2
Fuzzy Sets

The main idea of fuzzy sets is to consider the degree of membership. A fuzzy set is described by a membership function that assigns to each member or element a membership degree. Usually, the range of this membership function is from 0 to 1. A degree of 1 represents complete membership to the set, and degree of 0 represents absolutely no membership to the set. A degree between 0 and 1 represents partial membership to the set.

We can define high fever as a temperature higher than 102 . Even if most doctors will agree that the threshold is at about 102 , this does not mean that a patient with a body temperature of 101.9 does not have a high fever while another patient with 102 does indeed have a high fever. Therefore, instead of using this rigid definition, each body temperature is associated with a certain degree. For example, we show a possible description of high fever using membership degree as follows

The degree of membership can also be represented by a continuous function.

2.1 Membership Functions

Let be a subset of . Each element can either belong to or not belong to a set . This kind of set can be defined by the characteristic function

That is to say, the characteristic function maps elements of to elements of the set , which is formally expressed by .

Zadeh [162] proposed a concept of so-called fuzzy set by extending the range of the characteristic function to the unit interval . A fuzzy set in is defined to be a set of ordered pairs

where is called the membership function of . The value is regarded as the degree of membership of in . In other words, it indicates the degree to which belongs to . Any subset of can also be regarded as a fuzzy set in by taking the membership function as the characteristic function of . In this case, we write by regarding as a fuzzy set in . When is a singleton , we also write .

2.2 -level Sets

An interesting and important concept related to fuzzy sets is the -level set. Let be a fuzzy set in with membership function . The range of the membership function is denoted by . Throughout this book, we shall assume that the range contains 1. However, the range is not necessarily equal to the whole unit interval .

For , the -level set of is defined by

Since the range is assumed to contain 1, it follows that the -level sets are non-empty for all . Notice that the -level set is not defined by (2.1). The -level set will be defined in a different way that will be explained afterward.

Given any satisfying , it is easy to see

The strict inclusion can happen.

Notice that the expression (2.1) does not include the -level set. If we allowed the expression (2.1) taking , the -level set of would be the whole -dimensional Euclidean space . Defined in this way, the -level set would not be helpful for real applications. Therefore, we are going to invoke a topological concept to define the -level set. The support of a fuzzy set in is the crisp set defined by

The -level set of is defined to be the closure of the support , i.e.

For the concept of closure, refer to Definition 1.2.5.

Let be a subset of . Recall the notation . Then, we see that for any . Also, the -level set is given by

Recall that is a closed subset of when . Now, suppose that is a closed subset of . Then, we have for any . In particular, for any , since the singleton is a closed subset of , it follows that for any because of (2.6).

Let be a convex set in (refer to Definition 1.4.1). Then, we see that

are also convex sets in for . Therefore, we can extend the above concept to define the convexity of a fuzzy set in by replacing the characteristic function with the membership function.

Definition 2.2.5 does not include the convexity of the -level set. The following proposition can guarantee the convexity of the -level set.

For and a sequence in , recall that means that the sequence is increasing and converges to . For , recall that means that the sequence is decreasing and converges to .

Let be a fuzzy set in . Then, for , the strong -level set of is denoted and defined by

The family of -level sets is nested in the sense of for . The nestedness of -level sets says that

Regarding , we have the following interesting results.

For , we also define

Let denote the set of all rational numbers. It is well known that is dense in . In other words, given any , there exists such that can be arbitrarily close to . More precisely, given any , there exists that depends on satisfying . Then, we have the following interesting results.

Let be a fuzzy set in with membership function . The inverse function of does not necessarily exist. However, we can consider the inverse image of any subset of defined by

Given any , we also write to denote the inverse image of singleton . More precisely, we have

The set difference is defined as (i.e. implies and ).

Let be a fuzzy set in . The decomposition theorem says that the membership function can be expressed in terms of its -level sets .

Next, we are going to see how we can define when two fuzzy sets are identical. The concept of identical fuzzy sets is an important issue in applications. Recall that the range of the membership function of a fuzzy set is not necessarily equal to the whole unit interval . In other words, the membership function is not always an onto function.

2.3 Types of Fuzzy Sets

Now, we are going to consider some special structures of fuzzy sets by classifying the family of all fuzzy sets into many different sub-families in which the -level sets own elegant structure is useful in applications.

The fuzzy vector with core value can be regarded as a fuzzification of vector . Recall that the membership function of is given by

It is clear to see that for all and . We also say that is a crisp vector with value .

For , Proposition 2.3.2 says that each -level set is a bounded closed interval with degree , which explains the name of fuzzy interval in . Next, we also propose two kinds of fuzzy intervals.

Proposition 2.3.5 says that if is a standard fuzzy interval, then it is also a canonical fuzzy...