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Journal of Applied Mathematics
Volume 2013, Article ID 494857, 9 pages
http://dx.doi.org/10.1155/2013/494857
Research Article

On IVF Approximating Spaces

1Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
2College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
3College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 4 March 2013; Accepted 21 June 2013

Academic Editor: Jinde Cao

Copyright © 2013 Dingwei Zheng et al. 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

We introduce the concept of IVF approximating spaces and obtain decision conditions that every IVF topological space is an IVF approximating space.

1. Introduction

Rough set theory was proposed by Pawlak [1] as a mathematical tool to handle imprecision and uncertainty in data analysis. It 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 [25].

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 (see [2]).

As a generalization of Zadeh’s fuzzy set, interval-valued fuzzy (IVF, for short) sets were introduced by Gorzałczany [6] and Türksen [7]. Mondal and Samanta [8] defined topology of IVF sets and studied their properties.

By replacing crisp relations with IVF relations, Sun et al. [9] introduced IVF rough sets based on an IVF approximation space, defined IVF information systems, and discussed their attribute reduction. Gong et al. [10] presented IVF rough sets based on approximation spaces and studied the knowledge discovery in IVF information systems.

Topological structure is an important base for knowledge extraction and processing. Therefore, an interesting and natural research topic in rough set theory is to study the relationship between rough sets and topologies.

The purpose of this paper is to investigate IVF approximating space, that is, a particular type of IVF topological spaces where the given IVF topology coincides with the IVF topology induced by some reflexive IVF relation.

2. Preliminaries

Throughout this paper, “interval-valued fuzzy” is denoted briefly by “IVF.” denotes a nonempty set called the universe. denotes , and denotes . denotes the family of all IVF sets in . denotes for each .

For any (), we define

Obviously,   for each .

Definition 1 (see [6, 7]). For each , one define where .

Definition 2 (see [6, 7]). An IVF set in is defined by a mapping .
Denote Then (resp., ) is called the lower (resp., upper) degree at which belongs to . (resp., ) is called the lower (resp., upper) IVF set of .

The set of all IVF sets in is denoted by .

Let . represents the IVF set which satisfies for each . We denoted by .

We recall some basic operations on as follows [6, 7]: for any and ,(1) for each ,(2) for each ,(3) for each ,(4) for each ,(5) for each .

Moreover, where .(6) for each .

Obviously,

Definition 3 (see [8]). is called an IVF point in , if there exist and such that We denote by .

If , then

Remark 4. .

Definition 5 (see [8]). is called an IVF topology on , if(i),(ii),(iii).
The pair is called an IVF topological space. Every member of is called an IVF open set in . Its complement is called an IVF closed set in .
An IVF topology is called Alexandrov, if (ii) in Definition 5 is replaced by(ii)′.
We denote  .
The interior and closure of denoted, respectively, by and , are defined as follows:

Proposition 6 (see [8]). Let be an IVF topology on . Then, for any ,(1), ,(2),(3), ,(4), ,(5), ,(6), .

3. IVF Approximation Spaces and IVF Rough Sets

Recall that is called an IVF relation on if .

Definition 7 (see [9]). Let be an IVF relation on . Then, is called(1)reflexive, if for each ,(2)symmetric, if for any ,(3)transitive, if for any .

Let be an IVF relation on . is called preorder if is reflexive and transitive (see [11]).

Definition 8 (see [9]). Let be an IVF relation on . The pair is called an IVF approximation space. For each , the IVF lower and the IVF upper approximations of with respect to , denoted by and , are two IVF sets and are, respectively, defined as follows:

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

Remark 9. Let be an IVF approximation space. Then,(1)for each , (2)for each , .

Proposition 10 (see [9]). Let be an IVF approximation space. Then, for each ,

Proposition 11. Let be an IVF approximation space. Then, for any , , and ,(1), ,(2), ,(3), ,(4), ,(5), .

Proof. (1) and (2) are obvious.
(3) For each ,  by Proposition 10, Then, .
Pick . Since ,
(4) For each , by we have .
By (3) and , we have Then, .
(5) For each ,  by Proposition 10, Then, .
Similarly, we can prove that .

Theorem 12. Let be an IVF relation on , and let be an IVF topology on . If one of the following conditions is satisfied, then is preorder.(1) is the interior operator of .(2) is the closure operator of .

Proof. By Propositions 6(4) and 11(3), (1) and (2) are equivalent. We only need to prove that (2) implies the reflexivity and transitivity of .
By Remark 9(1),   for any . Note that is the closure operator of . Then, for each , Thus, is reflexive.
For any , denote , and by Remark 4, Remark 9(1) and, Proposition 11(5), Then, is transitive.

Theorem 13. Let be an IVF approximation space. Then,

Proof. (1) By Proposition 11(3), (ILR) and (IUR) are equivalent. We only need to prove that the reflexivity of is equivalent to (IUR).
Assume that is reflexive. For any and , by the reflexivity of , . Then, Thus, .
Conversely, assume that (IUR) holds. For each , pick . By (IUR), we have . By Remark 9(1), Then, . Thus, is reflexive.
(2) By Proposition 11(3), (ILS) and (IUS) are equivalent. We only need to prove that the symmetry of R is equivalent to (IUS).
For any , by Remark 9(1), and . So, the symmetry of R is equivalent to (IUS).
(3) By Proposition 11(3), (ILT) and (IUT) are equivalent. We only need to prove that the transitivity of R is equivalent to (IUT).
Assume that is transitive. Then,   for any . Denote . Then, for any and , So, .
Conversely, assume that (IUT) holds. For any , by (IUT), By Remark 9(1),
Hence, is transitive.

