In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the membership function of a
fuzzy set is a generalization of the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
for classical
sets. In
fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...
, it represents the
degree of truth as an extension of
valuation. Degrees of truth are often confused with
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
, 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.
Definition
For any set
, a membership function on
is any function from
to the
real unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...