Fact-checked by Grok 2 weeks ago

Fuzzy set

A fuzzy set is a generalization of a classical set in which elements have degrees of membership that are real numbers in the interval [0, 1], rather than strictly belonging or not belonging to the set.^[1] This membership is defined by a function μ_A: X → [0, 1], where X is the universal set and μ_A(x) represents the grade of membership of x in the fuzzy set A, with 0 indicating non-membership and 1 full membership.^[2] Introduced by Lotfi A. Zadeh in 1965, fuzzy set theory emerged as a mathematical framework to address imprecision and uncertainty in complex systems, such as those in the life sciences, economics, and engineering, where classical binary logic fails to capture gradual transitions in class membership—for instance, concepts like "tall" or "young."^[1] Zadeh's work built on advances in network and system theory, proposing fuzzy sets as a tool for modeling ill-defined classes without sharply delineated boundaries.^[2] Key operations on fuzzy sets include extensions of classical set operations: the complement of A is defined by μ_{A^c}(x) = 1 - μ_A(x); the intersection by μ_{A ∩ B}(x) = min(μ_A(x), μ_B(x)); and the union by μ_{A ∪ B}(x) = max(μ_A(x), μ_B(x)), enabling the manipulation of partial truths and overlaps.^[1] These concepts form the basis for fuzzy logic, which integrates with fuzzy sets to handle approximate reasoning and has evolved into a core component of soft computing alongside neural networks and evolutionary algorithms.^[3] Fuzzy set theory has found wide applications across disciplines, including control engineering for systems like fuzzy controllers in appliances and automotive systems; decision theory and operations research for multi-criteria optimization under uncertainty; artificial intelligence and expert systems for pattern recognition and robotics; and medicine for diagnostic modeling with imprecise data.^[3] Since its inception, the theory has advanced through extensions like type-2 fuzzy sets for handling linguistic uncertainties and interval-valued fuzzy sets, influencing fields from management science to environmental modeling.^[4]

Fundamentals

Definition

In fuzzy set theory, the concept of a fuzzy set was introduced by Lotfi A. Zadeh in 1965 as a mathematical framework to model vagueness and imprecision inherent in natural language and real-world data, extending classical set theory beyond binary membership.^[1] Unlike crisp sets, where elements either belong or do not belong to a set, fuzzy sets allow for degrees of membership, enabling graded representations of uncertainty.^[1] Formally, a fuzzy set A defined on a universe of discourse X—the universal set or domain containing all possible elements under consideration—is characterized by a membership function \mu_A: X \to [0,1], where \mu_A(x) denotes the degree to which element x \in X belongs to A, with values ranging from 0 (no membership) to 1 (full membership).^[1] This function assigns a real number in the unit interval to each element, quantifying partial belongingness; for instance, \mu_A(x) = 0.7 indicates a moderate degree of membership.^[1] The fuzzy set itself can be represented as the set of ordered pairs A = \{ (x, \mu_A(x)) \mid x \in X, \mu_A(x) > 0 \}, omitting pairs with zero membership for compactness, though the full definition includes all elements of X.^[1] To illustrate, consider a finite universe X = \{1, 2, 3\} representing possible temperatures in degrees Celsius. A fuzzy set A for "high temperatures" might be defined as A = \{(1, 0.9), (2, 0.5), (3, 0)\}, where 1 has high membership (0.9), 2 has moderate membership (0.5), and 3 has none (0).^[1] This example highlights how fuzzy sets capture continuum-like gradations, assuming familiarity with basic crisp set theory where membership is strictly 0 or 1.^[1]

Membership functions

The membership function of a fuzzy set A, denoted \mu_A: X \to [0,1], assigns to each element x in the universe X a real value representing the degree of membership of x in A, where 0 indicates no membership and 1 indicates full membership.^[1] These functions generalize the characteristic functions of crisp sets by allowing intermediate grades of belonging, and they can be defined over discrete or continuous universes, depending on the nature of X.^[1] For non-empty fuzzy sets, membership functions are typically normalized such that the supremum of \mu_A(x) over X equals 1, ensuring the set has at least one element with complete membership; such sets are called normal fuzzy sets.^[5] Common forms of membership functions include triangular, trapezoidal, Gaussian, and sigmoid shapes, chosen for their simplicity and interpretability in modeling gradual transitions. The triangular membership function, widely used due to its computational efficiency, is defined for parameters a < b < c as

\mu_A(x) = \max\left( \min\left( \frac{x - a}{b - a}, \frac{c - x}{c - b} \right), 0 \right),

where the function rises linearly from 0 at a to 1 at b, then falls linearly to 0 at c.^[6] The trapezoidal function extends this with a flat peak between two points, defined similarly using four parameters for the base and top. Gaussian functions, inspired by probability distributions, take the form \mu_A(x) = e^{-\frac{(x - c)^2}{2\sigma^2}}, providing smooth, bell-shaped curves parameterized by center c and width \sigma. Sigmoid functions, \mu_A(x) = \frac{1}{1 + e^{-k(x - c)}}, model S-shaped transitions, useful for cumulative effects.^[6] A fuzzy set A is convex if, for all x, y \in X and \lambda \in [0,1],

\mu_A(\lambda x + (1 - \lambda) y) \geq \min(\mu_A(x), \mu_A(y)),

meaning the membership degree at any convex combination of points is at least as high as the minimum of the individual degrees; this property ensures that the \alpha-cuts of A are convex sets for all \alpha \in (0,1].^[1] Normality, as noted earlier, requires \sup_{x \in X} \mu_A(x) = 1, distinguishing normal sets from subnormal ones where the maximum membership is less than 1, often arising in approximations or intersections.^[7] Membership functions enable the representation of linguistic variables, where vague terms like "young" or "tall" are modeled as fuzzy sets over domains such as age or height; for instance, "young" might use a decreasing function starting at 1 for ages around 20 and approaching 0 beyond 40, capturing subjective gradations in natural language.^[8]

