Journal of Applied Mathematics

Volume 2014 (2014), Article ID 405802, 7 pages

http://dx.doi.org/10.1155/2014/405802

Research Article

Fuzzy Approximating Spaces

Bin Qin

School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China

Received 11 October 2013; Accepted 9 May 2014; Published 15 June 2014

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2014 Bin Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Relationships between fuzzy relations and fuzzy topologies are deeply researched. The concept of fuzzy approximating spaces is introduced and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained.

1. Introduction

Rough set theory, proposed by Pawlak [1], is a new mathematical tool for data reasoning. It may be seen as an extension of classical set theory and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems, and many other fields [2–5].

The basic structure of rough set theory is an approximation space. Based on it, lower and upper approximations can be induced. Using these approximations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules. A key notion in Pawlak rough set model is equivalence relations. The equivalence classes are the building blocks for the construction of these approximations. In the real world, the equivalence relation is, however, too restrictive for many practical applications. To address this issue, many interesting and meaningful extensions of Pawlak rough sets have been presented. Equivalence relations can be replaced by tolerance relations [6], similarity relations [7], binary relations [8, 9], and so on.

Various fuzzy generalizations of rough approximations have been proposed [10–14]. The most common fuzzy rough set is obtained by replacing the crisp relations with fuzzy relations on the universe and crisp subsets with fuzzy sets. Dubois and Prade [10] first proposed the concept of rough fuzzy sets and fuzzy rough sets and pointed out that a rough fuzzy set is a special case of a fuzzy rough set. Now, fuzzy rough sets have been used to solve practical problems such as data mining [15], approximate reasoning [5], and medical time series.

An interesting and natural research topic in rough set theory is to study the relationship between rough sets and topologies. Many authors studied topological properties of rough sets [16–21]. It is known that the pair of lower and upper approximation operators induced by a reflexive and transitive relation is exactly the pair of interior and closure operators of a topology [16, 22].

The purpose of this paper is to investigate further topological properties of fuzzy rough sets.

The remaining part of this paper is organized as follows. In Section 2, we recall some basic concepts about fuzzy sets and fuzzy topologies. In Section 3, fuzzy rough approximation operators are further investigated. In Section 4, relationships between fuzzy approximation spaces and fuzzy topologies are established. In Section 5, the concept of fuzzy approximating spaces is introduced and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained. Conclusions are in Section 6.

2. Preliminaries

Throughout this paper, denotes a nonempty finite set, denotes , and denotes the set of all fuzzy sets in . For , denotes the constant fuzzy set in .

For all , we denote

Obviously, for some .

A fuzzy set is called a fuzzy point in , if it takes the value for each except one, say, . If its value at is , we denote this fuzzy point by , where the point is called its support and is called its height (see [23, 24]).

Specially,

Remark 1. Consider

Definition 2 (see [25]). is called a fuzzy topology on , if

 (i) ,, (ii) , (iii) .

The pair is called a fuzzy topological space. Every member of is called a fuzzy open set in . Its complement is called a fuzzy closed set in .

We denote .

It should be pointed out that if (i) in Definition 2 is replaced [26] by

then is a fuzzy topology in the sense of Chang [26]. We can see that a fuzzy topology in the sense of Lowen must be a fuzzy topology in the sense of Chang. In this paper, we always consider the fuzzy topology in the sense of Lowen.

A fuzzy topology is called Alexandrov [27] if (ii) in Definition 2 is replaced by

  .

Definition 3 (see [28]). Let be a relation on . , denote

Then and are called the predecessor and successor neighborhood of , respectively.

3. Fuzzy Approximation Spaces and Fuzzy Rough Approximation Operators

Recall that is called a fuzzy relation on if .

Definition 4 (see [14, 29]). Let be a fuzzy relation on . Then is called

(1)reflexive, if for any ,(2)symmetric, if for any ,(3)transitive, if for any .

Let be a fuzzy relation on . is called preorder if is reflexive and transitive. is called equivalence if is reflexive, symmetric, and transitive.