Corollary 14. Let be an IVF approximation space. If is preorder, then

Proof. This holds by Theorem 13.

4. Relationships between IVF Relations and IVF Topologies

Let be an IVF relation on . We denote

4.1. IVF Topologies Induced by IVF Relations

Theorem 15. Let be an IVF relation on . If is reflexive, then is an IVF topology on .

Proof. (i) By Proposition 11(1), . Then, .
By Theorem 13(1), . Then, . So, .
(ii) Let . By Proposition 11(4), Then, . Thus, .
(iii) Let . Then, for each . By Proposition 11(2), By Theorem 13(1), .
Then, , and so .
Thus, is an IVF topology on .

Definition 16. Let be an IVF relation on . If is reflexive, then is called the IVF topology induced by on .

Theorem 17. Let be a reflexive IVF relation on , and let be the IVF topology induced by on . Then, the following properties hold:(1),(2)for each , (3)for each , .

Proof. (1) This is obvious.
(2) For each , by Proposition 11(2), we have
By Propositions 6(4) and 11(3),
By Theorem 13(1),
(3) For each , by Remark 9(2) and Theorem 13(1), . Then, . By Proposition 6(4), So, .

Theorem 18. Let be a reflexive IVF relation on , and let be the IVF topology induced by on . If is transitive, then(1),(2) is the interior operator of ,(3) is the closure operator of .

Proof. (1) Obviously, By Corollary 14, . Then, .
(2) It suffices to show that for each , where .
Since , by (1), .
By Theorem 13(1), . Then, .
By (1), . Then, .
(3) This holds by (2), Proposition 6(4), and Proposition 11(3).

Example 19. Let , and let be a reflexive IVF relation on . is defined as follows: Pick
We have
Then, is not transitive.
Since we have . Then, . Thus,
Obviously, . By Proposition 11(3), Then, . Thus, .

4.2. IVF Relations Induced by IVF Topologies

Definition 20. Let be an IVF topology on . Define an IVF relation on by for each . Then, is called the IVF relation induced by on .
An IVF topology on is said to satisfy the following: (C1), (C2).

Theorem 21. Let be an IVF topology on , and let be the IVF relation induced by on . Then, the following properties hold.(1) is reflexive.(2)If satisfies axiom and , then

Proof. (1) For each , Then, is reflexive.
(2) Since , we have . For each , by Remark 4, axiom, and Proposition 11, Then, for each , Hence, .
By Propositions 6(4) and 11(3), so

Proposition 22. Let be an IVF topology on . If satisfies and axioms, then(1) is the closure operator of  ,(2) is the interior operator of   ,(3)for each ,  ,(4) is Alexandrov.

Proof. (1) For each , by Remark 4, axiom, and axiom, Then, for each , Hence, . Thus, is the closure operator of .
(2) This holds by (1), Proposition 6(4) and, Proposition 11(3).
(3) For each , by (2), Remark 9(2), and Proposition 6(2), Then  , and so .
(4) Let . By (2), for each , By (2) and Proposition 11(4), So .
Hence, is Alexandrov.

Proposition 23. Let be a preorder IVF relation on . Then, satisfies and axioms.

Proof. For any and , by Theorem 18(3) and Proposition 11(5), Thus,   satisfies axiom.
For any , by Proposition 11(4) and Theorem 18, Thus, satisfies axiom.

5. IVF Approximating Spaces

As can be seen from Section 4, a reflexive IVF relation yields an IVF topology. In this section, we consider the reverse problem; that is, under which conditions can an IVF topology be associated with an IVF relation which produces the given IVF topology?

Definition 24. Let be an IVF topological space. If there exists a reflexive IVF relation on such that , then is called an IVF approximating space.

Theorem 25. Let be an IVF topology on . Let be the IVF relation induced by , and let be the IVF topology induced by on . If satisfies and axioms, then .

Proof. By Theorem 21(1), is reflexive. For any , put . By Remark 4 and Proposition 11(2), Then, So, is transitive.
So, is preorder. For each , by Theorem 18, Since satisfies and axioms, by Proposition 22(1), . So, .
Thus, .

Theorem 26. Let be an IVF topology on . Then, the following are equivalent.(1) satisfies and axioms.(2)For any , and , (3)There exists a preorder IVF relation on such that is the closure operator of .(4)There exists a preorder IVF relation on such that is the interior operator of .(5) is the closure operator of .(6) is the interior operator of .

Proof. (1) (2) is obvious.
(1) (3). Suppose that satisfies and axioms. Pick . By Proposition 22(1), is the closure operator of . By Theorem 12(2), is preorder.
(3) (4). Let be the closure operator of for some preorder IVF relation on . For each , by Propositions 6(4) and 11(3), Thus, is the interior operator of .
(4) (6). Let be the interior operator of for some preorder IVF relation on . For each , by Remark 9(1), Then, . Note that is the interior operator of . Then, is the interior operator of .
(6) (5). This holds by Propositions 6(4) and 11(3).
(5) (1). For any and , by Proposition 11(5), Thus, satisfies axiom.
For any , by Proposition 11(4), Thus, satisfies axiom.

Theorem 27. Let be an IVF topological space. If one of the following conditions is satisfied, then is an IVF approximating space.(1) satisfies and axioms.(2)For any and , (3)There exists a preorder IVF relation on such that is the closure operator of .(4)There exists a preorder IVF relation on such that is the interior operator of .(5) is the closure operator of .(6) is the interior operator of .

Proof. These hold by Theorems 25 and 26.

Example 28. is an IVF approximating space.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11061004), the Natural Science Foundation of Guangxi (no. 2013GXNSFBA019016), and Guangxi University Science and Technology Research Project (no. 2013ZD020).

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