Definition
A fuzzy set is a pair where is a set and
For each the value is called the grade of membership of in For a finite set the fuzzy set is often denoted by
Let Then is called not included in the fuzzy set if is called fully included if and is called a fuzzy member if The set is called the support of and the set is called its kernel. The function is called the membership function of the fuzzy set
Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure of a given kind; usually it is required that be at least a poset or lattice. These are usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in are then called -valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.
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