Source-linked AI summary
A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets
Takaaki Fujita, Florentin Smarandache
TL;DR
Real-world uncertainty motivates a broad range of generalized set frameworks, while recurring ideas across four major families create a need for synthesis. The book addresses this need through a comprehensive survey intended to organize existing developments and stimulate further extensions and applications.
Problem
The literature contains many uncertainty-oriented frameworks and recurring concepts across Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets, motivating an organized synthesis.
Method
The book provides a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets.
Results
The survey presents an organized overview of existing developments across the four set families.
Takeaways & Limitations
The unified exposition is intended to stimulate new insights, conceptual extensions, and further applications across disciplines.
Abstract
from arXiv · showhide
Real-world phenomena often exhibit vagueness, partial truth, and incomplete information. To model 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], Penta-Partitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Within these frameworks, a wide range of notions has been proposed and studied, particularly in the settings of fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic set theories. This extensive literature underscores both the significance of these theories and the breadth of their application areas. As a result, many ideas, constructions, and structural patterns recur across these four major families of uncertainty-oriented models. In this book, we provide a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. Our goal is to give readers a systematic overview of existing developments and, through a unified exposition, to stimulate new insights, further conceptual extensions, and additional applications across a wide range of disciplines.
A Dynamic Survey of Fuzzy,
The material identifies the work as a survey of uncertainty-oriented set frameworks, with Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets among its central topics.
- The work surveys Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets.
4 Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set
The listed material introduces a section on uncertain and functional sets, alongside other uncertain sets and future works.
- The framework includes an Uncertain Set section.
- It also includes a Functional Set section and discussion of other uncertain sets and future works.
Introduction
The introduction presents generalized set frameworks for modeling vagueness, partial truth, and incomplete information, then situates Plithogenic Sets within a broad family of uncertainty models. It motivates a large-scale survey by the recurrence of concepts and structures across these frameworks.
- Generalized set frameworks address vagueness, partial truth, and incomplete information across many uncertainty-oriented models.
- Fuzzy Sets associate each element with one membership degree, while Intuitionistic Fuzzy Sets use membership and non-membership functions.
- Neutrosophic Sets represent truth, indeterminacy, and falsity without requiring the three values to sum to 1.
- Plithogenic Sets add attribute values, appurtenance degrees, and contradiction functions to support context-sensitive aggregation of heterogeneous evaluations.
- With t = 1, plithogenic scalar-contradiction variants provide extensions of classical paradigms, and setting pCF identically to zero recovers the corresponding non-plithogenic models.
- The book surveys four major set families because their extensive literature contains recurring ideas and structural patterns across diverse application domains.
Preliminaries
The preliminaries introduce set-based frameworks for representing graded membership, nonmembership, indeterminacy, and approximation-based uncertainty. They define fuzzy, intuitionistic fuzzy, neutrosophic, and rough-set notions and illustrate them with concrete examples.
- Fuzzy Set: A fuzzy set assigns each element a single membership degree in [0, 1].The membership function maps the universe X to [0, 1], with values representing degree of belonging.
- Fuzzy Set: Fuzzy examples represent comfort and premium-customer status through triangular or trapezoidal membership functions.The examples include temperature comfort and monthly spending, with intermediate membership values such as 0.5 and full membership at selected points.
- Intuitionistic Fuzzy Set: An intuitionistic fuzzy set assigns membership and nonmembership degrees whose sum is at most 1, leaving a hesitation degree.The hesitation degree is 1 − µA(x) − νA(x); ordinary fuzzy sets are recovered when nonmembership equals 1 − membership.
- Intuitionistic Fuzzy Set: Intuitionistic fuzzy examples model cardiovascular risk and service satisfaction using paired membership and nonmembership assessments.The examples distinguish clearly positive, clearly negative, and intermediate cases, with the remaining margin representing uncertainty or lack of information.
