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


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


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


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


  • 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