Page. 126 ![Page Image](https://bcdn.docswell.com/page/VJNYW3KV78.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Example 3.26.4 (Nonstandard plithogenic evaluation of renewable–energy projects). Consider two renewable projects P = {xwind , xsolar }, and a plithogenic attribute v = “sustainability criterion” with value set P v = {emissions, cost}. Work in a hyperreal field ∗R and fix a positive infinitesimal δ > 0; write [0, 1](δ) = [−δ, 1 + δ]. Set s = 1, so that pdf : P × P v −→ [0, 1](δ) returns a single (hyperreal) membership degree, and let pCF : P v × P v → [0, 1](δ) . Interpretation: pdf (x; u) is the (nonstandard) membership of project x in the plithogenic property “sustainable” under criterion u; pCF (u1 , u2 ) measures the contradiction between two criteria. Assume, for small infinitesimal ε with 0 < |ε|  δ, pdf (xwind ; emissions) = 0.94 + ε, pdf (xwind ; cost) = 0.68 − ε, pdf (xsolar ; emissions) = 0.90 − ε, pdf (xsolar ; cost) = 0.72 + ε, so each value lies in [0, 1](δ) and is “infinitesimally close” to a classical membership. Let the contradiction degree between criteria be c := pCF (emissions, cost) = 0.30 ∈ [0, 1](δ) , and choose internal t–norm/t–conorm T∗ (a, b) = min(a, b), S∗ (a, b) = max(a, b) (on hyperreals). The DCF–weighted aggregator of Definition of NSP is e c b := (1 − c) T∗ (a, b) + c S∗ (a, b), a∧ so for xwind with both criteria we obtain  e 0.30 (0.68 − ε). µNSP xwind ; {emissions, cost} = (0.94 + ε) ∧ Here T∗ = min(0.94 + ε, 0.68 − ε) = 0.68 − ε, so S∗ = max(0.94 + ε, 0.68 − ε) = 0.94 + ε,  µNSP xwind ; {emissions, cost} = (1 − 0.30)(0.68 − ε) + 0.30(0.94 + ε) = 0.7 · 0.68 + 0.3 · 0.94 + (−0.7 + 0.3)ε = 0.758 − 0.4 ε ∈ [0, 1](δ) . Its standard part is  st µNSP (xwind ; {emissions, cost}) = 0.758, interpreted as the classical aggregate sustainability score. The infinitesimal correction −0.4ε preserves additional hyperreal resolution (e.g. micro–uncertainty in expert judgments) that can be exploited by nonstandard analysis tools, while remaining plithogenic through the contradiction–weighted blend of criteria. # Page. 127 ![Page Image](https://bcdn.docswell.com/page/YE9PX9VVJ3.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets 3.27 Refined Plithogenic Set A Refined Plithogenic Set represents elements by multi-component truth, indeterminacy, falsity degrees with contradiction-aware weighting across attribute values and contexts. Refined Fuzzy Sets [403, 845, 846], Refined Intuitionistic Fuzzy Sets [844, 847–849], and Refined Neutrosophic Sets [119, 404, 850, 851] are also well-known in the existing literature. Definition 3.27.1 (Refined Plithogenic Set). Let U be a universe, v an attribute with value-set Pv (finite or countable), and let p, q, r ∈ N≥0 , s := p + q + r ≥ 1. Fix index sets IT = {1, . . . , p}, II = {1, . . . , q}, IF = {1, . . . , r}. A refinement signature is the triple (p, q, r). A refined plithogenic appurtenance (RPA) attached to v is a map pdf : U × Pv −→ [0, 1]s ,  pdf(x, a) = Ti (x, a)i∈IT , Ij (x, a)j∈II , Fk (x, a)k∈IF , where each listed component lies in [0, 1]. A plithogenic contradiction function is pCF : Pv × Pv −→ [0, 1]t , t ∈ N≥0 , satisfying reflexivity pCF (a, a) = 0 and symmetry pCF (a, b) = pCF (b, a). The Refined Plithogenic Set (RPS) determined by (U, v, Pv ; pdf, pCF ) is the data  RPS = U, v, Pv , (p, q, r), pdf, pCF . Semantics: for x ∈ U and a ∈ Pv , Ti (x, a) are refined truth-memberships, Ij (x, a) refined indeterminacymemberships, and Fk (x, a) refined falsity-memberships (when present), all evaluated for attribute value a. No further coupling is imposed at this level, except the usual range constraints [0, 1]; modelspecific constraints (e.g. intuitionistic or neutrosophic) appear as specializations below. Remark 3.27.2 (Typical scalarization/aggregation (optional)). In decision tasks, one often fixes a dominant value a∗ ∈ Pv and an aggregation Φ : [0, 1]s × [0, 1]t → [0, 1] that is monotone in each argument and Φ(·, 0) reduces to the underlying refined score. A plithogenic inclusion grade of x at a∗ is  µpl (x | a∗ ) := Φ pdf(x, a∗ ), pCF (a∗ , ·) , with unions/intersections realized via a chosen pair of t-conorm/t-norm at the aggregated level. The overview of refined fuzzy, intuitionistic fuzzy, neutrosophic, quadripartitioned neutrosophic, and pentapartitioned neutrosophic sets is presented in Table 3.35. Write sup T := supi∈IT Ti , etc. By choosing (p, q, r) and constraints as follows, the refined fuzzy, refined intuitionistic fuzzy, and refined neutrosophic families are obtained as exact special cases (take t = 0 or pCF ≡ 0 when “no contradiction”). A concrete example of this concept is provided below. Example 3.27.3 (Hospital triage with refined evidence channels). Consider emergency-room pneumonia triage. Let the universe be U = {pA , pB } (two patients). Let the attribute be v = “evidence source” with value–set Pv = {Img, Lab, Sym} = (chest imaging, laboratory markers, symptoms). Refinement signature (p, q, r) = (2, 1, 1) so that each refined appurtenance vector has (T1 , T2 ; I1 ; F1 ) ∈ [0, 1]4 . For x ∈ U and a ∈ Pv , the refined plithogenic degree pdf(x, a) is: # Page. 128 ![Page Image](https://bcdn.docswell.com/page/GE8D29P4ED.