- Rough Set: Rough sets approximate a subset through lower and upper approximations induced by equivalence classes.The lower approximation contains elements whose classes are entirely inside the target, while the upper approximation contains elements whose classes intersect it.
Dynamic Reviews and Results of Uncertain Sets
The chapter surveys relationships among fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic sets, emphasizing Plithogenic Sets’ attribute-based membership and contradiction management. Examples and reductions show how contradiction-aware aggregation supports context-sensitive evaluation across multiple uncertain-set extensions.
- Foundations: Plithogenic Sets generalize fuzzy, intuitionistic fuzzy, and neutrosophic sets through attribute values, appurtenance degrees, and contradiction functions.The framework supports context-sensitive aggregation of heterogeneous and conflicting evaluations.
- Foundations: The Degree of Appurtenance Function maps objects and attribute values to s-component memberships, while the Degree of Contradiction Function maps value pairs to t contradiction channels.The classical definition omits the power set, although formulations vary across studies.
- Contradiction-aware aggregation: Contradictions at value and pole levels jointly attenuate or emphasize components to produce a single plithogenic score for each candidate–value pair.The aggregation can encode a dominant context while preserving multi-polar information.
- Applications: Four-pole plithogenic evaluation lets decision makers emphasize benefit in high-reliability contexts while retaining neutrality, risk, and compliance information.The framework uses contradiction degrees and context-dependent weights to combine these poles.
- Applications: Contradiction-aware scalarization provides mathematically precise comparisons of designs in terms of safety, throughput, and cost under a safety-first context.The full multi-polar information remains represented while contradictory poles and objectives are downweighted.
- Extensions and decisions: The framework supports reductions and extensions across linguistic, interval-valued, hesitant, and soft-set models, including special-case recovery and decision screening.Examples include interval-valued suitability, threshold-based inclusion, and supplier ranking under heterogeneous expert opinions.
Then there exists a bijection
The chapter establishes bijective correspondences showing that hesitant fuzzy, hesitant neutrosophic, spherical fuzzy, and spherical neutrosophic sets can be realized as particular plithogenic sets. These results use singleton attribute values, zero contradiction in the hesitant cases, and spherical constraints for the spherical models.
- Hesitant fuzzy correspondence: Every Hesitant Fuzzy Set is canonically realized as a Hesitant Plithogenic Set with (s, t) = (1, 1), one attribute value, and zero contradiction.The maps ΦF and ΨF are mutual inverses, so the correspondence is bijective.
- Hesitant neutrosophic correspondence: Every Hesitant Neutrosophic Set is realized as a Hesitant Plithogenic Set with (s, t) = (3, 1), one attribute value, and trivial contradiction.The construction preserves the degree-of-appurtenance function and yields a bijection between the two classes.
- Spherical plithogenic construction: A spherical plithogenic set assigns truth, indeterminacy, and falsity degrees while imposing a radius-based spherical constraint and a symmetric contradiction function.The framework represents attribute-valued elements through spherical plithogenic neutrosophic triples.
- Spherical specializations: When there is one attribute value and c ≡0, a spherical plithogenic set reduces to the spherical fuzzy and spherical neutrosophic settings.This specialization is summarized as a spherical fuzzy–type model with a single attribute and zero contradiction.
- Spherical generalization: Spherical Plithogenic Sets strictly generalize both spherical fuzzy sets and spherical neutrosophic sets through bijective special-case correspondences.The spherical fuzzy correspondence is explicitly established as a bijection, while the theorem also states the spherical neutrosophic correspondence.
Step 2: From spherical plithogenic (radius
The chapter develops spherical, T-spherical, soft rough, expert, dynamic, probabilistic, and triangular plithogenic models as extensions or reductions of established uncertainty frameworks. These constructions preserve contradiction-aware information while recovering several existing models as special cases.
- T-spherical plithogenic sets: T-spherical plithogenic sets strictly generalize spherical plithogenic sets because the quadratic constraint is recovered at t = 2, while higher orders permit additional sets.The inclusion is proper for t > 2.
