Fuzzy Sets

In mathematics, Fuzzy Sets are sets whose elements have degrees of membership. In the real world, objects are often classified into different categories. For such categories as tall man, high inflation rate, pretty woman etc., all of them convey linguistic vague information. The concept of membership of an object in such categories is not obvious and not precise. Thus, the application of classical two valued logic to the real world is limited in some cases. The idea of fuzzy sets proposed by Zadeh aims to deal with such information.
Fuzzy set theory is an extension of classical set theory. In classical set theory, an element either belongs to a set or does not belong to a set. In fuzzy set theory, an element may partially belong to a set. Fuzzy sets have gradations of set membership which is represented by a function referred to as a membership function, and so they resemble the kinds of categories ordinary people use in natural thought or communication. The formal presentation of the fuzzy set theory is as follows:

DEFINITION 1.
Let x ϵ U and let S be a subset of U.ʯ(x):U → [0,1] is called the membership function which represents the degree of x belonging to the subset S. U is called the universe of discourse. Then the fuzzy set A is defined to be a set of ordered pairs A = {(x, ʯ(x)) | x ϵ S, S Ϲ U }. The membership function is denoted by ʯ_A(x) for the fuzzy set A. The support of a fuzzy set A denoted as A_sup is the crisp set of all points x in U such that ʯ_A(x) > 0. A  fuzzy set A whose support A_supcontains a single point x in U with ʯ_A(x) = 1 is  referred to as a fuzzy singleton. A fuzzy set A whose support A_sup is the universe of discourse U with ʯ(x) = 1 is referred to as a fuzzy universe. It is denoted by  Z. If the universe of discourse U is a set of real numbers, the fuzzy sets defined  on U are called fuzzy numbers. The fuzzy set operations are defined via their membership functions.

DEFINITION 2.
Let A₁ and A₂ be fuzzy sets in U and let B be a fuzzy set in V.

The operators Ʌ and V can be any kind of triangular norms and triangular conorms, respectively, for example, product, sum, max, or min. A Linguistic variable can be regarded as a variable whose values are defined  in linguistic terms (e.g., negative large, negative small, positive small, and positive large). These terms which are imprecise and ill-defined can be represented  by fuzzy sets. In fact, the use of fuzzy sets provides a basis for the systematic  manipulation of such linguistic variables or such linguistic terms.

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