Fuzzy Set

A fuzzy set is a pair (A,m) where A is a set and m : A \rightarrow [0,1].

For each x\in A, m(x) is called the grade of membership of x in (A,m). For a finite set A = {x1,...,xn}, the fuzzy set (A,m) is often denoted by {m(x1) / x1,...,m(xn) / xn}.

Let x \in A. Then x is called not included in the fuzzy set (A,m) if m(x) = 0, x is called fully included if m(x) = 1, and x is called fuzzy member if 0 < m(x) < 1. The set \{x\in A\mid m(x)>0\} is called the support of (A,m) and the set \{x\in A\mid m(x)=1\} is called its kernel.

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This kind of generalizations was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.