Relation to crisp sets

Crisp sets, also known as classical or ordinary sets, are defined by a characteristic function \chi_A: X \to \{0,1\}, where X is the universal set, \chi_A(x) = 1 if x \in A, and \chi_A(x) = 0 otherwise.^[1] This binary assignment enforces a strict dichotomy of membership or non-membership for every element.^[9] Fuzzy sets generalize crisp sets by replacing the characteristic function with a membership function \mu_A: X \to [0,1], allowing degrees of membership that capture partial belonging.^[1] Crisp sets emerge as a special case of fuzzy sets when the membership function takes only values in \{0,1\}, thus recovering the binary logic of classical set theory.^[9] A key connection between fuzzy and crisp sets is provided by alpha-cuts, also called level sets, which decompose a fuzzy set into a family of crisp sets. The weak (or closed) alpha-cut at level \alpha \in [0,1] is defined as A_\alpha = \{ x \in X \mid \mu_A(x) \geq \alpha \}, representing elements with membership at least \alpha.^[1] The strong (or open) alpha-cut is A^\alpha = \{ x \in X \mid \mu_A(x) > \alpha \}, capturing elements with membership strictly greater than \alpha.^[9] These alpha-cuts exhibit nestedness: for \alpha > \beta, the weak alpha-cuts satisfy A_\alpha \subseteq A_\beta, and similarly for strong alpha-cuts A^\alpha \subseteq A^\beta.^[9] This monotonicity reflects the gradual nature of fuzzy membership, enabling the representation of fuzzy sets through nested crisp subsets that approximate thresholds of commitment.^[1]

Operations and Properties

Set operations

Fuzzy set operations generalize the classical set operations of union, intersection, and complement to accommodate partial membership degrees, allowing for a more nuanced representation of uncertainty and vagueness.^[10] In the foundational formulation, these operations are defined pointwise on the membership functions of the sets involved.^[10] The union of two fuzzy sets A and B, denoted A \cup B, has a membership function given by

\mu_{A \cup B}(x) = \max(\mu_A(x), \mu_B(x))

for all x in the universe, which selects the highest degree of membership at each point.^[10] An alternative formulation uses the probabilistic sum, defined as

\mu_{A \cup B}(x) = \mu_A(x) + \mu_B(x) - \mu_A(x) \mu_B(x),

which models a non-interaction assumption between the sets.^[11] The intersection of A and B, denoted A \cap B, is defined by

\mu_{A \cap B}(x) = \min(\mu_A(x), \mu_B(x)),

capturing the lowest common membership degree.^[10] Another common variant is the algebraic product:

\mu_{A \cap B}(x) = \mu_A(x) \mu_B(x),

which interprets membership degrees probabilistically.^[11] The complement of a fuzzy set A, denoted \bar{A}, is standardly given by

\mu_{\bar{A}}(x) = 1 - \mu_A(x),

inverting the membership to represent non-belonging.^[10] A parameterized generalization, the Sugeno complement, takes the form

\mu_{\bar{A}}(x) = \frac{1 - \mu_A(x)}{1 + k \mu_A(x)}

for a parameter k > -1, allowing adjustment of the complement's strictness; when k = 0, it reduces to the standard complement.^[12] These min-max operations satisfy key properties analogous to crisp sets, including commutativity (A \cup B = B \cup A, A \cap B = B \cap A), associativity ((A \cup B) \cup C = A \cup (B \cup C), similarly for intersection), distributivity (A \cup (B \cap C) = (A \cup B) \cap (A \cup C), and vice versa), and idempotence (A \cup A = A, A \cap A = A).^[10] They also obey De Morgan's laws: \overline{A \cup B} = \bar{A} \cap \bar{B} and \overline{A \cap B} = \bar{A} \cup \bar{B}.^[10] More generally, fuzzy set operations are framed using triangular norms (t-norms) for intersection and triangular conorms (t-conorms) for union, which are monotonic, associative binary operations on [0,1] with identity elements 1 and 0, respectively.^[11] The minimum serves as a t-norm, while the maximum is a t-conorm; other examples include the Łukasiewicz t-norm T_L(a, b) = \max(a + b - 1, 0) and its dual t-conorm S_L(a, b) = \min(a + b, 1).^[11] Complements pair with t-norms and t-conorms to satisfy generalized De Morgan laws, ensuring consistency in fuzzy algebra.^[11]

Cardinality and measures

The scalar cardinality of a fuzzy set A, also known as the sigma-count or power, generalizes the notion of set size by summing the membership degrees of its elements. For a fuzzy set A defined on a finite discrete universe X = \{x_1, \dots, x_n\}, the scalar cardinality is given by \sigma(A) = \sum_{i=1}^n \mu_A(x_i), where \mu_A(x_i) is the membership function value for each x_i \in [0,1].90411-9) This measure was introduced by De Luca and Termini as a foundational tool for quantifying the "strength" or effective size of fuzzy sets.90411-9) For fuzzy sets over a continuous universe X, the scalar cardinality extends naturally to the Lebesgue integral \sigma(A) = \int_X \mu_A(x) \, dx, providing a continuous analog that integrates membership densities across the domain. The relative cardinality normalizes the scalar cardinality by the size of the universe, yielding \sigma(A)/|X| for finite X, which serves as a measure of the fuzzy set's support or average membership degree relative to the total space.^[13] This relative form is particularly useful in contexts where the universe's scale varies, offering a normalized indicator between 0 and 1. The sigma-count itself, as the discrete summation \sum_{x \in X} \mu_A(x), forms the basis for these measures and aligns with intuitive expectations for fuzzy counting.90411-9) To capture more nuanced aspects of fuzzy size beyond the first-order sum, the fuzzy cardinality can be represented as a vector of higher-order sigma-counts, where the k-th component is s_k(A) = \sum_{x \in X} \mu_A(x)^k for k = 1, 2, \dots, providing moments that describe the distribution of membership degrees. This vectorial approach, developed by Kosko, allows for richer analysis, such as approximating the fuzzy set's cardinality via Poisson-like distributions using these moments. Key properties of the scalar cardinality include monotonicity under fuzzy inclusion: if A \subseteq B, then \sigma(A) \leq \sigma(B), reflecting that larger sets in the fuzzy sense have greater or equal size.^[14] Additionally, it exhibits additivity for disjoint fuzzy sets, where \sigma(A \cup B) = \sigma(A) + \sigma(B) if A \cap B = \emptyset, and more generally satisfies \sigma(A) + \sigma(B) = \sigma(A \cap B) + \sigma(A \cup B) under standard min-max operations.^[14] For example, consider a fuzzy set A on X = \{x_1, x_2, x_3\} with membership values \mu_A(x_1) = 0.8, \mu_A(x_2) = 0.6, \mu_A(x_3) = 0.4; the scalar cardinality is \sigma(A) = 0.8 + 0.6 + 0.4 = 1.8, indicating an effective size between 1 and 3.90411-9)

