For every set S which is a subset of some universal set X, a function χS:X→{0, 1}, can be defined such that χS(x) = 1 if x ∈ S, 0 otherwise. This function χS is known as the membership or characteristic function of the set S over X. Clearly from this definition the null set's characteristic function is zero for any x ∈ X, while the universal set's is just as obviously 1 for all x ∈ X.

The characteristic function of the intersection two sets A and B is simply the logical AND of the characteristic functions of A and B, i.e. χA ∩ B(x) = 1 when both χA(x) = 1 and χB(x) = 1 (i.e. just multiplying the two characteristic functions together). The characteristic function of the union of A and B is in the same way, the logical OR of the characteristic functions of A and B (i.e. adding the characteristic functions, but taking any sum greater than 1 to be 1). The characteristic function of the complement of a set A is A's characteristic function's logical negation, i.e. 0 when χA is 1 and 1 when it is zero. This device is helpful in linking logic and set theory.

The theories of fuzzy logic and fuzzy sets generalize the notion of the characteristic function by allowing it to take on any real number value between 0 and 1 as opposed to just 0 or 1.

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