A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets

A Dynamic Survey of Fuzzy,

• The survey’s keywords name Fuzzy Set, Intuitionistic Fuzzy Set, Neutrosophic Set, and Plithogenic Set.Source: and Extensional Sets · p. 3

and Extensional Sets

• The contents list triangular, trapezoidal, nonstandard, refined, subset-valued, and picture plithogenic sets.Source: and Extensional Sets · p. 5

1.1 Uncertain Set

• Applications of fuzzy sets and extensions span decision science, chemistry, control systems, and machine learning, while model choice depends on the application and its uncertainty parameters.Source: §1.1 · p. 6
• Fuzzy sets assign each element one membership degree, intuitionistic fuzzy sets a membership/non-membership pair [2], and neutrosophic sets truth, indeterminacy, and falsity degrees.Neutrosophic truth, indeterminacy, and falsity values need not sum to 1, allowing incomplete, inconsistent, or redundant information.Source: §1.1 · p. 6
• Neutrosophic modeling assigns indeterminacy explicit significance and is presented as generalizing several fuzzy variants and regret, grey-system, and three-way decision theories.Source: §1.1 · p. 6
• Plithogenic sets represent attribute values and their appurtenance degrees, using contradiction between values to support context-sensitive aggregation of heterogeneous or conflicting evaluations [10, 18, 19].This structure refines fuzzy, intuitionistic fuzzy, and neutrosophic models.Source: §1.1 · p. 7
• With scalar contradiction t = 1, plithogenic fuzzy, intuitionistic fuzzy, and neutrosophic models extend their classical counterparts; setting pCF to zero recovers the non-plithogenic models.The introduction summarizes these variants and their classical limits in Table 1.2.Source: §1.1 · p. 7, Table 1.2 · p. 7

1.2 Applied Area: Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic

• Fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic sets have been studied across fields including algebra, graph and hypergraph theory, probability, and decision-making.Tables 1.3–1.5 summarize extensions of classical concepts across these frameworks.Source: §1.2 · p. 7, Table 1.3 · p. 8, Table 1.4 · p. 9

1.3 Our Contribution in This Book

• The book surveys four major uncertainty-oriented set families to organize recurring ideas and encourage new insights, conceptual extensions, and applications across disciplines.The authors describe the survey as comprehensive and large-scale.Source: §1.3 · p. 8

2.1 Fuzzy Set

• A fuzzy set assigns each element a membership degree in [0, 1], representing how strongly it belongs to the set [1,38,274].Its membership function maps the universe to [0, 1].Source: §2.1 · p. 10

2.2 Intuitionistic fuzzy set

• An intuitionistic fuzzy set assigns membership and nonmembership degrees whose sum is at most 1, leaving hesitation; ordinary fuzzy sets arise when nonmembership equals 1 minus membership.Source: §2.2 · p. 11

2.3 Neutrosophic Set

• A neutrosophic set assigns independent truth, indeterminacy, and falsity degrees in [0, 1] to each element, explicitly accommodating indeterminacy.Source: §2.3 · p. 12

2.4 Rough Set

• Rough sets use equivalence classes to approximate a target: the lower approximation marks definite membership and the upper approximation possible membership under coarse information.Source: §2.4 · pp. 14, 16

2.5 Soft set

• A soft set is a parameterized family of subsets, represented by a mapping from parameters to subsets of the universe; Molodtsov introduced it in 1999 for decision-making.Source: §2.5 · p. 17
• Soft sets map each practical requirement to the subset of options that meet it, illustrated by Tokyo apartment selection across rental criteria.The example defines a universe of four apartments and uses requirements such as location, rent, size, amenities, and suitability.Source: §2.5 · p. 17
• A laptop buyer filtering for budget, a 14-inch screen, and long battery life faces a trade-off because no single item satisfies all three preferences.The example first filters by budget and screen size, then adds long battery life as a further preference.Source: §2.5 · pp. 17–18
• A hypersoft set maps tuples from the Cartesian product of attribute domains to subsets of the universe, enabling queries such as laptops matching a CPU–RAM–SSD profile.For each tuple of attribute values, the mapping returns the items corresponding to that combination.Source: §2.5 · pp. 18–19
• A SuperHyperSoft Set uses subsets of each attribute’s values as its query domain and maps their combinations to matching options.This supports filters such as any of several seasons, budgets up to Mid, or multiple traveler types.Source: §2.5 · pp. 19–20

