Journal of Applied Mathematics

Volume 2013 (2013), Article ID 241485, 15 pages

http://dx.doi.org/10.1155/2013/241485

Research Article

Soft Rough Approximation Operators and Related Results

Zhaowen Li,¹ Bin Qin,² and Zhangyong Cai³

¹School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

²School of Information and Statistics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

³Department of Mathematics, Guangxi Teachers College, Nanning, Guangxi 530023, China

Received 2 August 2012; Revised 21 January 2013; Accepted 22 January 2013

Academic Editor: Hui-Shen Shen

Copyright © 2013 Zhaowen Li 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.

Soft set theory is a newly emerging tool to deal with uncertain problems. Based on soft sets, soft rough approximation operators are introduced, and soft rough sets are defined by using soft rough approximation operators. Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model. This paper is devoted to investigating soft rough approximation operations and relationships among soft sets, soft rough sets, and topologies. We consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are investigated, and their related properties are given. We show that Pawlak's rough set model can be viewed as a special case of soft rough sets, obtain the structure of soft rough sets, give the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space.

1. Introduction

Most of traditional methods for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, many practical problems within fields such as economics, engineering, environmental science, medical science, and social sciences involve data that contain uncertainties. We cannot use traditional methods because of various types of uncertainties present in these problems.

There are several theories: probability theory, fuzzy set theory, theory of interval mathematics, and rough set theory [1], which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties (see [2]). For example, theory of probabilities can deal only with stochastically stable phenomena. To overcome these kinds of difficulties, Molodtsov [2] proposed a completely new approach, which is called soft set theory, for modeling uncertainty.

Presently, works on soft set theory are progressing rapidly. Maji et al. [3–5] further studied soft set theory, used this theory to solve some decision making problems, and devoted fuzzy soft sets combining soft sets with fuzzy sets. Roy et al. [6] presented a fuzzy soft set theoretic approach towards decision making problems. Jiang et al. [7] extended soft sets with description logics. Aktaş et al. [8] defined soft groups. Feng et al. [9, 10] investigated relationships among soft sets, rough sets, and fuzzy sets. Shabir et al. [11] investigated soft topological spaces. Ge et al. [12] discussed relationships between soft sets and topological spaces.

The purpose of this paper is to investigate soft rough approximation operators and relationships among soft sets, soft rough sets, and topologies.

The remaining part of this paper is organized as follows. In Section 2, we recall some basic concepts of rough sets and soft sets. In Section 3, we consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are introduced or investigated, and the fact that Pawlak’s rough set model can be viewed as a special case of soft rough sets is proved. In Section 4, we investigate the relationships between soft sets and topologies, obtain the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space. In Section 5, we give the related properties of soft rough sets and obtain the structure of soft rough sets. In Section 6, we prove that there exists a one-to-one correspondence between the set of all soft sets and the set of all formal contexts. Conclusion is in Section 7.

2. Overview of Rough Sets and Soft Sets

In this section, we recall some basic concepts about rough sets and soft sets.

Throughout this paper, denotes initial universe, denotes the set of all possible parameters, and denotes the family of all subsets of . We only consider the case where both and are nonempty finite sets.

2.1. Rough Sets

Rough set theory was initiated by [1] for dealing with vagueness and granularity in information systems. This theory handles the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. 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 (see [1, 13]).

Let be an equivalence relation on . The pair is called a Pawlak approximation space. The equivalence relation is often called an indiscernibility relation. Using the indiscernibility relation , one can define the following two rough approximations:

and are called the Pawlak lower approximation and the Pawlak upper approximation of , respectively. In general, we refer to and as Pawlak rough approximation operators and and as Pawlak rough approximations of .

The Pawlak boundary region of is defined by the difference between these Pawlak rough approximations; that is, . It can easily be seen that .

A set is Pawlak rough if its boundary region is not empty; otherwise, the set is crisp. Thus, is Pawlak rough if .

We may relax equivalence relations so that rough set theory is able to solve more complicated problems in practice. The classical rough set theory based on equivalence relations has been extended to binary relations [14].

Definition 1 (see [14]). Let be a binary relation on . The pair is called an approximation space. Based on the approximation space , we define a pair of operations ,   as follows:

where and .

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

is called a definable set if ; is called a rough set if .

2.2. Soft Sets

Definition 2 (see [2]). Let be a nonempty subset of . A pair is called a soft set over , if is a mapping given by . We denote by .

In other words, a soft set over is a parametrized family of subsets of the universe . For , may be considered as the set of -approximate elements of the soft set .

Definition 3 (see [3]). Let and be two soft sets over .

(1) is called a soft subset of , if and for each . We denote it by .(2) is called a soft super set of , if . We denote it by .

Definition 4 (see [3]). Let and be two soft sets over . and are called soft equal, if and for each . We denote it by .

Obviously, if and only if and .

Definition 5 (see [10, 12]). Let be a soft set over .

(1) is called full, if .(2) is called partition, if forms a partition of .

Obviously, every partition soft set is full.

Definition 6. Let be a soft set over .

(1) is called keeping intersection, if for any , there exists such that .(2) is called keeping union, if for any , there exists such that .(3) is called topological, if is a topology on .

Obviously, every topological soft set is full, keeping intersection and keeping union, and is keeping intersection (resp., keeping union) if and only if for any , there exists such that (resp., ).

Example 7. Let , , and let be a soft set over , defined as follows:

Obviously, is not partition. We have

Then, is full and keeping intersection. But

Thus, is not keeping union.

Example 8. Let , , and let be a soft set over , defined as follows:

Then, is keeping intersection and keeping union. But is not full.

Example 9. Let , , and let be a soft set over , defined as follows:

Then, is full and keeping union. But is neither keeping intersection nor partition.

