Neutrosophic Sets: Theory and Applications

• Neutrosophic sets are set-theoretic structures that assign three membership values—truth, indeterminacy, and falsity—to each element, allowing for modeling of incomplete and inconsistent information.
• They generalize classical, fuzzy, and intuitionistic fuzzy sets by treating indeterminacy as an independent component, thereby accommodating normalized, incomplete, and contradictory data regimes.
• Computational frameworks like the PYNS system and matrix-based representations enable practical applications in decision making, clustering, and topology by systematically processing neutrosophic operations.

Searching arXiv for the cited neutrosophic-set sources to ground the article and cite them accurately. Searching for foundational and survey-style neutrosophic set papers. Neutrosophic sets are set-theoretic structures that assign to each element three values—truth-membership, indeterminacy-membership, and falsity-membership—rather than a single membership degree. In the single-valued formulation commonly used in applications, a set $A$ on a universe $X$ is written $A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$ with $T_A,I_A,F_A:X\to[0,1]$; the 2026 survey presents the standard single-valued constraint $0\le T_A(x)+I_A(x)+F_A(x)\le 3$, while the PYNS software framework works with the same component-wise bounds but imposes no separate sum constraint beyond $0\le \mu,\sigma,\omega\le 1$ (Fujita et al., 12 Mar 2026, Nordo et al., 2024). The central idea is that truth, indeterminacy, and falsity are treated as independent or partially independent components, so neutrosophic sets can represent incomplete information, normalized information, and inconsistent information within one formalism (Smarandache, 2019).

1. Foundational definition and historical emergence

Neutrosophic sets originate in Florentin Smarandache’s broader program of Neutrosophy, which introduces neutrality or indeterminacy as a fundamental logical value alongside truth and falsity (Fujita et al., 12 Mar 2026). In the more general formulations recalled in the symbolic and theoretical literature, the three components may even be taken as subsets of a nonstandard interval such as $]0^-,1^+[$, reflecting the original philosophical breadth of the theory (Smarandache, 2015, Smarandache et al., 2016). For practical mathematics and applications, however, the dominant formulation is the single-valued one, in which each component is a real number in $[0,1]$ (Smarandache et al., 2016).

In this single-valued setting, a neutrosophic set records, for each element $x$, a triple $(T_A(x),I_A(x),F_A(x))$, where $X$0 is truth-membership, $X$1 is indeterminacy-membership, and $X$2 is falsity-membership (Smarandache, 2019). The literature repeatedly emphasizes that the decisive departure from intuitionistic fuzzy theory is not merely the presence of three symbols, but the explicit treatment of indeterminacy as a first-class component rather than a residual term (Fujita et al., 12 Mar 2026). This permits three canonical informational regimes: $X$3 for incomplete information, $X$4 for normalized or balanced information, and $X$5 for inconsistent or paraconsistent information (Voskoglou, 2023).

The theory also admits refinement. The 2019 generalization paper defines refined neutrosophic sets with multiple truth, indeterminacy, and falsity subcomponents $X$6, $X$7, and $X$8, with $X$9, thereby turning the basic triplet into a family of subdegrees (Smarandache, 2019). The symbolic theory develops the same idea at a literal level, allowing symbolic components $A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$0, $A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$1, and $A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$2, refined indeterminacies, and algebraic constructions such as neutrosophic quadruple numbers $A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$3 (Smarandache, 2015).

2. Relation to fuzzy-type set theories

Neutrosophic sets are typically presented as a unifying generalization of classical, fuzzy, intuitionistic fuzzy, and several later uncertainty models. The 2026 survey states that it is now “widely recognized” that neutrosophic sets generalize intuitionistic fuzzy sets, inconsistent intuitionistic fuzzy sets including picture fuzzy and ternary fuzzy sets, Pythagorean fuzzy sets, spherical fuzzy sets, and q-rung orthopair fuzzy sets (Fujita et al., 12 Mar 2026).

