Membership function (mathematics)

Definition

For any set X , a membership function on X is any function from X to the real unit interval [ 0 , 1 ] .

Membership functions on X represent fuzzy subsets of X . The membership function which represents a fuzzy set A ~ is usually denoted by μ A . For an element x of X , the value μ A ( x ) is called the membership degree of x in the fuzzy set A ~ . The membership degree μ A ( x ) quantifies the grade of membership of the element x to the fuzzy set A ~ . The value 0 means that x is not a member of the fuzzy set; the value 1 means that x 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.

Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure L ; 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.

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, ν from S, the set of subsets of some set, into [ 0 , 1 ] , such that ν is set-wise monotone and is normalized (i.e. ν ( ∅ ) = 0 , ν ( Ω ) = 1 ) . 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.

Application example

In Biological systems engineering, a membership function growth response model for defining optimality degrees of vapor pressure deficit in greenhouse cultivation of tomato .