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 Zadeh in the first paper on fuzzy sets (1965). 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.
Definition.
For any set formula_1, a membership function on formula_1 is any function from formula_1 to the real unit interval formula_4.
Membership functions represent fuzzy subsets of formula_1. The membership function which represents a fuzzy set formula_6 is usually denoted by formula_7 For an element formula_8 of formula_1, the value formula_10 is called the "membership degree" of formula_8 in the fuzzy set formula_12 The membership degree formula_13 quantifies the grade of membership of the element formula_8 to the fuzzy set formula_12 The value 0 means that formula_8 is not a member of the fuzzy set; the value 1 means that formula_8 is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.
Membership function of a fuzzy set
Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure formula_18 ; usually it is required that formula_18 be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions.
Capacity.
"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, formula_20 from S, the set of subsets of some set, into formula_4, such that formula_20 is set-wise monotone and is normalized (i.e. formula_23 This is a generalization of the notion of a probability measure, where the probability axiom of countable additivity is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "expected value" of an outcome given a certain capacity can be found by taking the Choquet integral over the capacity.