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.35: Refined fuzzy, intuitionistic fuzzy, neutrosophic, quadripartitioned neutrosophic, and pentapartitioned neutrosophic sets as special cases of the Refined Plithogenic Set. Target refined model Refined Fuzzy Set (RFS) [403, 845, 852] Refined Intuitionistic Fuzzy Set (RIFS) [847–849] Refined Neutrosophic Set (RNS) [404, 853–855] Refined Quadripartitioned Neutrosophic Set (RQNS) Refined Pentapartitioned Neutrosophic Set (RPNS) pA pB Constraints and recovery inside RPS Only truth components Ti ∈ [0, 1] with pCF ≡ 0. Any monotone aggregation of (Ti ) yields the usual refined fuzzy grade. Paired components (Ti , Ii ) satisfying Ti + Ii ≤ 1 for each i, with pCF ≡ 0. Classical refined intuitionistic fuzzy sets are recovered by componentwise operations on (Ti , Ii ). Refined truth, indeterminacy, and falsity components Ti , Ij , Fk ∈ [0, 1] with sup T + sup I + sup F ≤ 3. Setting pCF ≡ 0 recovers the standard refined neutrosophic framework (single–valued case when there is exactly one T , I, and F ). The families T, I, F are grouped into four sub–blocks (quadripartitions), each block obeying the usual neutrosophic bounds (for example, blockwise sums ≤ 1). With pCF ≡ 0, one obtains refined quadripartitioned neutrosophic sets. Truth, indeterminacy, and falsity components are organized into five sub–blocks (pentapartitions) satisfying the corresponding neutrosophic constraints; taking pCF ≡ 0 yields refined pentapartitioned neutrosophic sets as a special case. Img (0.80, 0.75; 0.10; 0.05) (0.45, 0.40; 0.25; 0.30) Lab (0.60, 0.55; 0.20; 0.10) (0.50, 0.48; 0.22; 0.28) Sym (0.70, 0.65; 0.15; 0.12) (0.60, 0.50; 0.25; 0.20) A symmetric contradiction map pCF : Pv × Pv → [0, 1] is chosen as   0 0.2 0.3 pCF = 0.2 0 0.1 (rows/cols ordered as Img, Lab, Sym). 0.3 0.1 0 Fix the dominant value δ = Img and use contradiction–aware weights w(a | δ) := 1 − pCF (a, δ), giving X w(Img) = 1, w(Lab) = 0.8, w(Sym) = 0.7, w(a | δ) = 2.5. a Define the δ–relative refined degree of x componentwise by the weighted mean P a∈Pv w(a | δ) (Ti , I1 , F1 )(x, a) P (Ti , I1 , F1 )(δ) (x) = (i = 1, 2). a∈Pv w(a | δ) Numerical aggregation (to three decimals): pA pB (δ) T1 0.80+0.48+0.49 = 0.708 2.5 0.45+0.40+0.42 = 0.508 2.5 (δ) T2 0.75+0.44+0.455 = 0.658 2.5 0.40+0.384+0.35 = 0.454 2.5 (δ) I1 0.10+0.16+0.105 = 0.146 2.5 0.25+0.176+0.175 = 0.240 2.5 (δ) F1 0.05+0.08+0.084 = 0.086 2.5 0.30+0.224+0.14 = 0.266 2.5 # Page. 129 ![Page Image](https://bcdn.docswell.com/page/LELM2W1V7R.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets For a concrete scalarization, take 2 Φ(T1 , T2 , I, F ) := 0.5 · T1 +T + 0.2 · (1 − I) + 0.3 · (1 − F ). 2 Then Φ(pA ) = 0.5·0.683+0.2·0.854+0.3·0.914 = 0.787, Φ(pB ) = 0.5·0.481+0.2·0.760+0.3·0.734 = 0.613. Hence, under imaging–dominant context, pA has a higher refined plithogenic support. Example 3.27.4 (Supplier selection with ESG–dominant policy). A retailer evaluates two suppliers U = {sA , sB }. Attribute v = “criterion” with Pv = {ESG, Qual, Deliv} (environmental–social–governance, product quality, delivery reliability). Use refinement signature (p, q, r) = (2, 1, 1) and record refined appurtenances: sA sB ESG (0.85, 0.80; 0.10; 0.05) (0.60, 0.55; 0.20; 0.20) Qual (0.75, 0.70; 0.12; 0.08) (0.80, 0.78; 0.10; 0.10) Contradiction matrix (symmetric):   0 0.2 0.3 pCF = 0.2 0 0.1 0.3 0.1 0 Deliv (0.60, 0.55; 0.15; 0.12) (0.75, 0.70; 0.12; 0.12) (rows/cols ordered as ESG, Qual, Deliv). With dominant value δ = ESG, weights are w(ESG) = 1, w(Qual) = 0.8, w(Deliv) = 0.7, X w(a | δ) = 2.5. a Aggregate componentwise: (δ) T1 sA sB 0.85+0.60+0.42 = 0.748 2.5 0.60+0.64+0.525 = 0.706 2.5 (δ) (δ) T2 0.80+0.56+0.385 = 0.698 2.5 0.55+0.624+0.490 = 0.666 2.5 I1 0.10+0.096+0.105 = 0.120 2.5 0.20+0.080+0.084 = 0.146 2.5 (δ) F1 0.05+0.064+0.084 = 0.079 2.5 0.20+0.080+0.084 = 0.146 2.5 2 Using the same scalarization Φ(T1 , T2 , I, F ) = 0.5 · T1 +T + 0.2(1 − I) + 0.3(1 − F ), we obtain 2 Φ(sA ) = 0.5·0.723+0.2·0.880+0.3·0.921 = 0.814, Φ(sB ) = 0.5·0.686+0.2·0.854+0.3·0.854 = 0.770. Under an ESG–dominant policy with contradiction–aware weighting, sA is preferred. 3.28 Subset–Valued Plithogenic Sets A subset–valued plithogenic set maps each item and attribute value to subsets of membership vectors, aggregating contradictions via degree-of-contradiction weights. Definition 3.28.1 (Subset–Valued Fuzzy Set (SVFS)). Let X be a nonempty universe and P([0, 1]) the powerset of [0, 1]. A subset–valued fuzzy set on X is a map A : X → P([0, 1]) \ {∅}, x 7−→ A(x), where A(x) ⊆ [0, 1] is the (possibly non-singleton) set of admissible membership degrees of x. When each A(x) is a singleton, one recovers an ordinary fuzzy set. # Page. 130 ![Page Image](https://bcdn.docswell.com/page/4JMY89WNJW.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.28.2 (Subset–Valued Neutrosophic Set (SVNS)). [841] Let X be a nonempty universe. A subset–valued neutrosophic set on X assigns to each x ∈ X a triple  A(x) = TA (x), IA (x), FA (x) , with TA (x), IA (x), FA (x) ∈ P([0, 1]) \ {∅}. Writing inf and sup for the usual bounds (with the convention on closedness as needed), one requires 0 ≤ inf TA (x) + inf IA (x) + inf FA (x) ≤ sup TA (x) + sup IA (x) + sup FA (x) ≤ 3 (∀x ∈ X). When each of TA (x), IA (x), FA (x) is a singleton, this reduces to a single–valued neutrosophic set. Definition 3.28.3 (Subset-Valued Plithogenic Set (SVPS)). A plithogenic context is a tuple  P S = P, v, P v, pdf, pCF , where • P is a universe of items; • v is a fixed attribute with value set P v; • pdf : P × P v → P([0, 1]s ) \ {∅} (for some fixed s ∈ N) is a degree–of–appurtenance mapping that assigns to each (x, u) ∈ P × P v a subset pdf (x; u) ⊆ [0, 1]s of admissible membership vectors; • pCF : P v ×P v → [0, 1] is the degree of contradiction, satisfying pCF (a, a) = 0 and pCF (a, b) = pCF (b, a) for all a, b ∈ P v. A subset–valued plithogenic set over P S is given by the data (P, v, P v, pdf, pCF ). For each item x ∈ P and each attribute value u ∈ P v, the subset pdf (x; u) ⊆ [0, 1]s collects all admissible membership vectors of x under u. 3 Example 3.28.4 (Subset–valued plithogenic set for medical treatment side–effect risk). Let the universe of items be P = {DrugA, DrugB}, and consider a plithogenic attribute v = “evidence type”, Pv = {Trial, Post}, where Trial = controlled clinical trials, Post = post–marketing surveillance reports. We take s = 2, with components µ1 = “severe side–effect risk”, µ2 = “mild side–effect risk”, so that each membership vector lies in [0, 1]2 . For each (x, u) ∈ P × Pv , the map pdf : P × Pv −→ P([0, 1]2 ) \ {∅} 3 In applications, plithogenic operations (e.g. conjunction or aggregation) are often defined by combining elements of pdf (x; u) via a fixed t–norm / t–conorm, weighted by the contradiction degree pCF . For the representation and reduction results in this section, only the subset–valued mapping pdf and the function pCF are needed. # Page. 131 ![Page Image](https://bcdn.docswell.com/page/PJR95GY479.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.36: Subset–valued plithogenic set as a common generalization. Model Image of pdf (x; u) s Specialization of SVPS Subset-valued fuzzy set (SVFS) P([0, 1]) \ {∅} 1 Subset-valued intuitionistic fuzzy set (SVIFS) P([0, 1]2 ) \ {∅} 2 Subset-valued neutrosophic set (SVNS) [841] P([0, 1]3 ) \ {∅} 3 Subset-valued plithogenic set (SVPS) P([0, 1]s ) \ {∅} s≥1 Single attribute value u0 ; set pdf (x; u0 ) = A(x); take pCF ≡ 0. Single value u0 ; pdf (x; u0 ) is a subset of IF pairs (µ, ν) satisfying the usual intuitionistic bounds; pCF ≡ 0. Single value u0 ; pdf (x; u0 ) is a subset of neutrosophic triples (T, I, F ) obeying neutrosophic constraints; pCF ≡ 0. General case: arbitrary s, multiple attribute values in Pv , and nontrivial contradiction function pCF : Pv × Pv → [0, 1]. assigns a subset of possible pairs (µ1 , µ2 ) obtained from different studies or expert groups. A possible specification is:  pdf (DrugA; Trial) = (0.10, 0.30), (0.15, 0.35) ,  pdf (DrugA; Post) = (0.20, 0.40), (0.25, 0.50) ,  pdf (DrugB; Trial) = (0.05, 0.20), (0.08, 0.25) ,  pdf (DrugB; Post) = (0.12, 0.28), (0.18, 0.35) . The plithogenic degree of contradiction between evidence types is modeled by pCF : Pv × Pv → [0, 1], pCF (u, u) = 0, pCF (u1 , u2 ) = pCF (u2 , u1 ), for example pCF (Trial, Post) = 0.4, pCF (Trial, Trial) = pCF (Post, Post) = 0. Interpretation. For each drug x and evidence type u, the subset pdf (x; u) collects several plausible risk profiles (µ1 , µ2 ) coming from different datasets or statistical models, rather than a single fixed pair. The plithogenic mechanism then aggregates these subset–valued degrees across Trial and Post using the contradiction weight pCF (Trial, Post): high disagreement between evidence sources (large pCF ) yields more cautious or conservative combined risk assessments. Thus (P, v, Pv , pdf, pCF ) forms a subset–valued plithogenic set describing real–world medical side–effect uncertainty. 3.29 Picture Plithogenic Set A Picture Plithogenic Set models neutral and other degrees with attribute-based contradiction–weighted aggregation, capturing complex opinions in uncertain environments. # Page. 132 ![Page Image](https://bcdn.docswell.com/page/PEXQKXGZJX.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Definition 3.29.1 (Picture Fuzzy Set (PFS)). [7] Let U be a nonempty universe. A picture fuzzy set A on U is a family n o A = hx, µA (x), ηA (x), νA (x)i : x ∈ U , where, for every x ∈ U , µA (x), ηA (x), νA (x) ∈ [0, 1] denote respectively the positive, neutral, and negative degrees of x with respect to A, subject to 0 ≤ µA (x) + ηA (x) + νA (x) ≤ 1. The remaining quantity πA (x) := 1 − µA (x) − ηA (x) − νA (x) is called the refusal degree of x in A. Definition 3.29.2 (Picture Neutrosophic Set (PNS)). [445] Let U be a nonempty universe. A picture neutrosophic set B on U is a family n o B = hx, TB (x), IB (x), FB (x)i : x ∈ U , where, for each x ∈ U , TB (x), IB (x), FB (x) ∈ [0, 1] denote respectively the truth, indeterminacy, and falsity degrees of x with respect to B, and 0 ≤ TB (x) + IB (x) + FB (x) ≤ 3. The term “picture” emphasizes that the triple (TB (x), IB (x), FB (x)) gives a three–fold view (positive / neutral / negative) of the evaluation, while mathematically B is a single–valued neutrosophic set. Definition 3.29.3 (Picture Plithogenic Set (PPlS)). Let P be a nonempty universe, v an attribute, and Pv a nonempty set of attribute values of v. Fix integers t ≥ 1 and r ≥ 0. A picture plithogenic set on (P, v, Pv ) is a tuple P Spic = (P, v, Pv , pdf, pCF ), where: • the picture plithogenic degree of appurtenance function pdf : P × Pv −→ [0, 1]t+1 is written as pdf (x, a) =  α1 (x, a), . . . , αt (x, a), η(x, a) , with each component in [0, 1]; here  α(x, a) := α1 (x, a), . . . , αt (x, a) is the underlying plithogenic membership vector, and η(x, a) := αt+1 (x, a) is the additional picture–neutral degree of x with respect to v = a; • the degree of contradiction function (DCF) pCF : Pv × Pv −→ [0, 1]r satisfies, for all a, b ∈ Pv , pCF (a, a) = 0 ∈ [0, 1]r , pCF (a, b) = pCF (b, a). # Page. 133 ![Page Image](https://bcdn.docswell.com/page/3EK95W4VED.