Distance, similarity, and disjointness

In fuzzy set theory, distance measures quantify the difference between two fuzzy sets A and B defined on a universe X, treating their membership functions \mu_A and \mu_B as vectors. The Hamming distance, a fundamental metric, is defined for a finite universe X = \{x_1, \dots, x_n\} as

d_H(A, B) = \frac{1}{n} \sum_{i=1}^n |\mu_A(x_i) - \mu_B(x_i)|,

which represents the average absolute deviation in membership degrees across all elements. This measure generalizes the classical Hamming distance for crisp sets by accounting for partial memberships.^[9] The Euclidean distance extends this to a root-mean-square form:

d_E(A, B) = \sqrt{\frac{1}{n} \sum_{i=1}^n (\mu_A(x_i) - \mu_B(x_i))^2},

emphasizing larger differences more heavily due to the squaring operation. Both distances are widely used in pattern recognition and clustering to assess how closely two fuzzy sets align.^[9] Similarity measures, conversely, capture the degree of resemblance between fuzzy sets, often derived from vector or set-theoretic perspectives. The Jaccard similarity index for fuzzy sets adapts the classical set similarity by incorporating fuzzy cardinalities, defined as

J(A, B) = \frac{|A \cap B|}{|A \cup B|},

where the cardinalities |A \cap B| and |A \cup B| are computed using the \sigma-count or other fuzzy counting methods based on the min and max operations for intersection and union, respectively. This index ranges from 0 (complete dissimilarity) to 1 (identical sets) and is particularly useful for comparing overlapping fuzzy concepts in information retrieval.^[15] The cosine similarity measure treats membership functions as vectors and computes the cosine of the angle between them:

\cos(\theta) = \frac{\sum_{x \in X} \mu_A(x) \mu_B(x)}{\sqrt{\sum_{x \in X} \mu_A(x)^2} \sqrt{\sum_{x \in X} \mu_B(x)^2}},

focusing on directional alignment rather than magnitude, making it robust for high-dimensional fuzzy data in decision-making applications.^[16] Two fuzzy sets A and B are considered disjoint if their intersection is the empty fuzzy set, generalizing crisp disjointness. Formally, this occurs when \sup_{x \in X} \min(\mu_A(x), \mu_B(x)) = 0, meaning no element has positive membership in both sets simultaneously. This condition ensures that the supports of A and B do not overlap in terms of membership degrees.^[10] Distance measures like Hamming and Euclidean satisfy key metric properties, including the triangle inequality: d(A, C) \leq d(A, B) + d(B, C) for any fuzzy sets A, B, C, enabling their use in fuzzy metric spaces for optimization tasks.^[9] Similarity measures, such as Jaccard and cosine, exhibit reflexivity (S(A, A) = 1) and symmetry (S(A, B) = S(B, A)), ensuring they behave intuitively as resemblance indicators without directional bias.^[17]

Measures of Uncertainty

Entropy

In fuzzy set theory, entropy serves as a measure of the fuzziness or uncertainty inherent in a fuzzy set, quantifying the degree of ambiguity in membership degrees, distinct from probabilistic information-theoretic entropy. This concept captures how far a fuzzy set deviates from a crisp set, where membership is binary, providing a tool to assess vagueness in decision-making and modeling imprecise information.^[18] One seminal measure of fuzziness is the De Luca and Termini entropy, defined for a fuzzy set A on a finite universe X = \{x_1, \dots, x_n\} as

H(A) = -\sum_{i=1}^n \mu_A(x_i) \log \mu_A(x_i) - \sum_{i=1}^n (1 - \mu_A(x_i)) \log (1 - \mu_A(x_i)),

where \mu_A(x_i) \in [0,1] is the membership degree of x_i in A, and the logarithm is typically base 2 for bits of uncertainty. This formula draws an analogy to Shannon entropy by treating membership and non-membership as symmetric probabilities, maximizing when all \mu_A(x_i) = 0.5, yielding H(A) = n \log 2, which represents maximal fuzziness. For crisp sets, where each \mu_A(x_i) is 0 or 1, H(A) = 0, indicating no uncertainty. Other early measures include Zadeh's entropy based on the variance of membership values and Kosko's ratio-based entropy, which emphasize different aspects of fuzziness.^[18]^[19] De Luca and Termini established axiomatic properties for such entropy measures, including non-negativity (H(A) \geq 0), with equality only for crisp sets; invariance under complementation (H(A) = H(A^c), where A^c has membership $1 - \mu_A(x)); and additivity for orthogonal fuzzy sets (disjoint supports where \mu_A(x) + \mu_B(x) \leq 1 for all x, such that H(A \cup B) = H(A) + H(B)). These properties ensure the measure behaves consistently under fuzzy operations and transformations.^[18] An alternative approach is Yager's entropy, which quantifies deviation from crispness by averaging the minimum distance to the boundaries 0 and 1:

H(A) = \frac{1}{n} \sum_{i=1}^n \bigl(1 - \max(\mu_A(x_i), 1 - \mu_A(x_i))\bigr).

This simplifies to the average of \min(\mu_A(x_i), 1 - \mu_A(x_i)) across elements, emphasizing the "hesitation" at each point; it reaches maximum H(A) = 0.5 for uniform membership 0.5 and is zero for crisp sets. Yager's measure satisfies similar axioms to De Luca and Termini, including non-negativity and complement invariance, but focuses more directly on the spread from binary extremes.^[20] These entropy measures find applications in quantifying vagueness for fuzzy modeling, such as in pattern recognition where H(A) = 0 confirms a clear classification and H(A) = \log 2 (for n=1) signals complete ambiguity in a single-element universe. In control systems, they help optimize membership functions by minimizing entropy to reduce uncertainty in decision rules.^[18]^[20] Recent developments as of 2025 have extended fuzzy entropy to hybrid systems, such as intuitionistic fuzzy sets combining membership and non-membership degrees, with new axiomatic characterizations ensuring robustness in multi-criteria decision-making under compounded uncertainties. Further extensions include applications to neutrosophic sets for handling indeterminacy.^[21]^[22]

