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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an ''L''-relation, which he studied in an
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
ic context. Fuzzy relations, which are now used throughout
fuzzy mathematics Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 ...
and have applications in areas such as
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
,
decision-making In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either ra ...
, and clustering , are special cases of ''L''-relations when ''L'' is the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, 1 In classical
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a
membership function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
valued in the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
unit interval , 1 Fuzzy sets generalize classical sets, since the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
s (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1.D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York. In fuzzy set theory, classical bivalent sets are usually called ''crisp sets''. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as
bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
.


Definition

A fuzzy set is a pair (U, m) where U is a set (often required to be non-empty) and m\colon U \rightarrow ,1/math> a membership function. The reference set U (sometimes denoted by \Omega or X) is called universe of discourse, and for each x\in U, the value m(x) is called the grade of membership of x in (U,m). The function m = \mu_A is called the membership function of the fuzzy set A = (U, m). For a finite set U=\, the fuzzy set (U, m) is often denoted by \. Let x \in U. Then x is called * not included in the fuzzy set (U,m) if (no member), * fully included if (full member), * partially included if The (crisp) set of all fuzzy sets on a universe U is denoted with SF(U) (or sometimes just F(U)).


Crisp sets related to a fuzzy set

For any fuzzy set A = (U,m) and \alpha \in ,1/math> the following crisp sets are defined: * A^ = A_\alpha = \ is called its α-cut (aka α-level set) * A^ = A'_\alpha = \ is called its strong α-cut (aka strong α-level set) * S(A) = \operatorname(A) = A^ = \ is called its support * C(A) = \operatorname(A) = A^ = \ is called its core (or sometimes kernel \operatorname(A)). Note that some authors understand "kernel" in a different way; see below.


Other definitions

* A fuzzy set A = (U,m) is empty (A = \varnothing) iff (if and only if) :: \forall x \in U: \mu_A(x) = m(x) = 0 * Two fuzzy sets A and B are equal (A = B) iff ::\forall x \in U: \mu_A(x) = \mu_B(x) * A fuzzy set A is included in a fuzzy set B (A \subseteq B) iff ::\forall x \in U: \mu_A(x) \le \mu_B(x) * For any fuzzy set A, any element x \in U that satisfies ::\mu_A(x) = 0.5 :is called a crossover point. * Given a fuzzy set A, any \alpha \in ,1/math>, for which A^ = \ is not empty, is called a level of A. * The level set of A is the set of all levels \alpha\in ,1/math> representing distinct cuts. It is the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of \mu_A: ::\Lambda_A = \ = \ = \mu_A(U) * For a fuzzy set A, its height is given by ::\operatorname(A) = \sup \ = \sup(\mu_A(U)) :where \sup denotes the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
, which exists because \mu_A(U) is non-empty and bounded above by 1. If ''U'' is finite, we can simply replace the supremum by the maximum. * A fuzzy set A is said to be normalized iff ::\operatorname(A) = 1 :In the finite case, where the supremum is a maximum, this means that at least one element of the fuzzy set has full membership. A non-empty fuzzy set A may be normalized with result \tilde by dividing the membership function of the fuzzy set by its height: ::\forall x \in U: \mu_(x) = \mu_A(x)/\operatorname(A) :Besides similarities this differs from the usual normalization in that the normalizing constant is not a sum. * For fuzzy sets A of real numbers (''U'' ⊆ ℝ) with bounded support, the width is defined as ::\operatorname(A) = \sup(\operatorname(A)) - \inf(\operatorname(A)) :In the case when \operatorname(A) is a finite set, or more generally a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
, the width is just ::\operatorname(A) = \max(\operatorname(A)) - \min(\operatorname(A)) :In the ''n''-dimensional case (''U'' ⊆ ℝ''n'') the above can be replaced by the ''n''-dimensional volume of \operatorname(A). :In general, this can be defined given any measure on ''U'', for instance by integration (e.g.
Lebesgue integration In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
) of \operatorname(A). * A real fuzzy set A (''U'' ⊆ ℝ) is said to be convex (in the fuzzy sense, not to be confused with a crisp
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
), iff ::\forall x,y \in U, \forall\lambda\in ,1 \mu_A(\lambda + (1-\lambda)y) \ge \min(\mu_A(x),\mu_A(y)). : Without loss of generality, we may take ''x'' ≤ ''y'', which gives the equivalent formulation ::\forall z \in ,y \mu_A(z) \ge \min(\mu_A(x),\mu_A(y)). : This definition can be extended to one for a general
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''U'': we say the fuzzy set A is convex when, for any subset ''Z'' of ''U'', the condition ::\forall z \in Z: \mu_A(z) \ge \inf(\mu_A(\partial Z)) : holds, where \partial Z denotes the boundary of ''Z'' and f(X) = \ denotes the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of a set ''X'' (here \partial Z) under a function ''f'' (here \mu_A).