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Table 3.37: Reductions of picture fuzzy, picture neutrosophic, and plithogenic sets to the picture plithogenic framework. Source model Picture fuzzy set on P [7, 365] Picture neutrosophic set on P [445] Plithogenic set on P Representation as a picture plithogenic set Choose t = 2, r = 0, Pv = {a∗ } and set pdf (x, a∗ ) = (µ(x), ν(x), η(x)) (positive, negative, neutral). Choose t = 3, r = 0, Pv = {b∗ } and set pdf (x, b∗ ) = (T (x), I(x), F (x), 0); the first three coordinates give the neutrosophic triple and the neutral channel is fixed to 0. Given pdf0 : P × Pv → [0, 1]t , define pdf (x, a) = (pdf0 (x, a), 0); the extra neutral coordinate is always 0. Thus a picture plithogenic set is a plithogenic set whose membership vectors have t “ordinary” components together with one extra neutral component, stored in the (t + 1)–st coordinate. Table 3.37 presents a concise overview of how picture fuzzy sets, picture neutrosophic sets, and plithogenic sets can each be represented within the unified picture plithogenic framework. Theorem 3.29.4 (Picture Plithogenic Set as a common generalization). The class of picture plithogenic sets strictly generalizes the classes of picture fuzzy sets, picture neutrosophic sets, and plithogenic sets. More precisely: 1. Every picture fuzzy set on P can be represented as a picture plithogenic set with suitable (t, r), Pv , pdf , and pCF . 2. Every picture neutrosophic set on P can be represented as a picture plithogenic set with suitable (t, r), Pv , pdf , and pCF . 3. Every plithogenic set on P can be embedded into a picture plithogenic set by appending a neutral coordinate equal to zero. Proof. (1) Recovery of picture fuzzy sets. Let A be a picture fuzzy set on P : A= n o hx, µA (x), ηA (x), νA (x)i : x ∈ P , with µA (x), ηA (x), νA (x) ∈ [0, 1], 0 ≤ µA (x) + ηA (x) + νA (x) ≤ 1. Choose t := 2, r := 0, Pv := {a∗ } and define pCF (a∗ , a∗ ) := 0. Define pdf : P × Pv → [0, 1]t+1 = [0, 1]3 by   pdf (x, a∗ ) := α1 (x, a∗ ), α2 (x, a∗ ), η(x, a∗ ) := µA (x), νA (x), ηA (x) . Then for each x ∈ P , we have α1 (x, a∗ ) = µA (x), α2 (x, a∗ ) = νA (x), η(x, a∗ ) = ηA (x), # Page. 134 ![Page Image](https://bcdn.docswell.com/page/L73WK1DQ75.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets and the normalization 0 ≤ α1 (x, a∗ ) + η(x, a∗ ) + α2 (x, a∗ ) ≤ 1 is exactly the constraint of a picture fuzzy triple (µA (x), ηA (x), νA (x)). Thus A P Spic := (P, v, Pv , pdf, pCF ) is a picture plithogenic set whose coordinates recover A. Conversely, suppose we are given a picture plithogenic set (P, v, Pv , pdf, pCF ) with t = 2, r = 0, Pv = {a∗ }, and such that, for each x ∈ P ,  pdf (x, a∗ ) = α1 (x, a∗ ), α2 (x, a∗ ), η(x, a∗ ) ∈ [0, 1]3 satisfies Define 0 ≤ α1 (x, a∗ ) + η(x, a∗ ) + α2 (x, a∗ ) ≤ 1. µA (x) := α1 (x, a∗ ), νA (x) := α2 (x, a∗ ), ηA (x) := η(x, a∗ ). Then 0 ≤ µA (x) + ηA (x) + νA (x) ≤ 1, so A := n hx, µA (x), ηA (x), νA (x)i : x ∈ P o is a picture fuzzy set. Hence picture fuzzy sets are exactly those picture plithogenic sets with t = 2, r = 0, a singleton Pv , and the constraint α1 + η + α2 ≤ 1; this proves (1). (2) Recovery of picture neutrosophic sets. Let B be a picture neutrosophic set on P : n o B = hx, TB (x), IB (x), FB (x)i : x ∈ P , with TB (x), IB (x), FB (x) ∈ [0, 1], 0 ≤ TB (x) + IB (x) + FB (x) ≤ 3. Choose t := 3, r := 0, Pv := {b∗ }, and pCF (b∗ , b∗ ) := 0. Define pdf : P × Pv → [0, 1]t+1 = [0, 1]4 by   pdf (x, b∗ ) := α1 (x, b∗ ), α2 (x, b∗ ), α3 (x, b∗ ), η(x, b∗ ) := TB (x), IB (x), FB (x), 0 . Then, for each x ∈ P , α1 (x, b∗ ) = TB (x), α2 (x, b∗ ) = IB (x), α3 (x, b∗ ) = FB (x), η(x, b∗ ) = 0. Thus the first t = 3 components of pdf (x, b∗ ) reproduce the neutrosophic triple (TB , IB , FB ), and the extra neutral coordinate is identically 0. Hence B P Spic := (P, v, Pv , pdf, pCF ) is a picture plithogenic set that encodes B. # Page. 135 ![Page Image](https://bcdn.docswell.com/page/87DK3XPWJG.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets Conversely, suppose we are given a picture plithogenic set with t = 3, and such that r = 0, Pv = {b∗ },  pdf (x, b∗ ) = α1 (x, b∗ ), α2 (x, b∗ ), α3 (x, b∗ ), η(x, b∗ ) ∈ [0, 1]4 , with the additional constraint 0 ≤ α1 (x, b∗ ) + α2 (x, b∗ ) + α3 (x, b∗ ) ≤ 3, and where, for example, η(x, b∗ ) = 0 for all x. Define TB (x) := α1 (x, b∗ ), IB (x) := α2 (x, b∗ ), FB (x) := α3 (x, b∗ ). Then B as above is a picture neutrosophic set. Therefore picture neutrosophic sets correspond to picture plithogenic sets with t = 3, r = 0, singleton Pv , and neutral coordinate identically zero (or otherwise separated from the neutrosophic triple); this proves (2). (3) Embedding plithogenic sets. Let P S = (P, v, Pv , pdf0 , pCF ) be any plithogenic set with pdf0 : P × Pv → [0, 1]t ,  pdf0 (x, a) = α1 (x, a), . . . , αt (x, a) , and pCF : Pv × Pv → [0, 1]r reflexive and symmetric. Define pdf : P × Pv → [0, 1]t+1 by appending a neutral coordinate equal to zero:  pdf (x, a) := α1 (x, a), . . . , αt (x, a), η(x, a) , η(x, a) := 0. Then (P, v, Pv , pdf, pCF ) is a picture plithogenic set with the same attribute system and contradiction structure as P S, and with the underlying plithogenic part unchanged: αi (x, a) in P S ←→ αi (x, a) in P Spic for i = 1, . . . , t. Conversely, any picture plithogenic set (P, v, Pv , pdf, pCF ) for which the neutral coordinate satisfies η(x, a) ≡ 0 for all (x, a) ∈ P × Pv determines a plithogenic set by simply discarding the last coordinate:  pdf0 (x, a) := α1 (x, a), . . . , αt (x, a) . Thus plithogenic sets are exactly those picture plithogenic sets whose neutral coordinate is identically zero, proving (3). Combining (1), (2), and (3), we conclude that picture fuzzy sets, picture neutrosophic sets, and plithogenic sets appear as particular, well–defined subclasses of picture plithogenic sets obtained by suitable choices of the parameters (t, r), the value set Pv , the contradiction function pCF , and algebraic constraints on the coordinates of pdf . Therefore the picture plithogenic framework strictly generalizes all three structures. # Page. 136 ![Page Image](https://bcdn.docswell.com/page/VJPK4PNXE8.