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Fuzzy set

 
Sci-Tech Dictionary: fuzzy set
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(′fəz·ē ′set)

(mathematics) An extension of the concept of a set, in which the characteristic function which determines membership of an object in the set is not limited to the two values 1 (for membership in the set) and 0 (for nonmembership), but can take on any value between 0 and 1 as well.

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Wikipedia: Fuzzy set
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Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set.[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.[2] Classical bivalent sets are in fuzzy set theory 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 [3].

Contents

Definition

A fuzzy set is a pair (A,m) where A is a set and m : A \rightarrow [0,1].

For each x\in A, m(x) is called the grade of membership of x in (A,m). For a finite set A = {x1,...,xn}, the fuzzy set (A,m) is often denoted by {m(x1) / x1,...,m(xn) / xn}.

Let x \in A. Then x is called not included in the fuzzy set (A,m) if m(x) = 0, x is called fully included if m(x) = 1, and x is called fuzzy member if 0 < m(x) < 1.[4] The set \{x\in A\mid m(x)>0\} is called the support of (A,m) and the set \{x\in A\mid m(x)=1\} is called its kernel.

Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure L; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This generalization was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.[5]

Fuzzy logic

Main article: Fuzzy logic

As an extension of the case of multi-valued logic, valuations (\mu : \mathit{V}_o \to \mathit{W}) of propositional variables (Vo) into a set of membership degrees (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.[6]

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."[7]

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, normalized fuzzy set \tilde{\mathit{A}}\subseteq\mathbb{R} whose membership function is at least segmentally continuous and has the functional value μA(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).

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 μA(x) = 1. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.[8]

See also

External links

References

  1. ^ L. A. Zadeh (1965) "Fuzzy sets". Information and Control 8 (3) 338–353.
  2. ^ D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
  3. ^ Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, “FM-test: A Fuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis”, BMC Bioinformatics, 7 (Suppl 4): S7. 2006.
  4. ^ AAAI http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic
  5. ^ Goguen, Joseph A., 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
  6. ^ Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0863802621
  7. ^ "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.
  8. ^ "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28

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