Representations of a Comparison Measure Between Two Fuzzy Sets

: This paper analyzes the representation behaviors of a comparison measure between two compared fuzzy sets. Three types of restrictions on two fuzzy sets are considered in this paper: the two disjoint union fuzzy sets, the two disjoint fuzzy sets and the two general fuzzy sets. Differences exist among the numbers of possible representations of a comparison measure for the three types of fuzzy sets restrictions. The value of comparison measure is constant for the two disjoint union fuzzy sets. There are 42 candidate representations of a comparison measure for the two disjoint fuzzy sets. Of which 13 candidate representations with one or two terms can be used to easily calculate and compare a comparison measure.

Couso et al. [5] surveyed a large collection of axiomatic definitions from the literature regarding the notions of comparison measures between two compared FSs. One of the fundamental axioms of a comparison measure is as follows. For the two disjoint FSs A and B, if both comparison measures between A and empty set and that of B and empty set are less, then the degree of a comparison measure between A and B is less. From this axiomatic definition, we analyze the comparison measure behaviors of two FSs A and B in terms of the other simple comparison measures, especially for the intersection and the union of A and B, the empty set and the universal set. The representations of a comparison measure between two FSs can not only present the important components of a comparison measure but also analyze the comparison measure behaviors of two FSs in terms of other simple comparison measures. This paper focuses the representations of a comparison measure between two FSs.
To analyze the representation behaviors of a comparison measure between two FSs, three kinds of two FSs are considered in this paper: the two disjoint union FSs, the two disjoint FSs and the two general FSs. For the two FSs A and B, this paper deals with the representations of a comparison measure for the case that ∩ = ∅ and ∪ = , the case that ∩ = ∅ and the general FSs A and B.
The organization of this paper is as follows. Section 2 briefly reviews the FSs and the comparison measures between two compared FSs. We present representations of a comparison measure for the two disjoint union FSs in section 3, the two disjoint FSs in section 4 and the two general FSs in section 5. Finally, some concluding remarks and future research are presented.

FSs and Comparison Measures
We firstly review the basic notations of FSs. Let = { 1 , 2 , … , } be a non-empty universal set or referential set of real numbers ℛ.  which coincides with the result of G5. Therefore, the representation of ( , ) can not only present its important ingredients but also compare ( , ) in terms of other measures ( , ∅) and ( , ∅). For two FSs A and B, the adopted components of a comparison measure are A, B, the intersection and the union of A and B, the empty set and the universal set. To represent a comparison measure ( , ), the adopted comparison measures other than ( , ) are ( , ) of different FSs X and Y, ( , ) ≠ ( , ), , ∈ {∅, , , ∩ , ∪ , }.
The following sections list the representations of a comparison measure ( , ) for the two disjoint union FSs A and B, the two disjoint FSs A and B and the two general FSs A and B. More precisely, we consider the case that ∩ = ∅ and ∪ = for section 3, ∩ = ∅ for section 4 and the general FSs A and B for section 5.

Representations of a Comparison Measure for the Two Disjoint Union Fuzzy Sets
This section will present the representations of a comparison measure ( , ) for the two disjoint union   Among these 81 combinations, there are 42 candidate representations of ( , ) for type I. The number of terms ( , ), , ∈ {∅, , , ∪ , } of a candidate representation of ( , ) is 1, 2, 3 and 4, except for the constant term. There are 1, 12, 23 and 6 candidate representations of ( , ) for the number of terms being 1, 2, 3 and 4, respectively. The combination [8]- [1] is the one term ( , ) of a candidate representation of ( , ) as follows.   The combinations [1]- [3], [3]- [1] and [9]- [4] are listed in the type III. The fourth type IV is the duplicate representations of a comparison measure ( , ) which appear in type I. For example, the combination [1]- [5], we obtain that which is the same as that of combination [1]- [2]. There are 15 combinations in type IV described as follows.

Representations of a Comparison Measure for the Two General Fuzzy Sets
This section lists the representations of a comparison measure ( , ) for the two general FSs  Similarly, we have that the two general FSs, respectively. The fewer the number of restrictions placed on the two FSs, the more the number of possible representations of a comparison measure. For the two disjoint union FSs, the constant value of ( , ) = 1 3 implies that we cannot compare the comparison behaviors of the two disjoint union FSs A and B. Among the 81 combinations of the two disjoint FSs A and B, there are 42 candidate representations of ( , ), 15 duplicate representations of ( , ), 21 relationships between different ( , ), , ∈ {∅, , , ∪ , } other than ( , ) and 3 identical equations. There are 1 and 12 candidate representations of ( , ) for one and two terms ( , ), , ∈ {∅, , , ∪ , }, respectively. Applying these 13 candidate representations, we can easily calculate and compare the degree of a comparison measure ( , ) with ∩ = ∅.
In the future, we will analyze the representation behaviors of comparison measures for the generalization of FSs and the general forms of a comparison measure. In particular, the analysis can be extended to the intuitionistic fuzzy sets, hesitant fuzzy sets and neutrosophic sets. Thus, the representation analysis of comparison measures for the intuitionistic fuzzy sets is a subject of considerable ongoing research.
Author Contributions: Both J.H. Chen and H.C. Tang contributed equally towards the accomplishment of this research.