- Plithogenic soft rough sets: Plithogenic soft rough sets provide a common framework that specializes to both plithogenic rough sets and plithogenic soft sets.The reductions identify the corresponding lower and upper approximations with the original structures.
- Plithogenic Soft Expert Sets: A Plithogenic Soft Expert Set subsumes fuzzy, intuitionistic fuzzy, and neutrosophic soft expert sets through suitable choices of s and t.The reductions use s = 1, 2, and 3 with t = 0, respectively.
- Dynamic and probabilistic extensions: Dynamic and probabilistic plithogenic sets extend the framework to time-varying memberships, time-varying contradictions, and random multi-component degrees.Dynamic models represent snapshots over time, while probabilistic models support uncertain future conditions and subsume probabilistic fuzzy, intuitionistic fuzzy, and neutrosophic sets.
- Triangular and trapezoidal models: Triangular and trapezoidal plithogenic models represent applications with contradictory evaluations, including renewable-energy projects and supplier sustainability.The triangular framework also contains classical triangular fuzzy, intuitionistic, and neutrosophic sets as exact special cases.
Unifying Framework of Fuzzy, Intuitionistic, Neutrosophic, Plithogenic, and Other Set
The chapter proposes a unified uncertain-set framework that represents elements through model-specific membership-degree tuples and recovers several established uncertainty-set families. It also extends this perspective to functorial sets and related frameworks.
- Uncertain Set Framework: The framework defines an uncertain set as a model-specific labeling of a nonempty base set by membership-degree tuples.Once the base set and uncertain model are fixed, the set can be identified with its membership function.
- Recovery of Existing Sets: By selecting different uncertain models and domains, the framework unifies fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and other uncertainty-set frameworks.The text explicitly states that these structures are reproduced by suitable model choices.
- Functorial Sets: Functorial sets assign sets to category objects and transport elements along morphisms, extending the framework beyond uncertainty-oriented set concepts.The chapter describes functorial transport as a generalization covering uncertainty frameworks and features beyond uncertainty.
- Functorial Examples: Examples demonstrate functorial transport for city logistics, cross-timezone calendars, and geospatial containment while preserving composition.The logistics example explicitly verifies that transporting along two routes equals transport along their composite.
- Related Frameworks: The chapter identifies near, weighted, Z-, D-, and multiple sets as related frameworks that can also be generalized within the functorial-set perspective.These auxiliary concepts support approximate matching, prioritization, reliability-aware information, evidence fusion, or multiple membership grades.
Funding
The study reports no financial support from external organizations or grants.
- Funding: The study was conducted without financial support from external organizations or grants.No external funding source is reported.
Data Availability
The research is theoretical and mathematical rather than empirical or computational.
- Data Availability: No empirical data or computational analysis was used because the research is purely theoretical and mathematical.The authors encourage future data-oriented or experimental work.
Ethical Statement
No ethical approval was required because the study involved neither human participants nor animals.
- Ethical Statement: The study involved no experiments with human participants or animals, so ethical approval was not required.The statement concerns the study's research procedures.
Code Availability
No code or software was developed for this study, so it provides no software artifact.
- No code was developed for this study.
- No software was developed for this study.
- The study therefore provides no software artifact.
Clinical Trial
This study did not involve clinical trials.
- The study did not involve any clinical trials.
- No clinical-trial component was included in the study.
- Clinical-trial evidence is not part of this study.
Disclaimer (Others)
The work presents theoretical ideas and frameworks that have not yet been empirically validated. It invites practical exploration and refinement while noting that errors may remain and that the interpretations belong to the authors.
- The work presents theoretical ideas and frameworks that have not yet been empirically validated.
- Readers are encouraged to explore practical applications and further refine the concepts.
- The authors state that any errors or oversights are unintentional.
- The perspectives and interpretations expressed are solely those of the authors and may not reflect affiliated institutions.