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3.4 Fuzzy Set and Fuzzy Logic
A fuzzy set is a generalization of classical set theory that assesses the grades of
membership of elements in a set. In classical set theory, an element can only either
belong or not belong to a set; however, fuzzy set theory determines the membership of
an element in a set characterized by a membership function [85].
The framework of fuzzy sets provides an intuitive way to work with problems
where imprecision is caused by the lack of clear criteria of membership rather than
random uncertainties. In the real world, a class of objects may not have a crisp
boundary to define its members. Take “the class of all real numbers which are much
greater than 1” mentioned in [85] as an example: while 10,000 may belong to the set for
sure, ambiguity arises in the cases of smaller numbers such as 10. In this case, “a
number much greater than 1” defines a fuzzy variable, or a linguistic variable, and the
grade of membership of each real number may depend on the circumstance of different
applications.
Overview of fuzzy set theory
Let X be a space, and any element in X is denoted by x, i.e. xX. A fuzzy set A
in X is specified by a membership function f A (x) corresponding to each of the elements
x in X, where f A (x)[0,1] characterizes the “grade of membership” of x in A. The closer
the value of f A (x) to 1, the higher the grade of membership of x in A.
Empty set:
The fuzzy set A is empty if and only if its membership function is zero in X.
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