Fuzzy categories

Fuzzy categories provide a framework for modeling uncertainty in relational and compositional structures within fuzzy set theory, extending classical category theory to handle graded memberships and vague relations. While not a direct entropy-like measure, they quantify uncertainty through degrees of morphism validity and composition, offering tools for analyzing ambiguity in hierarchical or networked systems.^[23] Formally, a fuzzy category consists of a class of objects, for each pair of objects A and B, a fuzzy set Hom(A,B) whose elements are potential arrows from A to B with membership degrees in [0,1], and a composition operation that assigns to each pair of compatible arrows f ∈ Hom(A,B) and g ∈ Hom(B,C) a composite arrow g ∘ f ∈ Hom(A,C), satisfying fuzzy versions of associativity and identity axioms. This framework was pioneered by Joseph Goguen in his work on non-Cantorian set theory, where fuzzy categories provide a categorical foundation for handling inexact concepts and relations.^[24] The degree of a morphism f: A → B is given by its membership value μ_{Hom(A,B)}(f) ∈ [0,1], which quantifies the extent to which f qualifies as an arrow between A and B, reflecting uncertainty in relational validity. Fuzzy identities exist for each object A, with the identity morphism id_A satisfying μ_{Hom(A,A)}(id_A) = 1, ensuring full membership as the neutral element for composition. Composition in fuzzy categories is defined using a t-norm, such as the minimum operator, where the membership of the composite satisfies μ_{Hom(A,C)}(g ∘ f) ≥ T(μ_{Hom(A,B)}(f), μ_{Hom(B,C)}(g)) for a t-norm T (e.g., min(μ_f, μ_g)), promoting a form of fuzzy associativity that preserves degrees in a monotonic way and bounds uncertainty propagation.^[23]^[25] Examples of fuzzy categories include fuzzy posets, where the objects are elements of a set, and the hom-set Hom(p, q) is a fuzzy set representing a fuzzy preorder relation R with μ_{Hom(p,q)}(p ≤ q) = R(p, q) ∈ [0,1]; composition corresponds to transitivity via the t-norm, such as μ(g ∘ f) = min(μ_f, μ_g), allowing measurement of uncertain ordering. In fuzzy topology, fuzzy categories model open sets with fuzzy memberships, where morphisms represent fuzzy continuous functions, enabling the study of topological properties like compactness and connectedness in a graded manner to capture spatial uncertainty.^[23]^[26] The development of fuzzy categories emerged in the late 1960s as part of broader efforts to integrate fuzzy set theory with abstract algebraic structures, with foundational contributions from Goguen's 1969 paper on the logic of inexact concepts and subsequent extensions in the 1970s and 1980s to more general lattices and quantales. As of 2025, applications continue in areas like fuzzy concept analysis for knowledge representation under uncertainty.^[25]^[27]^[28]

Extensions

L-fuzzy sets

L-fuzzy sets represent a generalization of fuzzy sets, where the membership degrees are elements of an arbitrary complete lattice L rather than the unit interval [0,1]. Formally, given a nonempty universe X and a complete lattice L equipped with a partial order \leq, an L-fuzzy set A on X is defined as a mapping A: X \to L, where A(x) denotes the grade of membership of x \in X in A. This structure was introduced by Joseph A. Goguen in 1967 to provide a unified framework for handling vagueness in various algebraic settings beyond numerical truth values.^[29] The operations on L-fuzzy sets are defined pointwise using the lattice structure of L. For two L-fuzzy sets A, B: X \to L, the intersection is given by

(A \wedge B)(x) = A(x) \wedge B(x),

where \wedge is the lattice meet (infimum), and the union by

(A \vee B)(x) = A(x) \vee B(x),

with \vee as the lattice join (supremum). These operations preserve the lattice properties, ensuring that the collection of all L-fuzzy sets on X, denoted \mathrm{FS}_L(X), forms a complete lattice under pointwise ordering, where the infimum and supremum of a family \{A_i\} are \inf(A_i)(x) = \inf\{A_i(x)\} and \sup(A_i)(x) = \sup\{A_i(x)\}, respectively. Complement operations require additional structure on L, such as an involution, but are not universally defined in the basic framework.^[29]^[30] Examples illustrate the flexibility of L-fuzzy sets. When L = \{0, 1\} with $0 \leq 1, the meet \wedge as logical AND (min), and join \vee as OR (max), L-fuzzy sets recover classical crisp sets, where A(x) = 1 indicates full membership and $0 indicates none. For L = [0,1] ordered by the standard \leq, with \wedge = \min and \vee = \max, they coincide with Zadeh's original fuzzy sets. More advanced cases include L = [0,1]^n under the product order, allowing multidimensional membership grades to model complex concepts like "good country" across attributes such as education and economy. Probabilistic interpretations can arise when L = [0,1] is equipped with a different monoidal structure, such as multiplication for conjunction.^[29]^[30] Key properties of L-fuzzy sets include inclusion and equality relations: A \subseteq B if and only if A(x) \leq B(x) for all x \in X, and A = B if A(x) = B(x) for all x. The \alpha-cuts, defined as A_\alpha = \{x \in X \mid \alpha \leq A(x)\} for \alpha \in L, are crisp subsets satisfying nesting (\alpha \leq \beta implies A_\beta \subseteq A_\alpha) and recoverability (A is the supremum of its \alpha-cuts). These ensure adjointness in residuated structures when L admits implications, facilitating logical inferences.^[29]^[30] Applications of L-fuzzy sets lie primarily in modeling multi-valued logics and non-statistical uncertainty, such as in pattern recognition where boundaries are inherently vague. Goguen's framework supports abstract treatments of classification problems by embedding them in lattice-based semantics, enabling operations like weighting without probabilistic assumptions. They have been extended to algebraic structures, including L-fuzzy topologies and ideals in semigroups, for applications in decision-making and computational intelligence.^[29]^[31]