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets A concrete example of this concept is provided below. Example 3.29.5 (Public transport policy evaluation). Consider a city that evaluates two candidate public transport policies P = {p1 = “new bus lanes”, p2 = “fare discount”}. Let v be the attribute citizen attitude, with value set Pv = {a+ = support, a− = oppose}. We take t = 2, r = 1 and interpret, for each (x, a) ∈ P × Pv ,  pdf (x, a) = α1 (x, a), α2 (x, a), η(x, a) , where α1 is the positive degree, α2 the negative degree, and η a picture–neutral degree (“undecided / mixed”). For instance, based on a survey we may set pdf (p1 , a+ ) = (0.7, 0.1, 0.1), pdf (p1 , a− ) = (0.2, 0.6, 0.1), pdf (p2 , a+ ) = (0.6, 0.2, 0.1), pdf (p2 , a− ) = (0.1, 0.5, 0.2). To encode plithogenic contradiction between support and oppose we define pCF (a+ , a+ ) = pCF (a− , a− ) = 0, pCF (a+ , a− ) = pCF (a− , a+ ) = 0.9. Thus the picture plithogenic set policy P Spic = P, v, Pv , pdf, pCF  models, in a single framework, positive / negative / neutral citizen opinions together with their strong contradiction between “support” and “oppose”. Example 3.29.6 (Online product recommendation). Let P = {x1 , x2 , x3 } be three candidate smartphones on an e–commerce platform. Let v be the attribute customer sentiment category with Pv = {a1 = battery, a2 = camera}. We again choose t = 2, r = 1 and write pdf (x, a) = (α1 (x, a), α2 (x, a), η(x, a)), where α1 is the positive sentiment degree, α2 the negative sentiment degree, and η the neutral (“mixed / no clear opinion”) degree for product x on aspect a. Example membership assessments: pdf (x1 , a1 ) = (0.8, 0.1, 0.05), pdf (x1 , a2 ) = (0.4, 0.3, 0.2), pdf (x2 , a1 ) = (0.5, 0.3, 0.1), pdf (x2 , a2 ) = (0.7, 0.1, 0.1), pdf (x3 , a1 ) = (0.3, 0.5, 0.1), pdf (x3 , a2 ) = (0.4, 0.4, 0.1). # Page. 137 ![Page Image](https://bcdn.docswell.com/page/2EVVX2K3EQ.jpg) Chapter 3. Dynamic Reviews and Results of Uncertain Sets To model that “battery” and “camera” are only mildly contradictory as aspects, we define pCF (a1 , a1 ) = pCF (a2 , a2 ) = 0, pCF (a1 , a2 ) = pCF (a2 , a1 ) = 0.3. Then the picture plithogenic set phone P Spic = P, v, Pv , pdf, pCF  captures, for each phone and each aspect, the positive / negative / neutral picture of customer opinions together with a contradiction–aware interaction between the aspects “battery” and “camera” when aggregating overall recommendations. # Page. 138 ![Page Image](https://bcdn.docswell.com/page/57GLVRQYEL.jpg) Chapter 4 Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set In this chapter, we present a unifying framework for Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and other uncertainty-handling sets. 4.1 Uncertain Set An Uncertain Set is any set-theoretic model assigning graded, possibly multi-component membership degrees to elements, generalizing fuzzy, intuitionistic, neutrosophic, plithogenic and related uncertainty frameworks unified [856]. Definition 4.1.1 (Uncertain Set). Definition ??. Fix [856] Let U be the collection of all Uncertain Models as in U ∈ U, Dom(U ) ⊆ [0, 1]r for some integer r ≥ 1, and let X be a nonempty base set (universe of discourse). An Uncertain Set of type U on X is a pair AU = (X, µ), where µ : X −→ Dom(U ) assigns to each element x ∈ X a U –membership degree µ(x) ∈ Dom(U ). Equivalently, once the base set X and the Uncertain Model U are fixed, we may identify the Uncertain Set with its membership function and simply write AU : X −→ Dom(U ), x 7−→ µ(x), and view the collection of all Uncertain Sets of type U on X as the function space X Dom(U ) = { µ | µ : X → Dom(U ) }. In this sense, an Uncertain Set is a U –labeling of the base set X by membership–degree tuples taken from Dom(U ). 137 # Page. 139 ![Page Image](https://bcdn.docswell.com/page/4EQY6VK6JP.jpg) Chapter 4. Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set Remark 4.1.2 (Recovery of classical fuzzy–type sets). Let X be a nonempty set and let AU = (X, µ) be an Uncertain Set of type U . 1. (Fuzzy Set) Take U = Fuzzy with Dom(U ) = [0, 1] = [0, 1]1 . Then an Uncertain Set of type U is exactly a fuzzy set in the sense of Zadeh, since µ : X → [0, 1] is the usual fuzzy membership function. 2. (Intuitionistic Fuzzy Set) Take U = Intuitionistic Fuzzy with  Dom(U ) = (µ, ν) ∈ [0, 1]2 | µ + ν ≤ 1 ⊆ [0, 1]2 . Then AU = (X, µ) coincides with an intuitionistic fuzzy set, because for each x ∈ X,  µ(x) = µA (x), νA (x) ∈ [0, 1]2 satisfies µA (x) + νA (x) ≤ 1. 