#### Abstract

The interval set is a special set, which describes uncertainty of an uncertain concept or set with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations (i.e., upper approximation set () and lower approximation set ()) are presented, respectively. The disadvantages of using upper-approximation set () or lower-approximation set () as approximation sets of the uncertain set (uncertain concept) are analyzed, and a new method for looking for a better approximation set of the interval set is proposed. The conclusion that the approximation set () is an optimal approximation set of interval set is drawn and proved successfully. The change rules of () with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval set is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.

#### 1. Introduction

Since the twenty-first century, researchers have done more and more research on uncertain problems [1]. It is an important research topic on how to effectively deal with uncertain data and how to acquire more knowledge and rules from the big data. At the same time, many methods for acquiring uncertain knowledge from uncertain information systems appeared gradually. In 1965, fuzzy sets theory was proposed by Zadeh [2]. In 1982, rough sets theory was proposed by Pawlak [3]. In 1990, quotient space theory was presented by L. Zhang and B. Zhang [4]. In 1993, interval sets and interval sets algebra were presented by Yao [5, 6].

Rough set theory is a mathematical tool to handle the uncertain information, which is imprecise, inconsistent, or incomplete. The basic thought of rough set is to obtain concepts and rules through classification of relational database and discover knowledge by the classification induced by equivalence relations; then approximation sets of the target concept are obtained with many equivalence classes. Rough set is a useful tool to handle uncertain problems, as well as fuzzy set theory, probability theory, and evidence theory. Because rough set theory has novel ideas and its calculation is easy and simple, it has been an important technology in intelligent information processing [7â€“9]. The key issue of rough set is building a knowledge space which is a partition of the domain and is induced by an equivalence relation. In the knowledge space, two certain sets named upper approximation set and lower approximation set are used to describe the target concept as its two boundaries. If knowledge granularity in knowledge space is coarser, then the border region of described target concept is wider and approximate accuracy is relatively lower. On the contrary, if knowledge granularity in knowledge space is finer, then the border region is narrower and approximate accuracy is relatively higher.

The interval set theory is an effective method for describing ambiguous information [10â€“12] and can be used in uncertain reasoning as well as the rough set [13â€“15]. The interval set not only can be used to describe the partially known concept, but also can be used to study the approximation set of the uncertain target concept. So, the interval set is a more general model for processing the uncertain information [16]. The interval set is described by two sets named upper bound and lower bound [17]. The elements in lower bound certainly belong to target concept, and the elements in upper bound probably belong to target concept. When the boundary region has no element, the interval set degenerates into a usual set [5], while, in a certain knowledge granularity space, target concept may be uncertain. To solve this problem, in this paper, the approximate representation of interval set is discussed in detail in Pawlakâ€™s approximation space. And then, the upper approximation set of interval set and lower approximation set of interval set are defined, respectively. The change rules of the approximation set of interval set with the different knowledge granularity in Pawlakâ€™s approximation space are analyzed.

In this paper, an approximation set of the target concept is built in a certain knowledge space induced by many conditional attributes, and we find that this approximation set may have better similarity degree with the target concept than that of or . Therefore, an interval set is translated into a fuzzy set at first in this paper. And then, according to the different membership degrees of different elements in boundary region, an approximation set of interval set is obtained by cut-set with some threshold. And then, the decision-making rules can be obtained through the approximation set instead of in current knowledge granularity space. In addition, the change rules of similarity between a target concept and its approximation sets are analyzed in detail.

The method used is getting the approximation of interval sets with a special approximation degree. With this method, we can use certain sets to describe an interval set in Pawlakâ€™s space. Our motivation is to get a mathematical theory model, which can be helpful to promote interval sets development in knowledge acquisition.

The rest of this paper is organized as follows. In Section 2, the related basic concepts and preliminary knowledge are reviewed. In Section 3, the concept of similarity degree between two interval sets is defined. The approximation set of interval set and 0.5-approximation set are proposed in Section 4. The change rules of similarity degree between the approximation sets and the target concept with the different knowledge granularity spaces are discussed in Section 5. This paper is concluded in Section 6.

#### 2. Preliminaries

In order to introduce the approximation set of interval set more easily, many basic concepts will be reviewed at first.

*Definition 1 (interval set [17]). *An interval set is a new collection, and it is described by two sets named upper bound and lower bound. The interval set can be defined as follows. Let be a finite set which is called universal set, and then let be the power set of â€‰ and let interval set be a subset of . In mathematical form, interval set is defined as . If , is a usual classical set.

In order to better explain the interval set, there is an example [17, 18] as follows. Let be all papers submitted to a conference. After being reviewed, there are 3 kinds of results. The first kind of results is the set of papers certainly accepted and represented by . The second kind of results is the set of papers that need to be further reviewed and represented by . The last kind of results is the set of papers rejected and represented by . Although every paper just can be rejected or accepted, no one knows the final result before further evaluation. Through reviewing, the set of papers accepted by the conference is described as .

*Definition 2 (indiscernibility relation [4, 19]). *For any attribute set , let us define one unclear binary relationship .

*Definition 3 (information table of knowledge expression system [4, 20]). *A knowledge expression system can be described as . is the domain, and is the set of all attributes. Subset is a set of conditional attributes, and is a set of decision-making attributes. is the set of attribute values. describes the range of attribute values where . is an information function which describes attribute values of object in .