Dynamic Reviews and Results of Uncertain Sets

• Plithogenic Sets generalize Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets by incorporating attribute-based membership and contradiction management.The chapter also presents an overview of the uncertain-set families it discusses.Source: Dynamic Reviews and Results of… · p. 22

3.1 Plithogenic Sets of Uncertain Values

• Classical Plithogenic Sets pair a Degree of Appurtenance Function with a Degree of Contradiction Function; the appurtenance definition varies across papers.The author states that the Classical Plithogenic Set definition used here does not employ the power set.Source: §3.1 · p. 22

3.2 m-Polar Plithogenic Set

• An m-polar Plithogenic Set assigns appurtenance vectors to multiple poles and uses contradictions between values and poles to aggregate scores for a chosen context.Its component count accommodates fuzzy, intuitionistic-type, and neutrosophic-type structures; the framework generalizes multipolar fuzzy, intuitionistic fuzzy, and neutrosophic sets.Source: §3.2 · pp. 26–28
• The hiring and vendor examples show contradictions shaping aggregated scores: pole compatibility determines hiring weights, while vendor weighting emphasizes benefit in a high_reliability-oriented context.For vendors, neutrality, risk, and compliance are still accounted for through the weights.Source: §3.2 · pp. 31, 33

3.3 Complex Plithogenic Set (CPS)

• The complex plithogenic set combines amplitude–phase memberships with symmetric, reflexive contradiction functions and compatibility-weighted complex means relative to a dominant value.Its contradiction fusion is specified as symmetric and componentwise monotone, with zero output when all contradiction inputs are zero.Source: §3.3 · pp. 35–36
• CPS examples yield a contradiction-aware EV-charging membership vector, an MRI-dominant medical-imaging confidence with phase approximately 29°, and a balanced-tier product appraisal vector.The product vector is Γ(balanced)(LaptopA) = (0.6834e^i26.22°, 0.5883e^i53.99°).Source: §3.3 · pp. 37–38

3.4 SuperHyperPlithogenic Set

• An (m, n)-SuperhyperPlithogenic Set represents nested super-elements with set-valued membership-degree vectors and symmetric contradiction degrees.Its definition uses m-level nested powersets and n-level fuzzy-contradiction degree sets.Source: §3.4 · p. 39
• Under high-risk dominance, hospital super-team A has reduced hyper-neutrosophic degree µ(high)(A) = (0.620, 0.2475, 0.265).The example assigns compatibility weights 0.3, 0.7, and 1 to low, moderate, and high risk, respectively.Source: §3.4 · pp. 40–41
• Under rush-hour dominance, the nested super-trip remains feasible at approximately 0.524 after contradiction-aware aggregation.The example models scenario-clustered feasibility for a trip bundle with free, moderate, and heavy congestion values.Source: §3.4 · p. 41
• Under dual-sourcing dominance, compatibility weights are 0.6 for single, 1 for dual, and 0.7 for multi sourcing, giving D = 2.3.Source: §3.4 · p. 42

3.5 Plithogenic Linguistic Set

• The linguistic-set definition uses contradiction-aware compatibility weights and a dominant linguistic value to form a weighted aggregated membership, including single-term and hesitant variants.It also defines term-level contradiction and a lifted version for hesitant sets.Source: §3.5 · p. 43
• Under a “Very Good” policy, single-term supplier quality aggregates to 0.20, while hesitant restaurant service-speed aggregation yields approximately 0.414.The examples use contradiction-based compatibility weights relative to the dominant linguistic value.Source: §3.5 · pp. 44–45
• The multi-component linguistic example aggregates ticket satisfaction as a contradiction-aware (T, I, F) triplet under the dominant “Very Good” context.The supplied passage states that a triplet is obtained but does not provide its values.Source: §3.5 · p. 45

