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Fuzzy Set

A fuzzy set A on a universe X is specified by a membership function

mu_A:X-->[0,1],
(1)

where mu_A(x) is the degree to which x belongs to A (Zadeh 1965). An ordinary, or crisp, set is the special case in which mu_A takes values only in {0,1}, so that mu_A is a characteristic function. The support and core of A are {x in X:mu_A(x)>0} and {x in X:mu_A(x)=1}, respectively, while its alpha-cut is A_alpha={x in X:mu_A(x)>=alpha}.

The standard union, intersection, and set complement are defined pointwise by

mu_(A union B)(x)=max{mu_A(x),mu_B(x)}
(2)
mu_(A intersection B)(x)=min{mu_A(x),mu_B(x)}
(3)
mu_(A^c)(x)=1-mu_A(x).
(4)

See also

Characteristic Function, Fuzzy Graph, Fuzzy Logic, Set

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References

Zadeh, L. A. "Fuzzy Sets." Information and Control 8, 338-353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X.

Cite this as:

Weisstein, Eric W. "Fuzzy Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FuzzySet.html

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