Model	Typical constraint	Relation to neutrosophic sets
Classical set	$A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$4	Special case
Fuzzy set	$A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$5	Special case
Intuitionistic fuzzy set	$A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$6	Special case with dependent $A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$7
IIFS / PFS / TFS	$A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$8	Special case
Pythagorean fuzzy set	$A=\{\langle x,T_A(x),I_A(x),F_A(x)\rangle:x\in X\}$9 with derived $T_A,I_A,F_A:X\to[0,1]$0	Special case after squared embedding
Spherical fuzzy set	$T_A,I_A,F_A:X\to[0,1]$1	Special case after squared embedding
q-rung orthopair fuzzy set	$T_A,I_A,F_A:X\to[0,1]$2 with derived $T_A,I_A,F_A:X\to[0,1]$3	Special case after $T_A,I_A,F_A:X\to[0,1]$4-power embedding

The basic embeddings are straightforward. Classical sets appear when membership is crisp and indeterminacy is zero. Fuzzy sets are recovered by setting $T_A,I_A,F_A:X\to[0,1]$5 and $T_A,I_A,F_A:X\to[0,1]$6. Intuitionistic fuzzy sets are recovered when $T_A,I_A,F_A:X\to[0,1]$7 and $T_A,I_A,F_A:X\to[0,1]$8 satisfy $T_A,I_A,F_A:X\to[0,1]$9 and indeterminacy is defined by $0\le T_A(x)+I_A(x)+F_A(x)\le 3$0 (Nordo et al., 2024, Fujita et al., 12 Mar 2026). The difference is that neutrosophic theory does not require $0\le T_A(x)+I_A(x)+F_A(x)\le 3$1 to be determined by $0\le T_A(x)+I_A(x)+F_A(x)\le 3$2 and $0\le T_A(x)+I_A(x)+F_A(x)\le 3$3, and does not require $0\le T_A(x)+I_A(x)+F_A(x)\le 3$4 (Smarandache, 2019).

A recurrent misconception is that neutrosophic sets collapse to intuitionistic fuzzy sets whenever $0\le T_A(x)+I_A(x)+F_A(x)\le 3$5. The 2019 paper explicitly rejects that conclusion: even when the underlying triples satisfy the intuitionistic-fuzzy normalization, neutrosophic operators and intuitionistic fuzzy operators can still yield different outputs because intuitionistic fuzzy operations do not manipulate indeterminacy explicitly, whereas neutrosophic operations do (Smarandache, 2019). This difference is central to the claim that neutrosophic sets are not merely a renaming of intuitionistic fuzzy sets.

3. Order, operators, and mappings

The order structure of single-valued neutrosophic sets is componentwise but asymmetric across the three coordinates. In the PYNS framework and the neutrosophic-topology extension, for $0\le T_A(x)+I_A(x)+F_A(x)\le 3$6 and $0\le T_A(x)+I_A(x)+F_A(x)\le 3$7, one writes $0\le T_A(x)+I_A(x)+F_A(x)\le 3$8 when $0\le T_A(x)+I_A(x)+F_A(x)\le 3$9, $0\le \mu,\sigma,\omega\le 1$0, and $0\le \mu,\sigma,\omega\le 1$1 for all $0\le \mu,\sigma,\omega\le 1$2; equality is mutual inclusion. The same sources define the neutrosophic empty set $0\le \mu,\sigma,\omega\le 1$3, the neutrosophic absolute set $0\le \mu,\sigma,\omega\le 1$4, complement by $0\le \mu,\sigma,\omega\le 1$5, and induced mappings $0\le \mu,\sigma,\omega\le 1$6 and $0\le \mu,\sigma,\omega\le 1$7, with images computed over fibres by $0\le \mu,\sigma,\omega\le 1$8 on truth and indeterminacy and $0\le \mu,\sigma,\omega\le 1$9 on falsity, and with the default triple $]0^-,1^+[$0 on empty fibres in the PYNS implementation (Nordo et al., 2024, Nordo et al., 2024).