3.6 q-rung orthopair Plithogenic sets

• The q-rung n-tuple framework generalizes q-rung orthopair fuzzy sets [512,513], with n = 2 giving orthopairs and general n giving orthovectors constrained by the q-sum bound n − 1.The definition also specifies a symmetric, reflexive contradiction degree on attribute values.Source: §3.6 · p. 46
• The triage and credit-risk examples apply contradiction-aware aggregation under “severe” and “high income stability” contexts, respectively, and report post-aggregation constraint checks.The supplied passages do not show the resulting assessment values.Source: §3.6 · pp. 47–48
• Under the “3-year warranty” context, the q = 4 orthopair example checks that the listed pairs and aggregate satisfy the constraint of being below 1.The passage gives b2 as satisfying µ4 + ν4 ≤ 1 and reports the aggregate constraint value as approximately 0.4705.Source: §3.6 · p. 48

3.7 Type-n Plithogenic Set

• Type-n Plithogenic Sets recursively nest memberships and contradiction functions, forming n-level hierarchies whose Type-1 base case is a classical Plithogenic Set [544].The framework models hierarchical uncertain attributes and relations; t = 0 denotes no explicit top-level contradiction channel.Source: §3.7 · pp. 48–49
• Under the “high intensity” context, the Type-2 neutrosophic healthcare evaluation is approximately (T, I, F) = (0.828, 0.122, 0.244).The example aggregates lower-level neutrosophic triples for therapy intensity and reports the resulting evaluation.Source: §3.7 · pp. 49–51
• Under the “global scope” context, the Type-3 fuzzy plithogenic logistics-reliability evaluation is µ ≈ 0.851.The example aggregates across weather, delivery mode, and vendor-scope levels.Source: §3.7 · p. 51
• Under the “intensive program” context, the Type-2 intuitionistic fuzzy university-admission assessment is (µ, ν) ≈ (0.810, 0.151), with µ + ν ≈ 0.961 ≤ 1.The membership object aggregates study-habit information into an intuitionistic fuzzy assessment.Source: §3.7 · p. 52

3.9 Interval-Valued Plithogenic Set

• Interval-Valued Plithogenic Sets use bounded interval-valued memberships and contradiction degrees, and capture typical interval-valued fuzzy and neutrosophic families as special cases.Source: §3.9 · pp. 56–57, Table 3.15 · p. 57
• [0.4600, 0.8477] is the e-commerce product’s overall interval-valued fuzzy suitability under the dominant Standard shipping context.Source: §3.9 · p. 58
• The hiring example applies interval-valued intuitionistic aggregation to candidate y relative to the dominant Permanent contract context.Source: §3.9 · pp. 58–59
• The irrigation example gives a plot’s interval-valued neutrosophic assessment relative to the dominant Balanced policy.Source: §3.9 · p. 59

3.10 Plithogenic OffSet

• Plithogenic offsets let appurtenance and contradiction degrees range below 0 and above 1, representing negative membership and over-membership.Source: §3.10 · p. 60
• The bank-credit offset example aggregates weak negative evidence for “Low” with support for “Medium” and over-membership for “High,” yielding a result slightly above 1.The example frames the aggregate as supporting an upgrade beyond the standard “High” threshold.Source: §3.10 · p. 61
• In the power-grid example, a solar-dominant context yields (T, I, F) ≈(1.00, 0.0486, 0.1671), with decisive endorsement of Solar, very low indeterminacy, and modest falsity.The stated offsets encode a policy or subsidy preference for Solar and counter-evidence for Coal.Source: §3.10 · p. 62

3.11 Plithogenic Cubic Set

• A Plithogenic Cubic Set combines interval-valued and point-valued data weighted by attribute-wise contradiction, with fuzzy, intuitionistic fuzzy, and neutrosophic canonical instances.Its contradiction function is symmetric and zero for identical attribute values; the interval-valued and point-valued components share a type.Source: §3.11 · pp. 62–63
• In the e-commerce PFCS example, contradiction-aware weighting toward “Premium” tightens the interval near high affinity and raises crisp membership.The example aggregates segment-wise fuzzy intervals and crisp memberships by a weighted mean.Source: §3.11 · p. 64
• For scholarship selection, research-centered aggregation yields ([0.6543, 0.8255], [0.0702, 0.1574]) and crisp (µ, ν) = (0.7447, 0.1372), reducing disagreement while keeping nonmembership low.The aggregated intuitionistic intervals respect the bound µ+ + ν+ ≤1.Source: §3.11 · p. 65
• In bike-station siting, prioritizing demand yields ([0.6905, 0.8600], [0.0500, 0.1257], [0.0900, 0.1690]) and crisp (T, I, F) = (0.7905, 0.0871, 0.1224).The interpretation reports high truth and low indeterminacy about suitability, with modest falsity persisting from cost and safety trade-offs.Source: §3.11 · p. 66