Fuzzy set operations

Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity. * For a given fuzzy set A, its complement \neg (sometimes denoted as A^c or cA) is defined by the following membership function: ::\forall x \in U: \mu_(x) = 1 - \mu_A(x). * Let t be a
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection (s ...
, and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets A, B, their intersection A\cap is defined by: ::\forall x \in U: \mu_(x) = t(\mu_A(x),\mu_B(x)), :and their union A\cup is defined by: ::\forall x \in U: \mu_(x) = s(\mu_A(x),\mu_B(x)). By the definition of the t-norm, we see that the union and intersection are
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
,
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
,
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, and have both a
null Null may refer to: Science, technology, and mathematics Computing * Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value * Null character, the zero-valued ASCII character, also designated by , often use ...
and an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. For the intersection, these are ∅ and ''U'', respectively, while for the union, these are reversed. However, the union of a fuzzy set and its complement may not result in the full universe ''U'', and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finite
family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of fuzzy sets recursively. * If the standard negator n(\alpha) = 1 - \alpha, \alpha \in , 1/math> is replaced by another strong negator, the fuzzy set difference may be generalized by ::\forall x \in U: \mu_(x) = n(\mu_A(x)). * The triple of fuzzy intersection, union and complement form a De Morgan Triplet. That is,
De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
extend to this triple. :Examples for fuzzy intersection/union pairs with standard negator can be derived from samples provided in the article about
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection (s ...
s. :The fuzzy intersection is not
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
in general, because the standard t-norm is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent. That is, iteratively taking the intersection of a fuzzy set with itself is not trivial. It instead defines the ''m''-th power of a fuzzy set, which can be canonically generalized for non-
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
exponents in the following way: * For any fuzzy set A and \nu \in \R^+ the ν-th power of A is defined by the membership function: ::\forall x \in U: \mu_(x) = \mu_(x)^. The case of exponent two is special enough to be given a name. * For any fuzzy set A the concentration CON(A) = A^2 is defined ::\forall x \in U: \mu_(x) = \mu_(x) = \mu_(x)^2. Taking 0^0 = 1, we have A^0 = U and A^1 = A. * Given fuzzy sets A, B, the fuzzy set difference A \setminus B, also denoted A - B, may be defined straightforwardly via the membership function: ::\forall x \in U: \mu_(x) = t(\mu_A(x),n(\mu_B(x))), :which means A \setminus B = A \cap \neg, e. g.: ::\forall x \in U: \mu_(x) = \min(\mu_A(x),1 - \mu_B(x)).N.R. Vemuri, A.S. Hareesh, M.S. Srinath
Set Difference and Symmetric Difference of Fuzzy Sets
in: Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic
:Another proposal for a set difference could be: ::\forall x \in U: \mu_(x) = \mu_A(x) - t(\mu_A(x),\mu_B(x)). * Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, giving ::\forall x \in U: \mu_(x) = , \mu_A(x) - \mu_B(x), , :or by using a combination of just , , and standard negation, giving ::\forall x \in U: \mu_(x) = \max(\min(\mu_A(x), 1 - \mu_B(x)), \min(\mu_B(x), 1 - \mu_A(x))). :Axioms for definition of generalized symmetric differences analogous to those for t-norms, t-conorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009). * In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.


