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DEFINITION


A fuzzy set is a pair (A, m) where A is a set and m : A ightarrow {Link without Title} . 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)
eq 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 Algebra or Structure L; usually it is required that L be at least a Poset or Lattice . The usual membership functions with values in are then called [0, 1 -valued membership functions. This generalization was first considered by Joseph Goguen (1967).


Fuzzy logic

As an extension of the case of 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.

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."

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 ilde{\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 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 ilde{\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 .


SEE ALSO




EXTERNAL LINKS




REFERENCES


  • Goguen, Joseph A. , 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

  • Zadeh, Lotfi A. ,

  • --- 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.