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

Membership functions represent fuzzy subsets of ${\displaystyle X}$ ^{[citation needed]}. The membership function which represents a fuzzy set ${\displaystyle {\tilde {A}}}$  is usually denoted by ${\displaystyle \mu _{A}.}$  For an element ${\displaystyle x}$  of ${\displaystyle X}$ , the value ${\displaystyle \mu _{A}(x)}$  is called the membership degree of ${\displaystyle x}$  in the fuzzy set ${\displaystyle {\tilde {A}}.}$  The membership degree ${\displaystyle \mu _{A}(x)}$  quantifies the grade of membership of the element ${\displaystyle x}$  to the fuzzy set ${\displaystyle {\tilde {A}}.}$  The value 0 means that ${\displaystyle x}$  is not a member of the fuzzy set; the value 1 means that ${\displaystyle x}$  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,^[1] a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure ${\displaystyle L}$  ^{[further explanation needed]}; usually it is required that ${\displaystyle L}$  be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions.

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, ${\displaystyle \nu }$  from S, the set of subsets of some set, into ${\displaystyle [0,1]}$ , such that ${\displaystyle \nu }$  is set-wise monotone and is normalized (i.e. ${\displaystyle \nu (\emptyset )=0,\nu (\Omega )=1).}$  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.

• Defuzzification
• Fuzzy measure theory
• Fuzzy set operations
• Rough set

• Zadeh L.A., 1965, "Fuzzy sets". Information and Control 8: 338–353. [1]
• Goguen J.A, 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174

• Fuzzy Image Processing