Operator conventions, however, are not uniform across the literature. The 2026 survey explicitly notes that different authors use slightly different conventions for union, intersection, and complement (Fujita et al., 12 Mar 2026). One common family of definitions, used in the 2019 generalization paper and treated by the survey as standard, sets

$]0^-,1^+[$1

so indeterminacy behaves dually to truth under union and intersection (Smarandache, 2019). By contrast, the PYNS framework implements

$]0^-,1^+[$2

so indeterminacy is aggregated in the same direction as truth (Nordo et al., 2024). The existence of both conventions is an established feature of the field rather than an accidental inconsistency.

This lack of complete standardization also has structural consequences. In the theory of single-valued neutrosophic filters, for example, one generally has $]0^-,1^+[$3 and $]0^-,1^+[$4, unlike the crisp case, and this failure is precisely why the usual complement-based characterization of ultrafilters does not fully carry over to the neutrosophic setting (Nordo et al., 2020). A neutral reading is that neutrosophic set theory retains the algebraic vocabulary of set theory while weakening several Boolean identities.

4. Variants and derived mathematical structures

A large part of the literature consists of systematic variants of the basic triplet model. The 2026 survey treats single-valued neutrosophic sets as the canonical $]0^-,1^+[$5-component case inside a wider hierarchy that includes interval-valued neutrosophic sets, refined neutrosophic sets, quadripartitioned and pentapartitioned neutrosophic sets, and plithogenic sets with arbitrary appurtenance dimension $]0^-,1^+[$6 and contradiction functions (Fujita et al., 12 Mar 2026). The interval-valued soft-set literature makes this concrete: an interval-valued neutrosophic soft set maps each parameter to an interval-valued neutrosophic set, so every parameter-object pair carries three intervals rather than three scalars (Deli, 2014).

Soft-set and matrix-based extensions form another strand. Neutrosophic soft sets map each parameter to a neutrosophic set on the universe, and neutrosophic soft matrices store these parameterized triplets in matrix form for computation; complement, union, intersection, AND-products, and OR-products are then defined componentwise on matrix entries (Deli et al., 2014). This line of work is aimed at decision procedures and storage structures rather than foundational set theory alone.

Refinement can also occur along relational or bipolar dimensions. Neutrosophic multi sets assign sequences $]0^-,1^+[$7, $]0^-,1^+[$8, and $]0^-,1^+[$9 to each element, and neutrosophic multi relations are defined as neutrosophic multi subsets of Cartesian products; their composition uses max–min on truth and min–max on indeterminacy and falsity (Broumi et al., 2015). Bipolar neutrosophic sets instead split each evaluation into positive and negative triples $[0,1]$0 and $[0,1]$1, thereby representing both a property and its counter-property (Deli et al., 2015). Interval Valued Bipolar Fuzzy Weighted Neutrosophic Sets further combine interval valuation, bipolarity, neutrosophic triples, and a fuzzy weight index $[0,1]$2 (Deli et al., 2016).

Topological and order-theoretic developments show that neutrosophic sets support derived structures beyond isolated membership assignments. The PYNS topology extension defines neutrosophic topologies as families of neutrosophic sets containing $[0,1]$3 and $[0,1]$4, closed under arbitrary neutrosophic unions and finite neutrosophic intersections, and introduces basis and sub-basis generation algorithms (Nordo et al., 2024). The filter literature develops single-valued neutrosophic filter bases, filters, ultrafilters, filter completion, and images of filters under induced mappings, explicitly to prepare for neutrosophic topology and convergence theory (Nordo et al., 2020).

5. Computational frameworks and algorithmic use

Executable support for neutrosophic sets has become more systematic. The PYNS framework is organized around three classes: NSuniverse for finite ordered universes, NSset for single-valued neutrosophic sets, and NSmapping for mappings between universes together with neutrosophic image and inverse image operators (Nordo et al., 2024). Universes are represented as finite ordered lists of strings, neutrosophic sets as dictionaries $[0,1]$5, and the framework exposes both compact angle-bracket notation and tabular display. It also overloads operators so that +, &, ~, -, <=, >=, == implement neutrosophic union, intersection, complement, difference, subset, superset, and equality (Nordo et al., 2024).