3. (Neutrosophic Set) Take U = Neutrosophic with  Dom(U ) = (T, I, F ) ∈ [0, 1]3 | 0 ≤ T + I + F ≤ 3 ⊆ [0, 1]3 . Then AU = (X, µ) is exactly a single–valued neutrosophic set, since  µ(x) = TA (x), IA (x), FA (x) ∈ [0, 1]3 with 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3 for all x ∈ X. 4. (Plithogenic Set) For a Plithogenic Model U = Plithogenic with degree–domain n  Dom(U ) = v, pdf(x, v), pCF(v1 , v2 ) o v ∈ Pv , pdf(x, v) ∈ [0, 1]s , pCF(v1 , v2 ) ∈ [0, 1]t ⊆ [0, 1]s+t+` , an Uncertain Set of type U on X reproduces a Plithogenic Set on X, since each µ(x) ∈ Dom(U ) encodes the Plithogenic degrees associated with x ∈ X. Thus, by choosing different Uncertain Models U ∈ U and their corresponding domains Dom(U ) ⊆ [0, 1]r , the general notion of an Uncertain Set in Definition 4.1.1 unifies fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and many other existing uncertainty–set frameworks. 4.2 Functional Set A functorial set assigns to each object of a category a corresponding set and transports its elements along morphisms via the functor [856]. These constructions not only encompass fuzzy, intuitionistic, neutrosophic, plithogenic, and related uncertainty frameworks, but also generalize set concepts that possess features beyond uncertainty. # Page. 140 ![Page Image](https://bcdn.docswell.com/page/KJ4W4MPM71.jpg) Chapter 4. Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set Definition 4.2.1 (Functorial Set). [856] Let C be a category and F : C −→ Set be a (covariant) endofunctor. For any object X ∈ Ob(C), an F -set over X is an element s ∈ F (X). We denote the collection of all F -sets over X simply by F (X). A morphism f : X → Y in C induces a pushforward F (f ) : F (X) −→ F (Y ), s 7→ F (f )(s). A concrete example of this concept is provided below. Example 4.2.2 (City logistics: purchase orders functor). Let C be the (small) category generated by the cities T = Tokyo, O = Osaka, F = Fukuoka and route–morphisms f : T → O, g : O → F, g ◦ f : T → F, together with identities idT , idO , idF . Define a functor F : C → Set by: F (T ) = {PO T −101 : (50), T −102 : (30)}, F (O) = {PO O−201 : (40)}, F (F ) = ∅, where “PO X − · : (q)” denotes a purchase order for destination X with quantity q. For a route h : X → Y , define F (h) : F (X) → F (Y ) by retagging the destination while preserving the item and quantity: F (h) : PO X −n : (q) 7−→ PO Y −n : (q). Then functoriality holds: F (idX ) = idF (X) (trivial retagging), F (g ◦ f ) = F (g) ◦ F (f ) (same two-step retagging). Concrete check on PO T −101 : (50): F (f ) : PO T −101 : (50) 7→ PO O−101 : (50), F (g) : PO O−101 : (50) 7→ PO F −101 : (50), so (F (g) ◦ F (f ))(PO T −101 : (50)) = PO F −101 : (50), which equals F (g ◦ f )(PO T −101 : (50)) by the definition of g ◦ f : T → F . Thus F models functorial transport of orders along the logistics network. Example 4.2.3 (Cross–timezone calendars: meeting slots functor). Let T be the category whose objects are fixed UTC offsets {0, +9, −5} (UTC, JST, EST), and whose morphisms are offset–changes given by hα→β : α → β, hβ→γ ◦ hα→β = hα→γ , hα→α = idα . Define G : T → Set by letting G(α) be the set of local meeting timestamps at offset α (as naive datetimes). For hα→β : α → β, define G(hα→β ) to convert local times: G(hα→β )(YYYY-MM-DD hh:mm at α) := YYYY-MM-DD hh:mm + (β − α) hours, now at β. Functoriality is immediate: G(idα ) = idG(α) , G(hβ→γ ◦ hα→β ) = G(hβ→γ ) ◦ G(hα→β ) because time–shift addition is associative. Numerical check: Slot s = 2025-11-08 10:00 at UTC ∈ G(0). # Page. 141 ![Page Image](https://bcdn.docswell.com/page/LE1Y489Y7G.jpg) Chapter 4. Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set Then G(h0→+9 )(s) = 2025-11-08 19:00 at JST, so G(h+9→−5 ) = shift by − 14 h,  G(h+9→−5 ) ◦ G(h0→+9 ) (s) = 2025-11-08 05:00 at EST. Directly, G(h0→−5 )(s) = 2025-11-08 05:00 at EST, hence G(h+9→−5 ◦ h0→+9 ) = G(h+9→−5 ) ◦ G(h0→+9 ) on s. Thus G functorially transports meeting slots across timezones. Example 4.2.4 (Geospatial containment: points–of–interest functor). Let (R, ⊆) be a poset of regions with inclusions as morphisms, forming a category: objects are regions X, morphisms are iX⊆Y : X → Y when X ⊆ Y . Consider the chain Ward = Shinjuku ⊆ City = Tokyo ⊆ Country = Japan. Define H : R → Set by H(X) = {POIs located in X}. For iX⊆Y : X → Y , define H(iX⊆Y ) : H(X) → H(Y ) as the inclusion on elements (same POI, now regarded inside the larger region). Concrete data: H(Shinjuku) = {POI1: park, POI2: museum, POI3: station}, H(Tokyo) = {POI1, POI2, POI3, POI4: gallery}. Then H(iShinjuku⊆Tokyo )(POI2) = POI2 ∈ H(Tokyo). Functoriality: H(idX ) = idH(X) (identity inclusion), H(iX⊆Z ) = H(iY ⊆Z ) ◦ H(iX⊆Y ) whenever X ⊆ Y ⊆ Z. Numerically,   H(iShinjuku⊆Japan )(POI3) = POI3 = H(iTokyo⊆Japan ) ◦ H(iShinjuku⊆Tokyo ) (POI3). Thus H functorially embeds POIs along region inclusions, preserving composition. 4.3 Other Uncertain Sets and Future Works This section presents additional uncertainty–oriented frameworks that are frequently studied alongside Fuzzy Sets, Intuitionistic Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets. Looking ahead, it is expected that further extensions and new theoretical developments will emerge by combining these auxiliary concepts with the Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic models examined in this paper. Note that these can also be generalized within the Functorial Set framework. • Near Set [857–859]: A near set groups objects that are perceptually or descriptively similar under tolerance relations, supporting approximate classification, retrieval, and matching tasks. Related concepts such as Fuzzy Near Sets [860–863] are also known in the literature. • Weighted Set [864, 865]: A weighted set equips each element with a positive importance coefficient, enabling prioritized aggregation, ranking, or decision analysis over items. Related concepts such as Weighted Fuzzy Sets [716, 866, 867], Weighted Intuitionistic Fuzzy Sets [868, 869], Weighted Rough Sets [870–872], and Weighted Neutrosophic Sets [873, 874] are also known in the literature. # Page. 142 ![Page Image](https://bcdn.docswell.com/page/GEWGXZQ1J2.jpg) Chapter 4. Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set • Z-Number [875–877]: A Z-number describes uncertain information by pairing a fuzzy restriction with a reliability measure, capturing both value and confidence simultaneously. Related concepts such as Intuitionistic Fuzzy Z-Numbers [878–881] and Neutrosophic Z-Numbers [882–886] are also known in the literature. • D-Number [887–889]: A D-number generalizes Dempster–Shafer evidence by allowing incomplete or non-exclusive masses, improving knowledge fusion under open or dynamic frames situations [890]. Neutrosophic D-Numbers are also known as a related concept in the literature. • Multiple Sets [891–893]: A Multiple Set assigns several membership grades, arranged in an n×k matrix, to each element for modeling vagueness and multiplicity. Neutrosophic Multiple Sets are also known as a related concept [12, 894]. # Page. 143 ![Page Image](https://bcdn.docswell.com/page/47ZL61WXJ3.jpg) Chapter 4. Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set # Page. 144 ![Page Image](https://bcdn.docswell.com/page/YJ6W2LGPJV.jpg) Disclaimer Funding This study was conducted without any financial support from external organizations or grants. Acknowledgments We would like to express our sincere gratitude to everyone who provided valuable insights, support, and encouragement throughout this research. We also extend our thanks to the readers for their interest and to the authors of the referenced works, whose scholarly contributions have greatly influenced this study. Lastly, we are deeply grateful to the publishers and reviewers who facilitated the dissemination of this work. Data Availability Since this research is purely theoretical and mathematical, no empirical data or computational analysis was utilized. Researchers are encouraged to expand upon these findings with data-oriented or experimental approaches in future studies. Ethical Statement As this study does not involve experiments with human participants or animals, no ethical approval was required. Conflicts of Interest The authors declare that they have no conflicts of interest related to the content or publication of this paper. Code Availability No code or software was developed for this study. 143 # Page. 145 ![Page Image](https://bcdn.docswell.com/page/GJ5M21VPJ4.jpg) Chapter 4. Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set Clinical Trial This study did not involve any clinical trials. Consent to Participate Not applicable. Use of Generative AI and AI-Assisted Tools I use generative AI and AI-assisted tools for tasks such as English grammar checking, and I do not employ them in any way that violates ethical standards. Disclaimer (Others) This work presents theoretical ideas and frameworks that have not yet been empirically validated. Readers are encouraged to explore practical applications and further refine these concepts. Although care has been taken to ensure accuracy and appropriate citations, any errors or oversights are unintentional. The perspectives and interpretations expressed herein are solely those of the authors and do not necessarily reflect the viewpoints of their affiliated institutions. # Page. 146 ![Page Image](https://bcdn.docswell.com/page/LE3WK1D6E5.jpg) Appendix (List of Tables) 1.1 1.2 1.3 1.4 1.5 2.1 2.2 Representative set extensions and the canonical information stored per element. . . . . Plithogenic scalar-contradiction variants (t = 1) and their classical limits. . . . . . . . Parallel extensions of classical concepts in fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 2 — Additional concepts across classical, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 3 — Further concepts across classical, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concise comparison of extended rough–set families . . . . . . . . . . . . . . . . . . . . Concise comparison of Soft Set, Hypersoft Set, and SuperHyperSoft Set. Here P(U ) denotes the power set of U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 Overview of the uncertain-set families in this chapter (Part I). . . . . . . . . . . . . . . Overview of the uncertain-set families in this chapter (Part II). . . . . . . . . . . . . . A catalogue of Plithogenic Set families by number of components s. . . . . . . . . . . Overview of Plithogenic Sets by the contradiction dimension t in the Degree of Contradiction Function pCF : P v × P v → [0, 1]t . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Cases for m ∈ {1, 2, 3, ...} (polar level), t = 0 (no DCF), and s ∈ {1, 2, 3, 4, 5}. Here m=1 = classical, m=2 = bipolar, m=3 = tripolar. . . . . . . . . . . . . . . . . . . . . 3.6 Cases for m (polar level), t = 1 (single DCF), and s ∈ {1, 2, 3, 4, 5}. Here m=1 = classical, m=2 = bipolar, m=3 = tripolar. . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Summary of Complex Plithogenic Set (CPS) cases for t ∈ {0, 1} and s ∈ {1, 2, 3}. Here Γ(δ) is the contradiction-aware complex membership (Def. 3.3.1), γi ∈ D are CDAF components, and w(a | δ) = 1 − Φ(pCF1 (a, δ), . . . , pCFt (a, δ)). For t = 1 we take Φ(z) = z; for t = 0 we set Φ ≡ 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Specializations of (m, n)-SuperhyperPlithogenic Sets . . . . . . . . . . . . . . . . . . . 