We imagine all the things we might want to talk about in a given discussion as collected together in a set X called a universe of discourse.  (In probabilistic or statistical work it might be called a sample space.)

By way of shorthand notation, let us single out two domains of values, the boolean domain ℬ = {0, 1} and the unit interval ℐ = [0, 1].

In many applications, especially computational or statistical ones, it is often useful to represent any given subset S of X by means of a function from X to ℬ known as the characteristic function or indicator function of S in X.

By way of notation, we write f_S : X → ℬ for the indicator function of S in X, and define it as follows:

f_S(x) = 1   if and only if   x is an element of S.

All that is standard notions from ordinary set theory.

Fuzzy sets, more properly, fuzzy subsets of a given set X will be defined in a way that generalizes the target values in the previous definition from the boolean values in ℬ to the real values in ℐ.