Disjoint fuzzy sets

In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets: Two fuzzy sets A, B are disjoint iff :\forall x \in U: \mu_A(x) = 0 \lor \mu_B(x) = 0 which is equivalent to : \nexists x \in U: \mu_A(x) > 0 \land \mu_B(x) > 0 and also equivalent to :\forall x \in U: \min(\mu_A(x),\mu_B(x)) = 0 We keep in mind that / is a t/s-norm pair, and any other will work here as well. Fuzzy sets are disjoint if and only if their supports are disjoint according to the standard definition for crisp sets. For disjoint fuzzy sets A, B any intersection will give ∅, and any union will give the same result, which is denoted as :A \,\dot\, B = A \cup B with its membership function given by :\forall x \in U: \mu_(x) = \mu_A(x) + \mu_B(x) Note that only one of both summands is greater than zero. For disjoint fuzzy sets A, B the following holds true: :\operatorname(A \,\dot\, B) = \operatorname(A) \cup \operatorname(B) This can be generalized to finite families of fuzzy sets as follows: Given a family A = (A_i)_ of fuzzy sets with index set ''I'' (e.g. ''I'' = ). This family is (pairwise) disjoint iff :\text x \in U \text i \in I \text \mu_(x) > 0. A family of fuzzy sets A = (A_i)_ is disjoint, iff the family of underlying supports \operatorname \circ A = (\operatorname(A_i))_ is disjoint in the standard sense for families of crisp sets. Independent of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity: :\dot\, A_i = \bigcup_ A_i with its membership function given by :\forall x \in U: \mu_(x) = \sum_ \mu_(x) Again only one of the summands is greater than zero. For disjoint families of fuzzy sets A = (A_i)_ the following holds true: :\operatorname\left(\dot\, A_i\right) = \bigcup\limits_ \operatorname(A_i)


Scalar cardinality

For a fuzzy set A with finite support \operatorname(A) (i.e. a "finite fuzzy set"), its cardinality (aka scalar cardinality or sigma-count) is given by :\operatorname(A) = \operatorname(A) = , A, = \sum_ \mu_A(x). In the case that ''U'' itself is a finite set, the relative cardinality is given by :\operatorname(A) = \, A\, = \operatorname(A)/, U, = , A, /, U, . This can be generalized for the divisor to be a non-empty fuzzy set: For fuzzy sets A,G with ''G'' ≠ ∅, we can define the relative cardinality by: :\operatorname(A,G) = \operatorname(A, G) = \operatorname(A\cap)/\operatorname(G), which looks very similar to the expression for
conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
. Note: * \operatorname(G) > 0 here. * The result may depend on the specific intersection (t-norm) chosen. * For G = U the result is unambiguous and resembles the prior definition.


Distance and similarity

For any fuzzy set A the membership function \mu_A: U \to ,1/math> can be regarded as a family \mu_A = (\mu_A(x))_ \in ,1U. The latter is a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
with several metrics d known. A metric can be derived from a norm (vector norm) \, \,\, via :d(\alpha,\beta) = \, \alpha - \beta \, . For instance, if U is finite, i.e. U = \, such a metric may be defined by: :d(\alpha,\beta) := \max \ where \alpha and \beta are sequences of real numbers between 0 and 1. For infinite U, the maximum can be replaced by a supremum. Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe: :d(A,B) := d(\mu_A,\mu_B), which becomes in the above sample: :d(A,B) = \max \ Again for infinite U the maximum must be replaced by a supremum. Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g., \varnothing and U. Similarity measures (here denoted by S) may then be derived from the distance, e.g. after a proposal by Koczy: :S = 1 / (1 + d(A,B)) if d(A,B) is finite, 0 else, or after Williams and Steele: :S = \exp(-\alpha) if d(A,B) is finite, 0 else where \alpha > 0 is a steepness parameter and \exp(x) = e^x.Ismat Beg, Samina Ashraf
Similarity measures for fuzzy sets
at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016
Another definition for interval valued (rather 'fuzzy') similarity measures \zeta is provided by Beg and Ashraf as well.