3.9 HyperPlithogenic Set specializations by number of DAF components . . . . . . . . . . 3.10 Reductions of the Plithogenic Linguistic Set . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Naming map for q-rung n-tuple plithogenic sets PSq,n . . . . . . . . . . . . . . . . . . 3.12 Naming for Type-n Plithogenic Sets when t = 0 (no explicit contradiction dimension) 3.13 Concrete instantiations for n = 2 and n = 3 with t = 0 . . . . . . . . . . . . . . . . . . 3.14 Taxonomy of Iterative MultiPlithogenic Sets by order and instance . . . . . . . . . . . 3.15 Typical interval–valued fuzzy/neutrosophic families captured as special cases of the Interval–Valued Plithogenic Set (IVPS) by choosing s equal to the number of interval components and t = 0 (no DCF) or t ≥ 1 (DCF–aware). . . . . . . . . . . . . . . . . . 3.16 Comparison of plithogenic set, overset, underset, and offset via the range of the degree of appurtenance function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Special cases of the plithogenic offset P Soff (s, t) for t = 1. . . . . . . . . . . . . . . . . 3.18 Classical cubic set families as special cases of the Plithogenic Cubic Set . . . . . . . . 3.19 Plithogenic SuperHyperSoft Set (PSHSS) as a common generalization of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and partitioned Neutrosophic SuperHyperSoft Sets. . . 3.20 PSHSS reductions to Plithogenic HyperSoft / Plithogenic Soft Sets and their fuzzy, intuitionistic fuzzy, and neutrosophic specializations. . . . . . . . . . . . . . . . . . . . 3.21 Roles of the parameters s and t in hesitant plithogenic memberships . . . . . . . . . . 3.22 Spherical fuzzy–type models as special cases of a Spherical Plithogenic Set (single attribute a∗ , pCF ≡ 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6 6 7 8 8 15 17 22 23 25 26 28 29 35 39 39 43 46 48 49 53 56 59 60 63 66 67 70 77 # Page. 147 ![Page Image](https://bcdn.docswell.com/page/8EDK3XPM7G.jpg) Appendix (List of Tables) 3.23 T-spherical fuzzy–type models as special cases of a T-Spherical Plithogenic Set (single attribute a∗ , pCF ≡ 0, order t ≥ 1, radius 1). . . . . . . . . . . . . . . . . . . . . . . . 83 3.24 Examples of rough-type models that are special cases of the plithogenic rough set P LR (A) = (P LR (A), P LR (A)) by choosing the DAF dimension s and (optionally) the DCF dimension t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.25 Plithogenic soft rough set as a common generalization of several soft rough models. . . 91 3.26 Linear Diophantine plithogenic set as a common generalization . . . . . . . . . . . . . 96 3.27 TreePlithogenic Set as a unifying model for tree–based fuzzy / intuitionistic fuzzy / neutrosophic sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.28 ForestPlithogenic Set as a unifying model for forest–based fuzzy / intuitionistic fuzzy / neutrosophic sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.29 Reductions of a Plithogenic Soft Expert Set (PSES) to several soft expert models. . . 105 3.30 Dynamic fuzzy/intuitionistic/neutrosophic families as special cases of the Dynamic Plithogenic Set (DPS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.31 Reductions of the Probabilistic Plithogenic Set (PPS). A PPS collapses to PFS/PIFS/PNS by fixing one attribute, taking pCF ≡ 0, and choosing s = 1, 2, 3 respectively. . . . . . 112 3.32 Classical triangular fuzzy/intuitionistic/neutrosophic sets as exact special cases of the Triangular Plithogenic Set by signature choice and pCF ≡ 0. . . . . . . . . . . . . . . 116 3.33 Classical trapezoidal fuzzy/intuitionistic/neutrosophic families as specializations of the Trapezoidal Plithogenic Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.34 Nonstandard plithogenic set as a common generalization . . . . . . . . . . . . . . . . . 124 3.35 Refined fuzzy, intuitionistic fuzzy, neutrosophic, quadripartitioned neutrosophic, and pentapartitioned neutrosophic sets as special cases of the Refined Plithogenic Set. . . . 127 3.36 Subset–valued plithogenic set as a common generalization. . . . . . . . . . . . . . . . . 130 3.37 Reductions of picture fuzzy, picture neutrosophic, and plithogenic sets to the picture plithogenic framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 * # Page. 148 ![Page Image](https://bcdn.docswell.com/page/V7PK4PNQJ8.jpg) Bibliography [1] Lotfi A Zadeh. 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Neutrosophic Sets and Systems, 86(1):20, 2025. # Page. 180 ![Page Image](https://bcdn.docswell.com/page/9E2941QG7R.jpg) This dynamic survey comprehensively examines numerous generalized set-theoretic frameworks designed to model and capture real-world uncertainty, which frequently involves vagueness, partial truth, and incomplete information. The work traces the evolution of these mathematical concepts, starting with classical models like Fuzzy Sets and Intuitionistic Fuzzy Sets. The survey then extends to more sophisticated and recent models, including Neutrosophic Sets, Plithogenic Sets, and Extensional Sets. Additionally, it covers related concepts such as Vague Sets, Hesitant Fuzzy Sets, Picture Fuzzy Sets, and Quadripartitioned Neutrosophic Sets. The aim is to provide a unified and up-to-date analysis of these diverse theories, highlighting their foundational principles and their capacity for rigorous mathematical modeling of complex uncertainty in various scientific domains.