In mathematics, the membership function of a fuzzy set is a generalization of the indicator function for classical sets. In fuzzy logic, it represents the degree of truth") as an extension of valuation&action=edit&redlink=1 "Valuation (logical) (not yet written)"). Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in loosely defined sets, not the probability of some event or condition. Membership functions were introduced by Zadeh in the first paper on fuzzy sets (1965). Zadeh, in his fuzzy set theory, proposed to use a membership function (with a range "Range (mathematics)") covering the interval "Interval (mathematics)") [0,1]) that operates in the domain of all possible values.

For any set, a membership function on is any function of on the real unit interval.

Membership functions represent fuzzy subsets of . The membership function representing a fuzzy set is usually denoted by . for an element of , the value is called the degree of membership of in the fuzzy set The degree of membership quantifies the degree of membership of the element to the fuzzy set The value 0 means that it is not a member of the fuzzy set; The value 1 means that it is a member of the fuzzy set. Values ​​between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.

Sometimes[1]​ a more general definition is used, where membership functions take values ​​in a "Structure (logical)") or arbitrary fixed algebra; It is usually required to be at least a partially ordered set or "Lattice (mathematics)"). The usual membership functions with values ​​on [0,1] are then called membership functions valued on [0,1].

See the article on Capacity of a Set") for a closely related definition in mathematics..

One application of membership functions is as capabilities in decision theory.

In decision theory, a capability is defined as a function, of S, the set of subsets of some set, in , such that it is jointly monotonic and normalized (i.e., This is a generalization of the notion of a probability measure, where the probability axiom of countable additivity is weakened. A capability is used as a subjective measure of the probability of an event, and the "expected value (mathematical)") of an outcome given a certain capability is can be found by taking the Choquet integral") over the capacity.