''L''-fuzzy sets

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable)
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
or
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
L of a given kind; usually it is required that L be at least a
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
or lattice. These are usually called ''L''-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in , 1are then called , 1valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh. A classical corollary may be indicating truth and membership values by instead of . An extension of fuzzy sets has been provided by Atanassov. An intuitionistic fuzzy set (IFS) A is characterized by two functions: :1. \mu_A(x) – degree of membership of ''x'' :2. \nu_A(x) – degree of non-membership of ''x'' with functions \mu_A, \nu_A: U \mapsto ,1/math> with \forall x \in U: \mu_A(x) + \nu_A(x) \le 1 This resembles a situation like some person denoted by x voting * for a proposal A: (\mu_A(x)=1, \nu_A(x)=0), * against it: (\mu_A(x)=0, \nu_A(x)=1), * or abstain from voting: (\mu_A(x)=\nu_A(x)=0). After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions. For this situation, special "intuitive fuzzy" negators, t- and s-norms can be defined. With D^* = \ and by combining both functions to (\mu_A,\nu_A): U \to D^* this situation resembles a special kind of ''L''-fuzzy sets. Once more, this has been expanded by defining picture fuzzy sets (PFS) as follows: A PFS A is characterized by three functions mapping ''U'' to , 1 \mu_A, \eta_A, \nu_A, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition \forall x \in U: \mu_A(x) + \eta_A(x) + \nu_A(x) \le 1 This expands the voting sample above by an additional possibility of "refusal of voting". With D^* = \ and special "picture fuzzy" negators, t- and s-norms this resembles just another type of ''L''-fuzzy sets.


Neutrosophic fuzzy sets

The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets. Neutrosophic fuzzy sets were introduced by Smarandache in 1998. Like IFS, neutrosophic fuzzy sets have the previous two functions: one for membership \mu_A(x) and another for non-membership \nu_A(x). The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate i_A(x). This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate i_A(x) value can be particularly useful when one cannot be very confident on the membership or non-membership values for item ''x''. In summary, neutrosophic fuzzy sets are associated with the following functions: :1. \mu_A(x) - degree of membership of ''x'' :2. \nu_A(x) – degree of non-membership of ''x'' :3. i_A(x) – degree of indeterminate value of ''x''


Pythagorean fuzzy sets

The other extension of IFS is what is known as Pythagorean fuzzy sets. Pythagorean fuzzy sets are more flexible than IFSs. IFSs are based on the constraint \mu_A(x) + \nu_A(x) \le 1, which can be considered as too restrictive in some occasions. This is why Yager proposed the concept of Pythagorean fuzzy sets. Such sets satisfy the constraint \mu_A(x)^2 + \nu_A(x)^2 \le 1, which is reminiscent of the Pythagorean theorem. Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of \mu_A(x) + \nu_A(x) \le 1 is not valid. However, the less restrictive condition of \mu_A(x)^2 + \nu_A(x)^2 \le 1 may be suitable in more domains.


Fuzzy logic

As an extension of the case of
multi-valued logic Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
, valuations (\mu : \mathit_o \to \mathit) of propositional variables (\mathit_o) into a set of membership degrees (\mathit) can be thought of as membership functions mapping predicates into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy
premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s from which graded conclusions may be drawn. This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
fields of
automated Automation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines ...
control and knowledge engineering, and which encompasses many topics involving fuzzy sets and "approximated reasoning." Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...
.


Fuzzy number and only number

A fuzzy number Fuzzy sets as a basis for a theory of possibility
" ''Fuzzy Sets and Systems''
is a fuzzy set that satisfies all the following conditions : * A is normalised ; * A is a convex set ; * \exists ! x^* \in A, \mu_(x^*)=1 ; * The membership function \mu_(x) is at least segmentally continuous. If these conditions are not satisfied, then A is not a fuzzy number . The core of this fuzzy number is a singleton; its location is: :: \, C(A) = x^* : \mu_A(x^*)=1 When the condition about the uniqueness of is not fulfilled, then the fuzzy set is characterised as a fuzzy interval. The core of this fuzzy interval is a crisp interval with: ::\,C(A) = \left min\ ; \max\ \right/math>. Fuzzy numbers can be likened to the
funfair A fair (archaic: faire or fayre) is a gathering of people for a variety of entertainment or commercial activities. Fairs are typically temporary with scheduled times lasting from an afternoon to several weeks. Types Variations of fairs incl ...
game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function). The kernel K(A) = \operatorname(A) of a fuzzy interval A is defined as the 'inner' part, without the 'outbound' parts where the membership value is constant ad infinitum. In other words, the smallest subset of \R where \mu_A(x) is constant outside of it, is defined as the kernel. However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity.


