Fuzzy Set

Definition

A fuzzy set is a function A: U → L where U is an ordinary set of elements called universe and L is a scale which is usually supposed to have the structure of a residuated lattice (see “Residuated Lattice”). The function A is at the same time called also membership function, i.e., fuzzy set is identified with its membership function.

If x ∈ U is an element then A(x) ∈ L is a membership degree of x in the fuzzy set A. It can also be interpreted as a degree of truth of the fact that the element x belongs to the fuzzy set A. If U is a universe and A a fuzzy set then it is convenient to write A ⫇ U.

Key Points

Fuzzy sets can be explicitly written in the form

$$ \left\{a/u|a\in L,u\in U\right\} $$

where a ∈ L is a membership degree of an element u ∈ U.

Given a residuated lattice (see “Residuated Lattice”) as the structure of truth values, the basic operations with fuzzy sets can be defined as follows:

$$ \begin{array}{lrl}\left(\mathrm{union}\right)&\quad \left(\mathrm{A}\cup...