3.12 Plithogenic Soft, HyperSoft, and SuperHyperSoft Set

• A Plithogenic SuperHyperSoft Set extends hierarchical multi-attribute soft constructions with contradiction-aware, possibly multivalued fuzzy, intuitionistic fuzzy, or neutrosophic appurtenance.Its images are determined plithogenically using appurtenance values and contradictions; the section also presents reductions to Plithogenic HyperSoft and Soft Sets and a common-generalization table.Source: §3.12 · pp. 66–67, Table 3.19 · p. 67
• The capsule example includes s⋆ in the autumn–premium–sport capsule at the 0.75 aggregation threshold, with a managerial override for the borderline case.The stated decision rule is to include s⋆ when µagg ≥ 0.75.Source: §3.12 · p. 69
• The timetabling example accepts the candidate slot because its aggregate truth-minus-falsity score is 0.5400, above the 0.50 threshold.The example requires µagg + νagg ≤ 1 and applies the rule µagg − νagg ≥ 0.50.Source: §3.12 · p. 70

3.13 Hesitant Plithogenic Set

• In the medical triage example, the (3, 1)-dimensional model captures competing neutrosophic assessments and contradictions among severity labels for contradiction-aware aggregation.The proposed triage policy can aggregate each hesitation set by weighted averaging or ordered selection to prioritize treatments or allocate resources.Source: §3.13 · pp. 71–72
• For supplier assessment, contradiction-weighted aggregation penalizes reliability scores more when their attribute values conflict with the chosen dominant value, enabling ranking or screening under hesitant opinions.The example first combines memberships within each hesitation set, then weights the resulting scores using pCF relative to the dominant value.Source: §3.13 · pp. 73–74
• Hesitant Plithogenic Sets generalize Hesitant Fuzzy Sets and Hesitant Neutrosophic Sets through suitable choices of dimension and attribute values.They assign each object–attribute-value pair a finite nonempty set of plithogenic membership vectors together with a contradiction function.Source: §3.13 · pp. 70, 74

Then there exists a bijection

• Every hesitant fuzzy set corresponds uniquely to a hesitant plithogenic set with dimension (s, t) = (1, 1), one attribute value, and zero contradiction; the same holds for hesitant neutrosophic sets with (3, 1).The maps in each case are mutual inverses.Source: Then there exists a bijection · pp. 76–77

3.14 Spherical Plithogenic Sets

• A spherical plithogenic set assigns neutrosophic triples to element–attribute-value pairs under a spherical constraint, with a symmetric contradiction function that vanishes on identical values.The definition specifies a radius λ > 0 and a nonempty attribute-value set.Source: §3.14 · p. 78
• In air-quality risk assessment, contradiction-weighted aggregation discounts evidence for “high” when combined with evidence for “low,” yielding a spherical, contradiction-aware representation across districts.The model treats “low” and “high” as maximally contradictory and “medium” as partially compatible with both.Source: §3.14 · p. 79
• For diabetes severity, spherical plithogenic aggregation reconciles conflicting expert evaluations while preserving the spherical constraint, producing an interpretable assessment for every patient.The example describes “mild” and “severe” as highly incompatible, while “moderate” is closer to each.Source: §3.14 · p. 80
• Spherical plithogenic sets strictly generalize both spherical fuzzy sets and spherical neutrosophic sets; spherical fuzzy sets are in bijection with a subclass of spherical plithogenic sets at radius λ = 1.Source: §3.14 · p. 80

Step 2: From spherical plithogenic (radius

• Every spherical fuzzy set embeds at radius 1 and every spherical neutrosophic set at radius √3; the inverse maps and richer attributes with nonzero contradiction establish strict extension.The inverse maps recover each set pointwise, while spherical plithogenic sets can also use multiple attribute values and nontrivial contradiction.Source: Step 2: From spherical plithogenic… · p. 83