Definition 5 (see [14, 29]). Let be a fuzzy relation on . The pair is called a fuzzy approximation space. Based on , the fuzzy lower and the fuzzy upper approximation of , denoted, respectively, by and , are defined as follows:

The pair is called the fuzzy rough set of with respect to .

and are called the fuzzy lower approximation operator and the fuzzy upper approximation operator, respectively. In general, we refer to and as the fuzzy rough approximation operators.

Remark 6. and .

Proposition 7 (see [30]). Let be a fuzzy relation on . Then, for any , and ,

(1), ,(2), ,(3), ,(4), ,(5).

Theorem 8 (see [14, 29, 30]). Let be a fuzzy relation on . Then consider the following.

    is reflexive , .

 , .

    is symmetric , .

 , .

    is transitive , .

 , .

Remark 9.   , ;

if is reflexive, then , .

Theorem 10. Let be a fuzzy relation on and let be a fuzzy topology on . If one of the following conditions is satisfied, then is preorder:

(1) is the closure operator of ,(2) is the interior operator of .

Proof. Let . Put . Note that is the interior operator of . Then

Thus is reflexive. By Remark 1, Remark 6, and Proposition 7(5),

Then is transitive. Hence is preorder.

The proof is similar to .

Proposition 11. Let be a fuzzy relation on . Then, for each with , consider the following.

 (1), . , . (2) If is reflexive, then , . , .

Proof. (a) Necessity. Suppose that . Note that ,

Then , . Since , , we have

Sufficiency. Suppose that holds. Let . Consider the following.

(i)If , then

(ii)If , then and so

Hence .

Thus .

(b) Necessity. Suppose that . Note that ,

Then , . Since , , we have

Sufficiency. Suppose that holds. Let . Consider the following.

(i)If , then and so

(ii)If , then and so

Hence .

Thus .

(2) holds by (1), the reflexivity of , and Theorem 8(1).

4. Relationships between Fuzzy Relations and Fuzzy Topologies

4.1. Fuzzy Topology Induced by Fuzzy Relations

Let be a fuzzy relation on . Denote

Remark 12. Let be a fuzzy relation on . Then consider the following.

(1), .(2)If is transitive, then .(3)If is reflexive, then .(4)If is preorder, then .

Theorem 13 (see [30]). Let be a preorder fuzzy relation on . Then consider the following.

(1) is a fuzzy topology on .(2) is the interior operator of .(3) is the closure operator of .

Theorem 14. Let be a fuzzy relation on . Then

(1) is an Alexandrov fuzzy topology on ,(2)if is reflexive, then ,

(3).(4), .

Proof. (1) By Remark 9(1), .

Let . Then , . By Proposition 7(4),

Hence . So is Alexandrov.

(2) , by Proposition 7(2),

By Proposition 7(3),

By the reflexivity of and Theorem 8(1),

(3) holds by Proposition 7(3).

(4) holds by (3) and Remark 9(1).

Definition 15. Let be a fuzzy relation on . is called the fuzzy topology induced by on .

Definition 16. Let be a fuzzy relation on . is called pseudoconstant, if there exists such that, for any ,

We write by or .

Obviously, every pseudoconstant fuzzy relation is an equivalence fuzzy relation.

Remark 17. (1) , implies .

(2) , .

(3) , .

(4) , .

The following theorem gives the topological structure of fuzzy approximation spaces.

Theorem 18. Let be a fuzzy approximation space. Then

(1), (2).

Proof. (1) holds by Proposition 11(1) and Remark 17(3).

(2) holds by Remark 17(1).

4.2. Fuzzy Relations Induced by Fuzzy Topologies

Definition 19. Let be a fuzzy topology on . Define

Then is called the fuzzy relation induced by on .

Theorem 20. Let be a fuzzy topology on and let be the fuzzy relation induced by on . Then

(1) is reflexive,(2)If whenever , then ,

Proof. (1) ,

Then is reflexive.

(2) , by Remark 1 and Proposition 7(2),

Then ,

Hence .

By Proposition 7(3),

So

Theorem 21. Let be a preorder fuzzy relation, let be the fuzzy topology by on , and let be the fuzzy relation induced by on . Then .

Proof. By Remark 6, Remark 12(4), and Theorem 13(3),

Then .