Pythagorean fuzzy sets

Pythagorean fuzzy sets, introduced by Ronald R. Yager in 2013, extend the framework of intuitionistic fuzzy sets by relaxing the constraint on membership and non-membership degrees.^[32] A Pythagorean fuzzy set A on a universe X is defined as A = \{ \langle x, \mu_A(x), \nu_A(x) \rangle \mid x \in X \}, where \mu_A: X \to [0,1] represents the degree of membership, \nu_A: X \to [0,1] represents the degree of non-membership, and these satisfy the condition \mu_A^2(x) + \nu_A^2(x) \leq 1 for all x \in X. This formulation draws inspiration from Pythagorean theorem-like relations in the unit square, allowing for a broader representation of uncertainty compared to stricter linear constraints.^[32] The degree of hesitancy, or indeterminacy, in a Pythagorean fuzzy set is given by \pi_A(x) = \sqrt{1 - \mu_A^2(x) - \nu_A^2(x)}, which quantifies the non-determinacy and satisfies $0 \leq \pi_A(x) \leq 1. This hesitancy measure arises naturally from the Pythagorean condition, providing a geometric interpretation where the point (\mu_A(x), \nu_A(x)) lies within or on the unit circle in the first quadrant.^[32] Basic set operations for Pythagorean fuzzy sets are defined to preserve the defining condition. For two Pythagorean fuzzy sets A and B, the union is \mu_{A \cup B}(x) = \sqrt{\mu_A^2(x) + \mu_B^2(x) - \mu_A^2(x) \mu_B^2(x)} and \nu_{A \cup B}(x) = \sqrt{\nu_A^2(x) \nu_B^2(x)}, while the intersection uses \mu_{A \cap B}(x) = \sqrt{\mu_A^2(x) \mu_B^2(x)} and \nu_{A \cap B}(x) = \sqrt{\nu_A^2(x) + \nu_B^2(x) - \nu_A^2(x) \nu_B^2(x)}.^[32] These algebraic operations, analogous to probabilistic unions and intersections, ensure that the resulting sets remain Pythagorean fuzzy sets. Pythagorean fuzzy sets generalize intuitionistic fuzzy sets, where the condition is the stricter \mu(x) + \nu(x) \leq 1.^[32] For instance, degrees \mu(x) = 0.6 and \nu(x) = 0.6 satisfy $0.6^2 + 0.6^2 = 0.72 \leq 1 for Pythagorean fuzzy sets but violate $0.6 + 0.6 = 1.2 > 1 for intuitionistic fuzzy sets, enabling the modeling of scenarios with higher combined membership and non-membership. Pythagorean fuzzy sets find primary applications in multi-criteria decision-making under uncertainty, where they facilitate aggregation of expert opinions represented as paired degrees.^[32] Yager highlighted their utility in scenarios where intuitionistic constraints limit expressiveness, such as in complex decision environments involving trade-offs between affirmation and negation.

Type-2 fuzzy sets

Type-2 fuzzy sets extend the concept of fuzzy sets by allowing the membership degrees to themselves be fuzzy sets, thereby capturing uncertainties in the membership functions that arise from linguistic ambiguities or noisy data. Introduced by Lotfi A. Zadeh in 1975, a type-2 fuzzy set \tilde{A} defined on a universe of discourse X is mathematically represented as

\tilde{A} = \int_{x \in X} \int_{u \in [0,1]} \mu_{\tilde{A}}(x,u) / (x,u),

where \mu_{\tilde{A}}(x,u) denotes the secondary membership function, which assigns a membership grade u \in [0,1] to the primary membership value for each element x \in X. The footprint of uncertainty (FOU) of \tilde{A}, which visualizes the region of uncertainty, is defined as

\mathrm{FOU}(\tilde{A}) = \bigcup_{x \in X} \mathrm{supp} \, \mu_{\tilde{A}}(x,\cdot) = \{(x,u) \in X \times [0,1] \mid \mu_{\tilde{A}}(x,u) > 0 \}.

This structure enables type-2 fuzzy sets to model "fuzziness about fuzziness," providing a more nuanced representation of vagueness compared to crisp or type-1 fuzzy memberships.^[33] A prominent subclass is the interval type-2 fuzzy set, where the secondary membership function takes binary values, effectively representing the primary membership as an interval [\underline{\mu}_{\tilde{A}}(x), \overline{\mu}_{\tilde{A}}(x)] \subseteq [0,1] with \mu_{\tilde{A}}(x,u) = 1 for u within the interval and 0 otherwise. This simplification bounds the type-2 set between an upper membership function \overline{\mu}_{\tilde{A}}(x) and a lower membership function \underline{\mu}_{\tilde{A}}(x), facilitating computational efficiency while still addressing membership uncertainties. In contrast, general type-2 fuzzy sets employ continuous secondary membership functions, allowing for smoother variations in uncertainty but at higher computational cost. These interval variants are widely used in practical implementations due to their balance of expressiveness and tractability. Operations on type-2 fuzzy sets, such as union and intersection, are derived via Zadeh's extension principle, extending type-1 operations to the secondary domain using supremum-minimum compositions. For example, assuming the maximum t-conorm for union, the membership of the union \tilde{A} \cup \tilde{B} is computed as \mu_{\tilde{A} \cup \tilde{B}}(x,u) = \sup \{ \min(\mu_{\tilde{A}}(x,v), \mu_{\tilde{B}}(x,w)) \mid \max(v,w) = u \}, with similar sup-t-norm formulations for more general cases. To make type-2 fuzzy sets operable in decision-making systems, type-reduction is essential, converting the type-2 set to a type-1 fuzzy set. The centroid type-reduction method, implemented via the iterative Karnik-Mendel (KM) algorithm, calculates the centroid as y_c = \frac{\int \dots \int [u \mu_{\tilde{A}}(x,u)] \, du \, dx}{\int \dots \int [\mu_{\tilde{A}}(x,u)] \, du \, dx} by switching between lower and upper bounds to find the balance point, offering a defuzzified representative that preserves uncertainty information.^[33] Type-2 fuzzy sets offer significant advantages in managing linguistic uncertainties, such as those in natural language processing or expert knowledge elicitation, where exact membership values are inherently imprecise; this leads to more robust fuzzy logic systems that outperform type-1 counterparts in environments with high variability or noise. Post-2000 advancements, particularly in computational algorithms, have made type-2 sets viable for real-world applications. Recent developments integrate type-2 fuzzy sets with deep learning in hybrid frameworks, enhancing control systems—for instance, clustering-based adaptive type-2 neuro-fuzzy models for real-time prediction of PID control efforts in dynamic processes, achieving improved accuracy and adaptability in uncertain conditions.^[34]

