Membership function (mathematics)

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.

Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition.

Membership functions were introduced by Aliasker Zadeh in the first paper on fuzzy sets (1965).

Aliasker Zadeh, in his theory of fuzzy sets, proposed using a membership function (with a range covering the interval (0,1)) operating on the domain of all possible values.

Membership functions represent fuzzy subsets of

The membership function which represents a fuzzy set

quantifies the grade of membership of the element

is not a member of the fuzzy set; the value 1 means that

is fully a member of the fuzzy set.

Sometimes,[1] a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure

[further explanation needed]; usually it is required that

See the article on Capacity of a set for a closely related definition in mathematics.

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

In decision theory, a capacity is defined as a function,

This is a generalization of the notion of a probability measure, where the probability axiom of countable additivity is weakened.