3.15 T-Spherical Plithogenic Set

• T-spherical plithogenic sets include spherical plithogenic sets at order t = 2 and strictly generalize them, with T-spherical fuzzy-type models arising as special cases.The special cases fix one attribute value, set contradiction to zero, and apply the t-spherical constraint; for t > 2, some sets violate the spherical quadratic constraint.Source: §3.15 · pp. 84–86, Table 3.23 · p. 84

3.16 Plithogenic Rough Set

• Plithogenic rough sets define lower and upper approximations from plithogenic relations that combine vector appurtenance and contradiction degrees, with aggregation monotone in each argument.The framework’s relation uses separate appurtenance and contradiction dimensions, and the approximation definitions include scalar special cases.Source: §3.16 · pp. 86–87
• In the hospital triage example, p1 lies on the boundary at 0.51, while p3 has upper-region coverage 0.36 and a lower penalty of 0.76.Source: §3.16 · p. 88
• B1 has upper value 0.56 and lower value 0.625: it is moderately similar to defective batches but fails certainty, so it is flagged for re–inspection; B2 is weakly in the upper region.Source: §3.16 · pp. 89–90

3.17 Plithogenic soft rough set

• Plithogenic soft rough sets combine parameterized soft memberships with parameter-specific plithogenic rough information to form graded lower and upper approximations.For each parameter, the approximations combine the soft membership with rough information derived from that parameter’s relation.Source: §3.17 · pp. 90–91
• 0.90 for e1 and 0.80 for e2 are S2’s lower-approximation values; S1 is moderate, especially under eco-compliance, and may need improvements or audits.The example interprets S2 as strongly inside the strategic-partner lower approximation and S1 as a borderline candidate.Source: §3.17 · p. 94
• Every plithogenic rough set arises as a special case of a plithogenic soft rough set under a singleton parameter identification.The construction makes the soft membership constant, takes the parameter relation to be the original rough relation, and uses an aggregation that projects to the rough component.Source: §3.17 · p. 95
• Every plithogenic soft set is also realized as a specialization, with identical lower and upper soft memberships.The construction chooses the aggregation to project to the soft-membership component, independently of the rough approximation.Source: §3.17 · pp. 95–96

3.18 Linear Diophantine Plithogenic set

• A linear Diophantine plithogenic set uses multiple membership components and integer coefficients constrained by linear Diophantine equations, with a reference-dependent residual degree.The section presents it as a common generalization of several existing Linear Diophantine models.Source: §3.18 · pp. 96–97
• Linear Diophantine plithogenic sets combine weighted-membership constraints with contradictions between criteria, and the EcoPhone battery example satisfies the constraint with a value of 0.975.The model is applied to multi-criteria smartphone selection.Source: §3.18 · p. 98

3.19 TreePlithogenic Set

• TreePlithogenic Sets [762] aggregate memberships across a rooted attribute tree using contradiction-aware hierarchical weighting; a hiring example gives a Fit score of approximately 0.721.A one-level tree with one attribute reduces to an ordinary plithogenic set.Source: §3.19 · pp. 99–100
• The triage and supplier examples yield contradiction-aware hierarchical scores of approximately 0.720 for severity and 0.696 for suitability, respectively.The supplier score is described as borderline “preferred” under the specified hierarchy and contradictions.Source: §3.19 · pp. 101–102

3.20 ForestPlithogenic Set

• Forest-level membership aggregates the plithogenic memberships from every tree represented in a node subset under common contradiction control; with exactly one tree, this reduces to a TreePlithogenic Set.The contradiction-aware forest readmission score for p is ≈0.648 (moderate–high), and asset a achieves ≈0.707 (marginally attractive overall).Source: §3.20 · pp. 102, 104, 105

3.21 Plithogenic Soft Expert Set

• A Plithogenic Soft Expert Set [772] attaches plithogenic membership and contradiction-aware fusion to expert–parameter–opinion triples, accommodating heterogeneous memberships and conflicting labels.The framework uses activated triples formed from parameters, experts, and agree/disagree opinions.Source: §3.21 · p. 105
• With no contradiction channel, Plithogenic Soft Expert Sets reduce to fuzzy, intuitionistic fuzzy, and neutrosophic soft expert sets for membership dimensions 1, 2, and 3, respectively.The reductions are stated for t = 0.Source: §3.21 · p. 106, Table 3.29 · p. 106
• The soft expert examples use Φ(a, b) = a(1 − b) to reduce membership under contradiction, giving strong low-cost support and partially penalizing neutral sustainability.Source: §3.21 · pp. 107–108
• In the course example, high job-market relevance with low contradiction strongly supports recommending c2, whereas moderate difficulty gives weaker support for c3.Source: §3.21 · p. 108