Fuzzy numbers

A fuzzy number is a fuzzy set on the real line ℝ that is convex, normal, upper semi-continuous, and possesses compact support, enabling it to represent approximate quantities in a mathematically tractable manner.^[35] This structure ensures the membership function μ_A: ℝ → [0,1] reaches a maximum value of 1 at some point (normality), the set of points with membership at least α forms a closed interval for each α ∈ (0,1] (convexity and upper semi-continuity), and the support {x ∈ ℝ | μ_A(x) > 0} is bounded (compact support).^[36] A classic example is the Gaussian fuzzy number centered at m with spread σ > 0, defined by μ_A(x) = exp(-(x - m)^2 / (2σ^2)), which models symmetric uncertainty around a mean value.^[37] Key properties of fuzzy numbers include bounded support, which confines the uncertainty to a finite interval, and unimodality, stemming from normality, where the membership function increases to a peak and then decreases.^[38] The α-cuts, denoted [A]_α = {x ∈ ℝ | μ_A(x) ≥ α} = [A^L(α), A^R(α)], are closed and bounded intervals for all α ∈ (0,1], with A^L(α) and A^R(α) being non-decreasing and non-increasing functions, respectively.^[39] These properties facilitate decomposition and reconstruction of fuzzy numbers via their α-cuts, supporting efficient computations.^[40] Arithmetic operations on fuzzy numbers are typically defined using Zadeh's extension principle, which extends crisp operations to fuzzy domains by preserving membership degrees.^[41] For addition, the membership function is μ_{A ⊕ B}(z) = \sup_{x + y = z} \min(μ_A(x), μ_B(y)), with similar sup-min formulations for subtraction, multiplication, and division, though the latter requires handling cases where the denominator's membership is zero.^[42] For specific types like triangular fuzzy numbers A = (a, b, c) with a ≤ b ≤ c and membership μ_A(x) = \max\left( \min\left( \frac{x - a}{b - a}, \frac{c - x}{c - b} \right), 0 \right), arithmetic simplifies via endpoint operations on α-cuts: the sum's α-cut is [a + d, c + f] where B = (d, e, f) is another triangular fuzzy number.^[43] Trapezoidal fuzzy numbers extend this with a flat peak [b, c] where μ_A(x) = 1, with α-cuts given by [a + (b - a)α, d - (d - c)α]; arithmetic uses interval addition on these α-cuts.^[44] More general types include LR fuzzy numbers, parameterized by a modal value m, left and right spreads α and β, and non-increasing shape functions L and R: [0,1] → [0,1] with L(0) = R(0) = 1, such that μ_A(x) = L((m - x)/α) for x ≤ m, μ_A(x) = 1 for the core if α = β = 0, and μ_A(x) = R((x - m)/β) for x ≥ m.^[45] These allow flexible modeling of asymmetric uncertainty, with operations computable in closed form when L and R are specified (e.g., linear for triangular).^[46] Fuzzy numbers find prominent applications in fuzzy arithmetic for optimization, where they model imprecise coefficients in linear programming to yield robust solutions under uncertainty.^[47] Recent extensions post-2023 incorporate advanced interval arithmetic to enhance computational efficiency and handle complex operations like division more accurately, using representations such as extensional fuzzy numbers for fuzzy sets.^[48]

Fuzzy relations and equations

A fuzzy binary relation R on sets X and Y is defined as a fuzzy set on the Cartesian product X \times Y, characterized by a membership function \mu_R(x, y) \in [0, 1] for each pair (x, y) \in X \times Y, representing the degree to which x is related to y. This generalizes crisp binary relations by allowing graded associations rather than binary membership.^[49] The composition of two fuzzy relations R on X \times Y and S on Y \times Z, denoted R \circ S, is typically performed using the max-min rule:

\mu_{R \circ S}(x, z) = \bigvee_{y \in Y} \left( \mu_R(x, y) \wedge \mu_S(y, z) \right),

where \vee denotes the maximum and \wedge the minimum; this operation captures the strongest possible chain of associations through intermediate elements y.^[49] The max-min composition is associative, enabling multi-step relational inferences.^[49] Fuzzy relations exhibit properties analogous to those of crisp relations, adapted to graded memberships. A relation R on X \times X is reflexive if \mu_R(x, x) = 1 for all x \in X, indicating full self-association.^[49] It is symmetric if \mu_R(x, y) = \mu_R(y, x) for all x, y \in X, reflecting bidirectional equality in degrees.^[49] Transitivity holds if \mu_{R \circ R}(x, z) \geq \mu_R(x, z) for all x, z \in X, meaning the direct association is at least as strong as the strongest indirect path.^[49] Fuzzy relation equations, such as X \circ R = B where X is an unknown fuzzy relation on W \times X, R on X \times Y, and B on W \times Y, are solved using max-min composition to find X that satisfies the equation.^[50] The greatest (maximal) solution is given by \hat{X} = B \circ R^-, where R^- is the pseudoinverse of R defined componentwise as R^-(y, x) = \bigwedge_{w} i(\mu_R(x, y), \mu_B(w, y)), with i(a, b) = 1 if a \leq b and i(a, b) = b otherwise (Gödel implication).^[50] Solvability requires that substituting \hat{X} back into the equation yields \hat{X} \circ R = B; if true, \hat{X} is the unique maximal solution, and the complete solution set consists of all X \leq \hat{X} that satisfy the equation, often comprising finitely many minimal solutions.^[50] These equations find applications in control systems for modeling relational constraints and in fuzzy inference for tasks like medical diagnosis, where R represents symptom-disease associations and B observed symptoms.^[50] Recent extensions include fuzzy relational databases using associative arrays to handle graded data storage and querying, improving flexibility in uncertain information systems.^[51] Additionally, studies on equation stability analyze fluctuation tolerance in solutions, such as symmetric interval solutions for fuzzy relation inequalities, enhancing robustness in uncertain systems.^[52]