*Definition 4 (upper approximation set and lower approximation set of rough set [3]). *A knowledge-expression system is described as . For any and , upper approximation set and lower approximation set of rough set on are defined as follows:
where is the classification of equivalence relation on . Upper approximation set and lower approximation set of rough set on can be defined in another form as follows:
where and is an equivalence class of on relation . is a set of objects which certainly belong to according to knowledge ; is a set of objects which possibly belong to according to knowledge . Let be called* boundary region* of target concept on relation . Let be called* positive region* of target concept on relation . Let be called* negative region* of target concept on relation . is a set of objects which just possibly belong to target concept .

*Definition 5 (similarity degree between two sets [20]). *Let and be two subsets of domain , which means . Defining a mapping , that is, , is the similarity degree between and , if satisfies the following conditions.(1)For any (boundedness).(2)For any (symmetry).(3)For any if and only if .

Any formula satisfying (1), (2), and (3) is a similarity degree formula between two sets. Zhang et al. [20] gave out a similarity degree formula where represents the number of elements in finite subset. Obviously, this formula satisfies (1), (2), and (3).

*Definition 6 (similarity degree between two interval sets). *Let be an interval set and let be also an interval set. Similarity degree between two interval sets can be defined as follows:
accords with Definition 5.

*Definition 7 (upper approximation set and lower approximation set of an interval set). *Let be an interval set. Let be an equivalence relation on domain . Upper approximation set of this interval set is defined as . Lower approximation set of this interval set is defined as .

Figures 1 and 2 are probably helpful to understand Definition 7. In Figure 1, the outer circle standing for a set and inner circle standing for a set represent an interval set , and each block represents an equivalence class. The black region represents , and the whole colored region (black and gray region) represents . In Figure 2, the outer circle standing for a set and inner circle standing for a set represent an interval set , and each block represents an equivalence class. The black region represents , and the whole colored region (black and gray region) represents .

#### 3. Approximation Set of an Interval Set

If stands for the upper approximation set of the interval set , then the similarity degree between and can be defined as follows: If stands for the lower approximation set of the interval set , then the similarity degree between and is defined as follows:

If the knowledge space keeps unchanged, is there a better approximation set of the target concept ? In this paper, the better approximation sets of target concept will be proposed. Let be a nonempty set of objects. Let , , and the membership degree of belonging to set is defined as Obviously, .

*Definition 8 (-approximation set of set [20]). *Let be a nonempty set of objects, and let knowledge space be . Let , and the membership degree belonging to set is
If , then is called -approximation set of set .

*Definition 9 (-approximation set of set ). *Let and ; then is called -approximation set of the interval set .

Figure 3 is probably helpful to understand Definition 9. In Figure 3, the outer circle standing for a set and inner circle standing for a set represent an interval set , and each block represents an equivalence class. The black region represents , and the whole colored region (black and gray region) represents .

#### 4. Approximation Set of an Interval Set

Lemma 10 (see [20]). *Let , and be all real numbers. If , then .*

Lemma 11 (see [20]). *Let , and be all real numbers. In the numbers, . If , then . If , then .*

In order to better understand the similarity degree between and , Theorems 12 and 13 are presented as follows.

Theorem 12. *Let be a finite domain, let be an interval set on , and let be an equivalence relation on . Then, .*

For example, let , , . Then, , , , , , .

And then we can have , , , .

*Proof. *According to Definition 6,

(1) There we first prove
For all , we have . That is,

Because is an equivalence relation on , the classifications induced by can be denoted as . Then, < . Obviously, , and then let . So, . Because the intersection sets between any two elements in are empty sets, we can get that
Becauseâ€‰â€‰ and the intersection set between any two elements inâ€‰â€‰, is empty, we have that + . So,

Because
we have . In the same way, according to and Lemma 10, we can easily get
Therefore,

(2) In a similar way with (1), we can have the inequality

From (1) and (2), we have . So, Theorem 12 has been proved completely.

Theorem 12 shows that the similarity degree between an interval set and its approximation set is better than the similarity degree between and its lower approximation set .

Theorem 13. *Let be a finite domain, let be an interval set on , and let be an equivalence relation on . If
**
then .*

For example, let ,â€‰â€‰, . Then, , , , , , ,

And then we can have ,, .

*Proof. *According to Definition 6,

(1) Let , and the intersection sets between any two elements in are empty sets. Because
it is obvious that . In the same way, we have . Then we have âˆ’ . Because the intersection sets between any two elements in are empty sets, we have , , and *âˆª*. Because the intersection sets between any two elements in are empty sets, and are held. So,

For
according to Lemma 11, we have
that is to say,
Therefore, we have

(2) In a similar way with (1), we can easily obtain the conclusion that
when

According to (1) and (2), the inequality is held. So, Theorem 13 has been proved successfully.

Theorem 13 shows that, under some conditions, the similarity degree between an interval set and its approximation set is better than the similarity degree between and its lower approximation set .

Theorem 14. *Let be a finite domain, an interval set on , and an equivalence relation on . If , then .*

For example, let ,, . Then, , , , ,, , and , .

And then we can have , , , . This example is in accordance with the theorem.

*Proof. *(1) For all , then , which means
Because < , we can easily get . Let