3.22 Dynamic Plithogenic Set

• DPS makes membership and contradiction time-dependent, with reflexive, symmetric contradiction, yielding static snapshots per instant and dynamic fuzzy, intuitionistic-fuzzy, and neutrosophic models as special cases.Source: §3.22 · p. 109, Table 3.30 · p. 109
• Examples track Premium satisfaction rising after redesign while air-quality risk shifts from low toward high and contradiction between moderate and high rises after a pollution episode.Source: §3.22 · pp. 110–111

3.23 Probabilistic Plithogenic Set

• PPS assigns random, possibly multicomponent plithogenic degrees and fuses attributes through contradiction-aware aggregation, with probabilistic fuzzy, intuitionistic-fuzzy, and neutrosophic sets as special cases.Source: §3.23 · pp. 111–112, Table 3.31 · p. 113
• Under strict regulation, S1’s environmental-performance membership for “high” rises to 0.65, while S2 is expected to remain consistently low-cost despite regulatory changes.Source: §3.23 · p. 114
• Probabilistic Plithogenic Sets model sustainable supplier suitability under uncertain future conditions, with expected values and quantiles supporting risk-averse evaluation.The contradiction-aware aggregator combines membership degrees, biasing toward min for compatible values and max for strongly contradictory values.Source: §3.23 · p. 114
• Probabilistic plithogenic triples aggregate support, indeterminacy, and counter-evidence across attributes to represent high-disease-risk status under measurement noise, with averages and quantiles providing summaries and conservative bounds.Source: §3.23 · pp. 115–116

3.24 Triangular Plithogenic Set

• Triangular Plithogenic Sets use triangular fuzzy numbers for appurtenance degrees and contradiction-aware aggregation; classical triangular fuzzy, intuitionistic, and neutrosophic sets arise as exact special cases.Source: §3.24 · pp. 116–117, Table 3.32 · p. 117
• For laptop selection, contradiction-aware aggregation toward dominant “high” utility can favor L3 when its low-utility component is small, and centroid defuzzification yields scalar scores.Source: §3.24 · p. 118
• Triangular Plithogenic Sets model renewable-project sustainability judgments by aggregating truth, indeterminacy, and falsity triangles while accounting for contradictions with lower values.The example evaluates projects against excellent sustainability as the dominant value.Source: §3.24 · p. 120

3.25 Trapezoidal Plithogenic Sets

• A Trapezoidal Plithogenic Set assigns truth, indeterminacy, and falsity trapezoids for each attribute value, then contradiction-weights defuzzified means to obtain a scalar inclusion grade.The model uses a symmetric contradiction function and a design parameter β; set operations are defined at the (T, I, F) level using chosen t-conorm, t-norm, and negator.Source: §3.25 · p. 121
• Supplier SA's trapezoidal plithogenic sustainability inclusion grade is 179/240 ≈ 0.746 (about 0.75).The passage attributes this high grade to high truth, low falsity, limited indeterminacy, and contradiction between the chosen value “high” and alternatives.Source: §3.25 · p. 123
• Patient P1's low-risk inclusion grade is 187/300 ≈ 0.623 (about 0.62), supporting patient ranking, follow-up intervals, or monitoring-resource allocation under cardiovascular risk uncertainty.The grade jointly reflects the trapezoidal truth, indeterminacy, and falsity profiles and contradiction between “low” and other risk categories.Source: §3.25 · p. 124

3.26 Nonstandard Plithogenic Sets

• Nonstandard plithogenic sets aggregate hyperreal memberships across attribute values using contradiction-weighted internal t-norm/t-conorm; the renewable-project example's degrees are infinitesimally close to classical membership values.Their membership values may lie in the hyperreal band [−δ, 1 + δ] for a positive infinitesimal δ.Source: §3.26 · pp. 125–126
• In nonstandard plithogenic aggregation, the wind criteria yield T*=0.68−ε and S*=0.94+ε, while the −0.4ε correction preserves additional hyperreal resolution.Source: §3.26 · p. 126