Fuzzy logic

Fuzzy logic extends classical two-valued logic to handle uncertainty and imprecision by allowing truth values in the continuous interval [0,1], where fuzzy sets provide the foundational semantics for propositions and operations. Introduced as a multivalued logic system, it enables approximate reasoning through graded truth degrees rather than binary true/false assignments.^[53] In fuzzy logic, an atomic proposition p of the form "x is A," where A is a fuzzy set, has a truth value τ(p) = μ_A(x), representing the degree to which x belongs to A. This assignment interprets the membership function μ_A as a fuzzy truth value, allowing propositions to express partial truths. Complex propositions are formed using logical connectives: negation ¬p is defined as the complement 1 - τ(p); conjunction p ∧ q uses a triangular norm (t-norm) T(τ(p), τ(q)), such as the minimum operator min(τ(p), τ(q)); disjunction p ∨ q employs a triangular conorm (t-conorm) S(τ(p), τ(q)), like the maximum max(τ(p), τ(q)); and implication p → q is often realized via the S-implication max(1 - τ(p), τ(q)), known as the Kleene-Dienes implication. These connectives generalize Boolean operations to fuzzy environments, preserving properties like monotonicity and boundary conditions. Inference in fuzzy logic relies on generalized rules adapted from classical logic. The generalized modus ponens, from premises "if p then q" with τ(p → q) and fact τ(p), infers q with degree min(τ(p), τ(p → q)) when using the minimum t-norm. More generally, the compositional rule of inference computes the output fuzzy set B' from input A' and relation R via the sup-t-norm composition: μ_{B'}(y) = \sup_x T(μ_{A'}(x), μ_R(x, y)), enabling relational propagation in rule-based systems.^[53]^[54] Fuzzy logic systems, particularly for control applications, include the Mamdani model, which uses fuzzy rules with min for aggregation and max for disjunction, followed by defuzzification to obtain crisp outputs—such as the centroid method, computing the center of gravity of the aggregated output membership function: z^* = \frac{\int z \mu(z) dz}{\int \mu(z) dz}. The Sugeno model, in contrast, employs linear or constant output functions in rules, simplifying computation with weighted average defuzzification: y = \frac{\sum_i w_i f_i}{\sum_i w_i}, where w_i is the firing strength and f_i the consequent function. These models facilitate practical implementations in uncertain environments. Historically, Zadeh outlined the calculus of fuzzy logic in 1973, laying the groundwork for approximate reasoning. Recent advancements integrate fuzzy logic with neural networks in hybrid neuro-fuzzy systems, enhancing AI applications like predictive modeling and control, as seen in 2024-2025 developments combining adaptive neuro-fuzzy inference with deep learning for energy optimization.^[55]^[56]^[57]^[53]

