Data Mining - Mehmed Kantardzic [236]
The notion of a set occurs frequently as we tend to organize, summarize, and generalize knowledge about objects. We can even speculate that the fundamental nature of any human being is to organize, arrange, and systematically classify information about the diversity of any environment. The encapsulation of objects into a collection whose members all share some general features naturally implies the notion of a set. Sets are used often and almost unconsciously; we talk about a set of even numbers, positive temperatures, personal computers, fruits, and the like. For example, a classical set A of real numbers greater than 6 is a set with a crisp boundary, and it can be expressed as
where there is a clear, unambiguous boundary 6 such that if x is greater than this number, then x belongs to the set A; otherwise, x does not belong to the set. Although classical sets have suitable applications and have proven to be an important tool for mathematics and computer science, they do not reflect the nature of human concepts and thoughts, which tend to be abstract and imprecise. As an illustration, mathematically we can express a set of tall persons as a collection of persons whose height is more than 6 ft; this is the set denoted by the previous equation, if we let A = “tall person” and x = “height.” Yet, this is an unnatural and inadequate way of representing our usual concept of “tall person.” The dichotomous nature of the classical set would classify a person 6.001 ft tall as a tall person, but not a person 5.999 ft tall. This distinction is intuitively unreasonable. The flaw comes from the sharp transition between inclusions and exclusions in a set.
In contrast to a classical set, a fuzzy set, as the name implies, is a set without a crisp boundary, that is, the transition from “belongs to a set” to “does not belong to a set” is gradual, and this smooth transition is characterized by membership functions (MFs) that give sets flexibility in modeling commonly used linguistic expressions such as “the water is hot” or “the temperature is high.” Let us introduce some basic definitions and their formalizations concerning fuzzy sets.
Let X be a space of objects and x be a generic element of X. A classical set A, A ⊆ X, is defined as a collection of elements or objects x ∈ X such that each x can either belong or not belong to set A. By defining a characteristic function for each element x in X, we can represent a classical set A by a set of ordered pairs (x, 0) or (x, 1), which indicates x ∉ A or x ∈ A, respectively.
Unlike the aforementioned conventional set, a fuzzy set expresses the degree to which an element belongs to a set. The characteristic function of a fuzzy set is allowed to have values between 0 and 1, which denotes the degree of membership of an element in a given set. If X is a collection of objects denoted generically by x, then a fuzzy set A in X is defined as a set of ordered pairs:
where μA(x) is called the membership function (MF) for the fuzzy set A. The MF maps each element of X to a membership grade (or membership value) between 0 and 1.
Obviously, the definition of a fuzzy set is a simple extension of the definition of a classical set in which the characteristic function is permitted to have any value between 0 and 1. If the value of the MF μA(x) is restricted to either 0 or 1, then A is reduced to a classic set and μA(x) is the characteristic function of A. For clarity, we shall also refer to classical sets as ordinary sets, crisp sets, non-fuzzy sets, or, simply, sets.
Usually X is referred to as the universe of discourse, or, simply, the universe, and it may consist of discrete (ordered or non-ordered) objects or continuous space. This can be clarified by the following examples. Let