3.27 Refined Plithogenic Set

• Refined plithogenic sets encode multiple truth, indeterminacy, and falsity components under contradiction-aware attribute weighting, with refined fuzzy, intuitionistic fuzzy, and neutrosophic families as special cases.In the examples, patient pA has higher refined support under imaging-dominant context, and supplier sA is preferred under an ESG-dominant policy.Source: §3.27 · pp. 127, 129

3.28 Subset–Valued Plithogenic Sets

• Subset-valued plithogenic sets retain sets of admissible membership vectors per item and attribute value, and aggregate them using contradiction weights.Singleton values recover ordinary fuzzy and single-valued neutrosophic sets [841]; in the medical-risk example, greater disagreement between evidence sources yields more cautious combined assessments.Source: §3.28 · pp. 129–131

3.29 Picture Plithogenic Set

• Picture plithogenic sets strictly generalize picture fuzzy, picture neutrosophic, and plithogenic sets: the first two are representable, and the last embeds by adding a zero neutral coordinate.Source: §3.29 · p. 133
• Picture neutrosophic sets correspond to picture plithogenic sets with t = 3, r = 0, singleton Pv, and neutral coordinate identically zero or separated from the neutrosophic triple.Plithogenic sets are exactly picture plithogenic sets whose neutral coordinate is identically zero.Source: §3.29 · p. 135
• The transport-policy example represents positive, negative, and neutral citizen opinions while assigning 0.9 contradiction between support and oppose.The neutral degree represents undecided or mixed attitudes.Source: §3.29 · p. 136
• The smartphone example encodes positive, negative, and neutral sentiment, with battery–camera contradiction set to 0.3 for recommendation aggregation.The neutral degree represents mixed sentiment or no clear opinion.Source: §3.29 · pp. 136–137

4.1 Uncertain Set

• Choosing an Uncertain Model and its membership-degree domain unifies fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and other uncertainty-set frameworks.An Uncertain Set labels a base set with membership-degree tuples.Source: §4.1 · pp. 138–139

4.2 Functional Set

• Functorial sets assign sets to category objects and transport elements along morphisms, encompassing uncertainty frameworks and set concepts beyond uncertainty.In the logistics example, purchase orders are retagged along routes, and the two-step transport agrees with transport along the composed route.Source: §4.2 · pp. 139–140

4.3 Other Uncertain Sets and Future Works

• Further theoretical developments may combine auxiliary concepts with fuzzy, intuitionistic, neutrosophic, and plithogenic models; these concepts can also be generalized within the Functorial Set framework.Source: §4.3 · p. 141
• Related frameworks include near sets [857–859], weighted sets [864, 865], Z-numbers, D-numbers, and multiple sets [891–893].Multiple sets assign each element several membership grades arranged in an n×k matrix; neutrosophic multiple sets are also known [12].Source: §4.3 · pp. 141–142

Funding

• The study received no financial support from external organizations or grants.Source: Funding · p. 144

Data Availability

• The study used no empirical data or computational analysis, and encourages future data-oriented or experimental extensions.The passage characterizes the research as purely theoretical and mathematical.Source: Data Availability · p. 144

Ethical Statement

• No ethical approval was required because the study involved no experiments with human participants or animals.Source: Ethical Statement · p. 144

• No code or software was developed for the study.Source: Code Availability · p. 144

Clinical Trial

• The study did not involve any clinical trials.Source: Clinical Trial · p. 145

Disclaimer (Others)

• The theoretical ideas and frameworks in the work have not yet been empirically validated.Source: Disclaimer (Others) · p. 145
• Readers are encouraged to explore practical applications and further refine the concepts.Source: Disclaimer (Others) · p. 145
• The authors state that care was taken with accuracy and citations, and any errors or oversights are unintentional.Source: Disclaimer (Others) · p. 145
• The perspectives and interpretations are solely the authors’ and do not necessarily reflect their affiliated institutions’ viewpoints.Source: Disclaimer (Others) · p. 145