References

[1]
https://www.sciencedirect.com/science/article/pii/S001999586590241X
[2]
[PDF] OF FUZZY SYSTEMS by L. A. Zadeb
Definition of a fuzzy set. Let. X = {x} denote a space of points (objects), with x denoting a generic element of. X. Then a fuzzy set A in X is a set of ...
[3]
Fuzzy set theory - Zimmermann - Wiley Interdisciplinary Reviews
May 25, 2010 · In this review, the basic mathematical framework of fuzzy set theory will be described, as well as the most important applications of this theory to other ...Abstract · HISTORY · MATHEMATICAL THEORY... · APPLICATIONS
[4]
A Review of Fuzzy Set Theoretic Approaches and Their Applications ...
Fuzzy set theory deals with characterizing and quantifying vague and imprecise information and as such can be of high value in environmental practice.
[5]
Fuzzy sets - ScienceDirect.com
View PDF; Download full issue. Search ScienceDirect. Elsevier · Information and Control · Volume 8, Issue 3, June 1965, Pages 338-353. Information and Cont…
[6]
Measurement of Membership Functions: Theoretical and Empirical ...
This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function.
[7]
Evaluation of membership functions for fuzzy logic controlled ...
The general membership functions under consideration are triangular, trapezoidal, Gaussian, bell, sigmoidal and polynomial types.Missing: common | Show results with:common
[8]
(PDF) Introduction to Fuzzy Sets - ResearchGate
A fuzzy set is “normal” if its core is not empty. •A fuzzy set Ais “convex ... be a fuzzy set deﬁned in a discrete universe X= x1,x2,...,xp . A=μA(x1) ...<|control11|><|separator|>
[9]
The concept of a linguistic variable and its application to ...
1975, Pages 43-80. Information Sciences. The concept of a linguistic ... 27. L.A. Zadeh. A fuzzy-set-theoretic interpretation of linguistic hedges. J ...
[10]
[PDF] Fuzzy Sets and Fuzzy Logic: Theory and Applications
The first part of Fuzzy Sets and Fuzzy Logic provides a very carefully crafted introduction to the basic concepts and techniques of fuzzy set theory. The ...<|control11|><|separator|>
[11]
[PDF] Fuzzy Sets* - - Delft Center for Systems and Control
INFOR~ATIO~ AND CONTROL 8, 338--353 (1965). Fuzzy Sets* -. L. A. ZADEH. Department of Electrical Engineering and Electronics l~esearch Laboratory,. University ...
[12]
Triangular norms and conorms - Scholarpedia
Mar 15, 2007 · Triangular norms and conorms are operations which generalize the logical conjunction and logical disjunction to fuzzy logic.
[13]
[PDF] Lecture 8 -- Operations on Fuzzy Sets
Feb 15, 2007 · Design fuzzy morphological operations. Use α-cut decomposition principle, and fuzzification of set operation by t-norms and t-conorms.
[14]
[PDF] On Cardinality of Fuzzy Sets - MECS Press
That is to say that the cardinality of a fuzzy set A, the so called sigma- count, is expressed as the sum of the membership values of A. This approach to ...
[15]
[PDF] Counting with Fuzzy Sets Proposition 2:
Oct 16, 2009 · BART KOSKO. Abstract-The notion of fuzzy set cardinality is examined. Zadeh's suggested measure of fuzzy cardinality, the sigma-count, is ...
[16]
New similarity measures of intuitionistic fuzzy sets based on the ...
Apr 23, 2018 · The Jaccard index is a statistic used for comparing the similarity and diversity of sample sets. In this study, we propose a new similarity ...Missing: original | Show results with:original
[17]
Cosine similarity measures for intuitionistic fuzzy sets and their ...
Cosine similarity measures [15], [16] are defined as the inner product of two vectors divided by the product of their lengths. This is nothing but the cosine of ...
[18]
Similarity in fuzzy systems
Aug 21, 2014 · All the similarity measures listed above satisfy the reflexivity, symmetry and boundedness property. These three properties are ...
[19]
A definition of a nonprobabilistic entropy in the setting of fuzzy sets ...
A functional defined on the class of generalized characteristic functions (fuzzy sets), called “entropy≓, is introduced using no probabilistic concepts.
[20]
ON THE MEASURE OF FUZZINESS AND NEGATION Part I
ON THE MEASURE OF FUZZINESS AND NEGATION Part I: Membership in the Unit Interval. RONALD R. YAGER. Pages 221-229 | Published online: 02 Apr 2008. Cite this ...
[21]
A Hybrid Intuitionistic Fuzzy Entropy–BWM–WASPAS Approach for ...
Feb 18, 2025 · This paper proposes a novel integrated approach for supplier selection in shipbuilding enterprises by combining intuitionistic fuzzy sets (IFSs) with the ...Missing: characterizations | Show results with:characterizations
[22]
[1410.1478] Fuzzy Categories - arXiv
Oct 6, 2014 · Since categories are graphs with additional "structure", one should start from fuzzy graphs in order to define a theory of fuzzy categories.
[23]
The Logic of Inexact Concepts - JA Goguen - PhilPapers
The logic of inexact concepts · J. A. Goguen · Synthese 19 (3-4):325-373 (1969) ... Categories. Degree Theories of Vagueness in Philosophy of Language ...
[24]
The logic of inexact concepts | Synthese
J. A. Goguen, 'Categories of Fuzzy Sets: Applications of Non-Cantorian Set ... Goguen, J.A. The logic of inexact concepts. Synthese 19, 325–373 (1969) ...
[25]
Some Categorical Aspects of Fuzzy Topology - ScienceDirect
Research in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect ...
[26]
(PDF) Categories of fuzzy sets - ResearchGate
Aug 7, 2025 · In this paper we look at two categories, the category of fuzzy subsets and a quotient category / of fuzzy sets.
[27]
[PDF] On L- Fuzzy Sets - Research India Publications
For any L-fuzzy subsets A and B of X, define A ≤ B if and only if. A(x) ≤ B(x) for all x ∈ X. Clearly ≤ is a partial order on FSL(X). Also, for any crisp.
[28]
L-FUZZY SETS, - DTIC
L-fuzzy sets are a theory of sets with fuzzy boundaries, used for non-statistical uncertainty, such as in pattern classification.
[29]
Pythagorean fuzzy subsets | IEEE Conference Publication
We look at the basic set operations for the Pythagorean fuzzy subsets. Published in: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS).
[30]
A hybrid clustering-based type-2 adaptive neuro-fuzzy forecasting ...
Apr 1, 2024 · We focus on the real-time prediction of intelligent and active PID control effort utilizing the hybrid clustering-based adaptive type-2 neuro-fuzzy framework.
[31]
Basic Concepts
Definition 2.3. A fuzzy number is a fuzzy convex and normal fuzzy subset in R with upper semicontinuous membership function and compact support. The family ...
[32]
A note on the core of fuzzy numbers - ScienceDirect
A fuzzy number u is called the supremum of A if u is an upper bound of A and u ≤ v for each upper bound v of A .
[33]
[PDF] The role of fuzzy sets in decision sciences: old techniques and new ...
Abstract. We try to provide a tentative assessment of the role of fuzzy sets in decision analysis. We discuss membership functions, aggregation operations, ...
[34]
[PDF] Convexity and Upper Semicontinuity of Fuzzy Sets - CORE
A fuzzy number we treat in this study is a fuzzy set tt : R 1 ~ [0, 1] which is normal, upper semicontinuous, and quasi-convex.
[35]
Basic theorems for fuzzy differential equations in the quotient space ...
Dec 2, 2014 · ... fuzzy-valued function of a real variable whose values are normal, convex, upper semicontinuous, and are compactly supported fuzzy sets in R n .
[36]
Existence, Uniqueness, and Characterization Theorems for ...
Mar 2, 2015 · A fuzzy number u is a fuzzy subset of the real line with a normal, convex, and upper semicontinuous membership function of bounded support. For ...
[37]
A concrete extension principle for fuzzy set theory - ScienceDirect.com
Jun 25, 2025 · Zadeh's extension principle (ZEP) has served as a fundamental approach for extending mathematical functions for classical crisp sets to the corresponding ...
[38]
[PDF] extension Principle
An extension principle developed by Zadeh. [1975] enables us to extend the domain of a function on fuzzy sets. It thus generalizes a common point-to- point ...
[39]
Computational method for fuzzy arithmetic operations on triangular ...
This paper presents original computational methods for implementing fuzzy arithmetic operations on triangular fuzzy numbers using the extension principle ...
[40]
Fuzzy Numbers and Their Arithmetic - Wiley Online Library
Jun 26, 2020 · A triangular fuzzy number is a fuzzy set whose graph is a triangle. Trapezoidal fuzzy numbers are, of course, fuzzy sets but since their ...<|control11|><|separator|>
[41]
[PDF] Gradual Numbers and their Application to Fuzzy Interval Analysis
The second idea is to use arithmetics on L-R fuzzy intervals. (Dubois and Prade 1978 [5]). This method can give a closed form result for the basic arithmetic ...
[42]
[PDF] fuzzy real algebra: some results - HAL
Didier Dubois and Henri Prade, Comment on 'Tolerance analysis using fuzzy sets' and 'A procedure for multiple-aspect decision making', Internat. J. Systems Sci.
[43]
[PDF] 5. Fuzzy Optimization - Cornell eCommons
Both probability distributions and fuzzy membership functions of these uncertain or qualitative variables can be included in quantitative optimization models.
[44]
[PDF] operations on extensional fuzzy numbers - arXiv
Oct 27, 2025 · Abstract. Fuzzy numbers are commonly represented with fuzzy sets. Their objective is to better represent imprecise data.
[45]
https://www.irit.fr/~Didier.Dubois/Papers0804/DFF_IEEE_TFS08.pdf
[46]
https://hal.science/hal-04215228/document
[47]
https://ecommons.cornell.edu/bitstream/1813/2804/16/05_chapter05.pdf
[48]
Outline of a New Approach to the Analysis of Complex Systems and ...
The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis.
[49]
The generalized modus ponens considered as a fuzzy relation
The generalized modus ponens (GMP) is an inference rule in fuzzy logic, with the following scheme: if S1 is P1 then S2 is P2, but S1 is Q1, hence S2 is Q2.
[50]
An experiment in linguistic synthesis with a fuzzy logic controller
January 1975, Pages 1-13 ... An experiment in linguistic synthesis with a fuzzy logic controller. Author links open overlay panelE.H. Mamdani, S. Assilian.
[51]
Fuzzy identification of systems and its applications to modeling and ...
Feb 28, 1985 · Abstract: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented in this paper.
[52]
Internet of things driven hybrid neuro-fuzzy deep learning building ...
Mar 25, 2025 · We have implemented an automated, DL-enabled approach to predict energy consumption with the goal to enable cost and schedule optimization.