fuzzy subset


Fuzzy set theory is based on the idea that vague notions as “big”, “near”, “hold” can be modelled by “fuzzy subsets”. The idea of a fuzzy subset T of a set S is the following: each element xS, there is a number p[0,1] such that px is the “probability” that x is in T.

To formally define a fuzzy set, let us first recall a well-known fact about subsets: a subset T of a set S corresponds uniquely to the characteristic functionMathworldPlanetmathPlanetmathPlanetmath cT:S{0,1}, such that cT(x)=1 iff xT. So if one were to replace {0,1} with the the closed unit interval [0,1], one obtains a fuzzy subset:

A fuzzy subset of a set S is a map s:S[0,1] from S into the interval [0,1].

More precisely, the interval [0,1] is considered as a complete latticeMathworldPlanetmath with an involution 1-x. We call fuzzy subset of S any element of the direct power [0,1]S. Whereas there are 2|S| subsets of S, there are 1|S| fuzzy subsets of S.

The join and meet operationsMathworldPlanetmath in the complete lattice [0,1]S are named union and intersectionMathworldPlanetmathPlanetmath, respectively. The operation induced by the involution is called complementPlanetmathPlanetmath. This means that if s and t are two fuzzy subsets, then the fuzzy subsets st,st,-s, are defined by the equations

(st)(x)=max{s(x),t(x)};(st)(x)=min{s(x),t(x)};-s(x)=1-s(x).

It is also possible to consider any latticeMathworldPlanetmath L instead of [0,1]. In such a case we call L-subset of S any element of the direct power LS and the union and the intersection are defined by setting

(st)(x)=s(x)t(x);(st)(x)=s(x)t(x)

where and denote the join and the meet operations in L, respectively. In the case an order reversing function ¬:LL is defined in L, the complement -s of s is defined by setting

-s(x)=¬s(x).

Fuzzy set theory is devoted mainly to applications. The main success is perhaps fuzzy control.

References

  • 1 Cignoli R., D Ottaviano I. M. L. and Mundici D.,Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, (1999).
  • 2 Elkan C., The Paradoxical Success of Fuzzy LogicMathworldPlanetmath. (November 1993). Available from http://www.cse.ucsd.edu/users/elkan/http://www.cse.ucsd.edu/users/elkan/ Elkan’s home page.
  • 3 Gerla G., Fuzzy logic: Mathematical tools for approximate reasoning, Kluwer Academic Publishers, Dordrecht, (2001).
  • 4 Goguen J., The logic of inexact conceptsMathworldPlanetmath, Synthese, vol. 19 (1968/69)
  • 5 Gottwald S., A treatise on many-valued logics, Research Studies Press, Baldock (2000).
  • 6 Hájek P., MetamathematicsMathworldPlanetmathPlanetmathPlanetmath of fuzzy logic. Kluwer (1998).
  • 7 Klir G. , UTE H. St.Clair and Bo Yuan, Fuzzy Set Theory Foundations and Applications, (1997).
  • 8 Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 0-7923-7435-5.
  • 9 Zadeh L.A., Fuzzy Sets, Information and Control, 8 (1965) 338­-353.
  • 10 Zadeh L. A., The concept of a linguistic variable and its application to approximate reasoning I, II, III, Information Sciences, vol. 8, 9(1975), pp. 199-275, 301-357, 43-80.
Title fuzzy subset
Canonical name FuzzySubset
Date of creation 2013-03-22 16:34:54
Last modified on 2013-03-22 16:34:54
Owner ggerla (15808)
Last modified by ggerla (15808)
Numerical id 11
Author ggerla (15808)
Entry type Definition
Classification msc 03E72
Classification msc 03G20
Synonym fuzzy set
Synonym L-subset
Related topic Logic
Related topic FuzzyLogic2