Membership function (mathematics)

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 ${\displaystyle X}$. 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 ${\displaystyle L}$ ; 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.
• Goguen J.A, 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174

• Fuzzy Image Processing