Fuzzy categories

The use of
set membership In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets o ...
as a key component of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
can be generalized to fuzzy sets. This approach, which began in 1968 shortly after the introduction of fuzzy set theory, led to the development of Goguen categories in the 21st century.Michael Winter "Representation theory of Goguen categories"
Fuzzy Sets and Systems ''Fuzzy Sets and Systems'' is a peer-reviewed international scientific journal published by Elsevier on behalf of the International Fuzzy Systems Association (IFSA) and was founded in 1978. The editors-in-chief (as of 2010) are Bernard De Baets ...
Volume 138, Issue 1, 16 August 2003, Pages 85–126
In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in ''L''-fuzzy sets.


Fuzzy relation equation

The fuzzy relation equation is an equation of the form , where ''A'' and ''B'' are fuzzy sets, ''R'' is a fuzzy relation, and stands for the composition of ''A'' with ''R'' .


Entropy

A measure ''d'' of fuzziness for fuzzy sets of universe U should fulfill the following conditions for all x \in U: #d(A) = 0 if A is a crisp set: \mu_A(x) \in \ #d(A) has a unique maximum iff \forall x \in U: \mu_A(x) = 0.5 #\mu_A \leq \mu_B \iff :::\mu_A \leq \mu_B \leq 0.5 :::\mu_A \geq \mu_B \geq 0.5 ::which means that ''B'' is "crisper" than ''A''. #d(\neg) = d(A) In this case d(A) is called the entropy of the fuzzy set ''A''. For finite U = \ the entropy of a fuzzy set A is given by :d(A) = H(A) + H(\neg), ::H(A) = -k \sum_^n \mu_A(x_i) \ln \mu_A(x_i) or just :d(A) = -k \sum_^n S(\mu_A(x_i)) where S(x) = H_e(x) is Shannon's function (natural entropy function) :S(\alpha) = -\alpha \ln \alpha - (1-\alpha) \ln (1-\alpha),\ \alpha \in ,1/math> and k is a constant depending on the measure unit and the logarithm base used (here we have used the natural base e). The physical interpretation of ''k'' is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
''k''''B''. Let A be a fuzzy set with a continuous membership function (fuzzy variable). Then :H(A) = -k \int_^\infty \operatorname \lbrace A \geq t \rbrace \ln \operatorname \lbrace A \geq t \rbrace \,dt and its entropy is :d(A) = -k \int_^\infty S(\operatorname \lbrace A \geq t \rbrace )\,dt.


Extensions

There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965, many new mathematical constructions and theories treating imprecision, inexactness, ambiguity, and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others try to mathematically model imprecision and uncertainty in a different way (; ; Deschrijver and Kerre, 2003).


See also

*
Alternative set theory In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set ...
*
Defuzzification Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems. Th ...
* Fuzzy concept *
Fuzzy mathematics Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 ...
*
Fuzzy set operations Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. The most widely used operations are called ''standard fuzzy set operations''; they comprise: fuzzy compl ...
*
Fuzzy subalgebra Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure. Definition Consider a first order l ...
*
Interval finite element In numerical analysis, the interval finite element method (interval FEM) is a finite element method that uses interval parameters. Interval FEM can be applied in situations where it is not possible to get reliable probabilistic characteristics of ...
* Linear partial information *
Multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
*
Neuro-fuzzy In the field of artificial intelligence, neuro-fuzzy refers to combinations of artificial neural networks and fuzzy logic. Overview Neuro-fuzzy hybridization results in a hybrid intelligent system that these two techniques by combining the human ...
*
Rough fuzzy hybridization {{No footnotes, date=April 2009 Rough fuzzy hybridization is a method of hybrid intelligent system or soft computing, where Fuzzy set theory is used for linguistic representation of patterns, leading to a ''fuzzy granulation'' of the feature spac ...
*
Rough set In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the ''lower'' and the ''upper'' approxima ...
*
Sørensen similarity index Sørensen () is a Danish- Norwegian patronymic surname meaning "son of Søren" (given name equivalent of Severin). , it is the eighth most common surname in Denmark. Immigrants to English-speaking countries often changed the spelling to ''Sorensen ...
* Type-2 fuzzy sets and systems *
Uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable ...


References


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