Fuzzy sets - Definition

Fuzzy sets are an extension of the classical set theory used in Fuzzy logic. A fuzzy set is characterized by a membership-degree function, which maps the members of the Universe into the unit interval [0,1]. The value 0 means that the member is not included in the given set, 1 describes a fully included member (this behaviour corresponds to the indicator function of classical sets). The values between 0 and 1 characterize fuzzy members.

For the universe X and given the membership-degree function f : X→[0,1], the fuzzy set A is defined as A = {(x, f(x)) | x ∈ X}.

The fuzzy set B, where B = {(3,0.3), (4,0.7), (5,1), (6,0.4)} would be enumerated as B = {0.3/3, 0.7/4, 1/5, 0.4/6} using standard fuzzy notation. Note that any value with a membership grade of zero does not appear in the set. The standard notation for finding the membership grade of the fuzzy set B at 6 is μB(6) = 0.4.