Fuzzy Set

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Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi Asker Zadeh (1965) as an extension of the classical notion of set. 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 (mathematics) valued in the real unit interval . 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.

Definition A fuzzy set is a pair (A, m) where A is a set and m : A \rightarrow . For each x\in A, m(x) is the grade of membership of x. x\in (A, m)\iff x\in A\wedge m(x)\neq 0. If A=\{x_1,...,x_n\} the fuzzy set (A, m) can be denoted \{m(z_1)/z_1,...,m(z_n)/z_n\}.

An element mapping to the value 0 means that the member is not included in the fuzzy set, 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.

Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed universal algebra or structure (mathematical logic) L; usually it is required that L be at least a poset or lattice (order). The usual membership functions with values in are then called -valued membership functions. This generalization was first considered by Joseph Goguen (1967).

Fuzzy logic 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 function (mathematics) mapping first-order logic 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.

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 automation 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 number A fuzzy number is a Convex set, normalizing constant fuzzy set \tilde{\mathit{A-->\subseteq\mathbb{R}whose membership function is at least segmentally continuous function 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 contestants weight, with closer guesses being more correct, and where the guesser "wins" if they guess near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

Fuzzy interval 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 set, normalizing_constant, at least segmentally continuous function.

See also

Fuzzy measure theory
Alternative set theory
Defuzzification
Fuzzy logic
Fuzzy set operations
Neuro-fuzzy
Rough set
Uncertainty
Rough fuzzy hybridization
Fuzzy subalgebra

External links

Uncertainty model Fuzziness
The Algorithm of Fuzzy Analysis
Fuzzy Image Processing
Zadeh's 1965 paper on Fuzzy Sets

References

Joseph Goguen, 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
Gottwald, Siegfried, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0863802621
Lotfi A. Zadeh,

1965, "Fuzzy sets," Information and Control 8: 338–353.
1975, "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.
1978, "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28.