The same line of work extends from individual sets to families. The 2024 topology enhancement adds an NSfamily class to represent finite families of neutrosophic sets over a common universe and to automate basis construction, sub-basis generation, topology generation, and closure tests under union and intersection (Nordo et al., 2024). This is significant because it turns previously abstract definitions—such as “the topology generated by a family” or “the basis of finite intersections of a sub-basis”—into programmable objects.

Computational neutrosophics is not limited to symbolic set manipulation. In decision making, tabular representations of soft sets have been replaced by neutrosophic triplets $[0,1]$6 in order to encode not only how strongly an attribute applies, but also the decision maker’s doubt about that assessment (Voskoglou, 2023). In clustering, a neutrosophic-domain method defines a density-based indeterminacy value $[0,1]$7, assigns lower indeterminacy to dense regions and higher indeterminacy to sparse regions, introduces a noise cluster, and then optimizes memberships so that outliers go to the noise cluster while boundary points receive similar memberships to main clusters (Rashno et al., 2018). These works show that neutrosophic sets now function both as abstract models and as computational primitives.

6. Applications, misconceptions, and open issues

The application range claimed in the literature is broad. The PYNS framework paper cites pure mathematics and topology, graph theory, decision making, medicine, statistics, image analysis, and machine learning as domains in which neutrosophic sets arise (Nordo et al., 2024). The 2016 “New Trends” volume adds data mining, e-learning, engineering, medical diagnosis, probability theory, topology, and social science, and emphasizes the rapid appearance of hybrid models such as rough neutrosophic sets, bipolar neutrosophic sets, and single-valued neutrosophic hesitant fuzzy sets (Smarandache et al., 2016). The 2026 survey correspondingly catalogues neutrosophic relations, graphs, matrices, linear algebra, topology, probability, optimization, decision making, clustering, and neutrosophic numbers, positioning neutrosophic sets as one of the four main “core” uncertainty frameworks alongside fuzzy, intuitionistic fuzzy, and plithogenic sets (Fujita et al., 12 Mar 2026).

Decision-making papers illustrate what the explicit indeterminacy coordinate contributes. In the soft-set-based soccer-player example, binary entries and even grey or triangular fuzzy values are replaced by neutrosophic triplets so that truth, indeterminacy, and falsity can be averaged separately; the method then ranks candidates by high truth, low falsity, and lower indeterminacy as lower risk (Voskoglou, 2023). In clustering, the neutrosophic treatment of indeterminacy is explicitly tied to local density, so dense regions receive low indeterminacy, while boundary and outlier points receive higher indeterminacy and are handled differently by the optimization procedure (Rashno et al., 2018).

Several misconceptions recur. One is that neutrosophic sets are only useful for contradiction; the standard formulation explicitly covers incomplete, normalized, and inconsistent information, so contradiction is only one regime among several (Fujita et al., 12 Mar 2026). Another is that the theory is purely philosophical; the existence of software frameworks, decision procedures, clustering algorithms, and matrix formalisms suggests otherwise (Nordo et al., 2024). A third is that a single canonical calculus exists; in fact, the survey states that operational rules are not unique, and the literature contains multiple legitimate operator families (Fujita et al., 12 Mar 2026).

The open issues are therefore largely methodological. The 2026 survey identifies expressive power versus complexity, parameter selection, standardization of operations, and redundancy among closely related models as persistent questions (Fujita et al., 12 Mar 2026). The 2016 volume similarly notes increased model complexity, many alternative operator definitions, and a lack of consensus on the “best” operators in some domains (Smarandache et al., 2016). A plausible implication is that neutrosophic sets are best understood not as a single fixed calculus, but as a flexible three-component paradigm whose precise algebra depends on the intended semantics of indeterminacy.