4.3. (CC) Axiom

The following condition for a fuzzy topology on is called (CC) axiom in [31],

(CC) axiom: for any and ,

Proposition 22. Let be a fuzzy topology on . If satisfied the (CC) axiom, then

(1) is the closure operator of ,(2) is a preorder fuzzy relation on ,(3), ,(4) is Alexandrov.

Proof. (1) , by Remark 1 and (CC) axiom,

Then ,

Thus . So is the closure operator of .

(2) holds by (1) and Theorem 10.

(3) , by (2), Proposition 7(3), and Remark 9(2),

Then .

(4) By (1) and Proposition 7(3), is the interior operator of .

Let . Then , . By Proposition 7(4),

So . Hence is Alexandrov.

Proposition 23. Let be a preorder fuzzy relation on . Then satisfies (CC) axiom.

Proof. For any and , by Proposition 7(6) and Proposition 22,

Theorem 24. Let be a fuzzy topology on , let be the fuzzy relation induced by on , and let be the fuzzy topology induced by on . If satisfies (CC) axiom, then .

Proof. By Theorem 20(1), is reflexive. , put . By (CC) axiom and Remark 1,

Then

Thus is transitive and so is preorder.

, by Remark 12 and Theorem 13(3),

By (CC) axiom and Proposition 22, . So . Thus

Hence .

Theorem 25. Let be a fuzzy topology on . Then the following are equivalent.

(1) satisfies (CC) axiom.(2)There exists a preorder fuzzy relation on such that is the closure operator of .(3)There exists a preorder fuzzy relation on such that is the interior operator of .(4) is the closure operator of .(5) is the interior operator of .

Proof. (1) (2). Suppose that satisfies (CC) axiom. Pick . By Proposition 22, is the closure operator of . By Theorem 10, is preorder.

(2) (3). Let be the closure operator of for some preorder fuzzy relation on . , by Proposition 7(3),

Thus is the interior operator of .

(3) (5). Let be the interior operator of for some preorder fuzzy relation on .

By Remark 6,

Then . Thus is the interior operator of .

(5) (4) holds by Proposition 7(3).

(4) (1). For any and , by Proposition 7(6),

Thus satisfies (CC) axiom.

Theorem 26. Let

Then there exists a one-to-one correspondence between and .

Proof. and are defined as follows:

By Theorem 21,

where is the composition of and and is the identity mapping on .

By Theorem 24,

where is the composition of and and is the identity mapping on .

Hence and are two one-to-one correspondences. This proves that there exists a one-to-one correspondence between and .

5. Fuzzy Approximating Spaces

As can be seen from Section 4, a fuzzy relation yields a fuzzy topology. In this section, we consider the reverse problem; that is, when can the given fuzzy topology coincide with the fuzzy topology induced by some fuzzy relation?

Definition 27. Let be a fuzzy topological space. If there exists a fuzzy relation on such that , then is called a fuzzy approximating space.

Theorem 28. Let be a fuzzy topological space. If one of the following conditions is satisfied, then is a fuzzy approximating space.

(1) satisfies (CC) axiom.(2)There exists a preorder fuzzy relation on such that is the closure operator of .(3)There exists a preorder fuzzy relation on such that is the interior operator of .(4) is the closure operator of .(5) is the interior operator of .

Proof. These hold by Theorems 24 and 25.

Theorem 29. Let be a fuzzy topological space. Then is a fuzzy approximating space if and only if there exists a fuzzy relation such that

Proof. This holds by Theorem 18(1).

Example 30. is a fuzzy approximating space.

6. Conclusions

In this paper, relationships between fuzzy relations and fuzzy topology were discussed. The fact that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies which satisfy (CC) axiom was proved. We introduced the concept of fuzzy approximating spaces and gave decision conditions that a fuzzy topological space is a fuzzy approximating space.

The results of this paper illustrate that fuzzy relations can be researched by means of topology. We hope that one can find applications of topological properties of fuzzy rough sets in information sciences. In future work, we will do similar exploration of fuzzy neighborhood spaces like this paper.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 11261005 and 11161029), the Natural Science Foundation of Guangxi (no. 2012GXNSFDA276040), Guangxi University Science and Technology Research Project (no. 2013ZD061), and the Natural Science Foundation for Young Scholar of Guangxi Province (2013GXNSFBA019020).

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