# Fuzzy set

### From Glossary

Given a universe set, , and a *membership
function*, , a fuzzy set is a collection of pairs:
. Often, the membership function is subscripted
by the set name, say . Generally, for all , , and
. In the context of uncertainty, the value is
used to model the statement of how *possible* it is for to
be in . For this reason, is sometimes called the
*possibility function* of . What we consider the usual set
(without a membership function) is called a *crisp set* in
fuzzy mathematics.

Fuzzy set operations, say on fuzzy sets and , with membership functions and , resp., are defined by the following:

*Union*: .*Intersection*: .*Complement*: .

One must be careful when using fuzzy sets to represent uncertainty (which is not the only type of interpretation – see fuzzy mathematical program). In particular, if , its complement is also . Thus, , despite the fact that (in ordinary set theory). Similarly, , despite the fact that . This illustrates the fact that need not equal even if as crisp sets.

While the fuzzy set is fundamental for fuzzy mathematical programming, other concepts in fuzzy mathematics also apply, such as fuzzy arithmetic, fuzzy graphs and fuzzy logic.