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In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh^{[1]} and Dieter Klaua^{[2]} in 1965 as an extension of the classical notion of set. At the same time, Salii (1965) defined a more general kind of structures called L-relations, which were studied by him in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, et al., 2000), decision-making (Kuzmin, 1982) and clustering (Bezdek, 1978), are special cases of L-relations when L is the unit interval [0, 1].
In classical set theory, 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 valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.^{[3]} 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.^{[4]}
It has been suggested by Thayer Watkins that Zadeh's ethnicity is an example of a fuzzy set because "His father was Turkish-Iranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijan in the Soviet Union...Lotfi was born in Baku in 1921 and lived there until his family moved to Tehran in 1931."^{[5]}
A fuzzy set is a pair (U, m) where U is a set and m\colon U \rightarrow [0,1].
For each x\in U, the value m(x) is called the grade of membership of x in (U,m). For a finite set U=\{x_1,\dots,x_n\}, the fuzzy set (U, m) is often denoted by \{m(x_1)/x_1,\dots,m(x_n)/x_n\}.
Let x \in U. Then x is called not included in the fuzzy set (U,m) if m(x) = 0, x is called fully included if m(x) = 1, and x is called a fuzzy member if 0 < m(x) < 1.^{[6]} The set \{x\in U\mid m(x)>0\} is called the support of (U,m) and the set \{x\in U\mid m(x)=1\} is called its kernel or core. The function m is called the membership function of the fuzzy set (U, m).
Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind; usually it is required that L be at least a poset 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 [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.^{[7]}
As an extension of the case of multi-valued logic, valuations (\mu : \mathit{V}_o \to \mathit{W}) of propositional variables (\mathit{V}_o) into a set of membership degrees (\mathit{W}) 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 premises from which graded conclusions may be drawn.^{[8]}
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 fields of automated control and knowledge engineering, and which encompasses many topics involving fuzzy sets and "approximated reasoning."^{[9]}
Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.
A fuzzy number is a convex, normalized fuzzy set \tilde{\mathit{A}}\subseteq\mathbb{R} whose membership function is at least segmentally continuous and has the functional value \mu_{A}(x)=1 at precisely one element.
This can be likened to the funfair 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).
A fuzzy interval is an uncertain set \tilde{\mathit{A}}\subseteq\mathbb{R} with a mean interval whose elements possess the membership function value \mu_{A}(x)=1. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.^{[10]}
The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation, and A · R stands for the composition of A with R.
^{[11]} Let A be a non-empty set and P(A) be the power set of A . The set function Cr is known as credibility measure if it satisfies following condition
Cr{B} indicates how frequently event B will occur.
^{[12]} Let A be a fuzzy variable with membership function u. Then for any set B of real numbers, we have
^{[13]} Let A be a fuzzy variable. Then the expected value is
^{[14]} Let A be a fuzzy variable with a continuous membership function. Then its entropy is
Where
There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965, a lot of 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 (Burgin and Chunihin, 1997; Kerre, 2001; Deschrijver and Kerre, 2003).
The diversity of such constructions and corresponding theories includes:
Axiom of choice, Mathematical logic, Category theory, Mathematics, Foundations of mathematics
Language, Semantics, Noam Chomsky, Sociolinguistics, Semiotics
Logic, Set theory, Statistics, Number theory, Mathematical logic
Sociology, Technology, History, History of science, Science
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Finite element method, Fuzzy number, Fuzzy set, Non-classical logic, Real numbers
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Computing with words and perceptions, Fuzzy set, Fuzzy logic, Soft computing, Vagueness