Example 10. Let , , and let be a soft set over , defined as follows:

Obviously, is partition. But

Thus, is neither keeping intersection nor keeping union.

Example 11. Let , and let be a soft set over , defined as follows

Obviously, is full, keeping intersection and keeping union. But is not topological.

From Examples 7, 8, 9, and 10 and 11, we have the following relationships:

3. Soft Rough Approximation Operators and Soft Rough Sets

Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model (see Example 4.6 in [10]), and defining soft rough sets and some related concepts needs using soft rough approximation operators based on soft sets. Thus, soft rough approximation operators deserve further research.

In this section, we consider a pair of soft rough approximation operators which are presented by Feng et al. in [9, 10], proposing three pairs of soft rough approximation operators and giving their properties. Four sorts of soft rough sets are defined by using four pairs of soft rough approximation operators.

3.1. Soft Rough Approximation Operators and

Definition 12 (see [9, 10]). Let be a soft set over . Then, the pair is called a soft approximation space. We define a pair of operators , as follows:

and are called the soft -lower approximation operator on and the soft -upper approximation operator on , respectively. In general, we refer to and as soft -rough approximations operator on .

and are called the soft -lower approximation and the soft -upper approximation of , respectively. In general, we refer to and as soft rough approximations of with respect to .

is called a soft -definable set if ; is called a soft -rough set if .

Moreover, the sets

are called the soft -positive region, the soft -negative region, and the soft -boundary region of , respectively.

Proposition 13 (see [9, 10]). Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

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

Proposition 14. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold.

(1)If is full, then

(a)  for  any   ; (b).

(2)If is keeping union, then

(a)for  any   , there  exists   such  that   ; (b)for  any   , there  exists     such  that .

(3)If is keeping intersection, then

(4)If is partition, then

(5)If is full and keeping union, then

Proof. (1)(a) By Proposition 13,  . Suppose that . Pick

Since is full, . So, for some . implies . Thus, , contradiction. Hence,

(1)(b) This holds by (1) and Proposition 13.

(2) This holds by Proposition 13.

(3) By Proposition 13,

Suppose that . Pick

Then, there exist such that and . Since is keeping intersection, then for some . This implies . Thus, , contradiction. Hence,

Therefore,

(4) Suppose that . Pick

Then, there exist such that and . Since is partition, then . This implies . So, , contradiction. Thus,

Therefore,

(5) Since is full and keeping union, then for some . For each and each , and , and then .

3.2. Soft Rough Approximation Operators and , and , and and

Definition 15. Let be a soft set over .

(1)Define a binary relation on by

for each . Then, is called the binary relation induced by on .(2)For each , define a successor neighborhood of in by

Since the following Proposition 16 is clear, we omit its proof.

Proposition 16. Let be a soft set over , and let be the binary relation induced by on . Then, the following properties hold.

(1) is a symmetric relation.(2)If is full, then is a reflexive relation.(3)If is partition, then is an equivalence relation.

Proposition 17. Let be a soft set over , and let be the binary relation induced by on . Then, the following properties hold.

(1)If with , then .(2)If is partition and with , then .(3)If is keeping union, then for each , there exists such that .

Proof. (1) This is obvious.

(2) Suppose that . Then, , and so for some . Since is partition and , then . Thus, . This implies . By (1),

(3) Suppose that . Then, , and so for some . By (1), . Thus, . This implies

Since is keeping union, then for some . Thus,

Definition 18. Let be a soft set over , and let be a soft approximation space. We define three pairs of soft rough approximation operations: as follows:

(1)

is called a soft -definable set if . is called a soft -rough set if . The sets

are called the soft -positive region, the soft -negative region, and the soft -boundary region of , respectively. Consider,

(2)

is called a soft -definable set if . is called a soft -rough set if . The sets

are called the soft -positive region, the soft -negative region, and the soft -boundary region of , respectively. Consider,

(3)

is called a soft -definable set if . is called a soft -rough set if . The sets

are called the soft -positive region, the soft -negative region, and the soft -boundary region of , respectively.

In general, we also refer to and ,   and , and and as soft rough approximations of with respect to , , , respectively.

It is not very difficult to prove Propositions 19, 20, and 21 (see [15]).

Proposition 19. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1). If is full, then

(2); . If is full, then

(3); . (4). (5). (6); . (7).

Proposition 20. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1)

(2);   . If is full, then

(3); . (4); .

Proposition 21. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1)If is full, then

(2); . If is full, then

(3); . (4). (5). (6); . (7). (8).

Example 22. Let , , and let be a soft set over , defined as follows:

Obviously, is not full. We have

Let , , and .

(1)We have

Thus,

(2)We have

Thus,

(3)We have

Thus,

(4)We have

Thus,

(5)We have

Thus,

3.3. The Relationships among Four Pairs of Soft Rough Approximation Operators

Lemma 23. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1)If is full, then

(2)If is full and keeping union, then

(3)If is partition, then

(a); (b). (c); (d).

Proof. (1) Suppose that . Then, . Since is full, then for some . By Proposition 17, . Thus, . This implies . Thus,

(2) If , then . If , by Proposition 14, .

Hence,

(3)(a) Suppose that . Then, for some . Since is partition and , then by Proposition 17. This implies . Thus,

By (1),

(3)(b) This is similar to the proof of (3) (a).

Suppose that . Then, . Since is full, then for some . Since is partition and , then by Proposition 17. This implies . Thus,

Hence, .

By Propositions 13 and 17, we have Lemma 24.

Lemma 24. Let be a soft set over , and let be a soft approximation space. If is keeping union, then for any ,

Lemma 25. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1)If is full, then

(2)If is partition, then

(a); (b).

Proof. (1) Suppose that . Then, . Since is full, then by Proposition 17. This implies . Thus,

Suppose that . Pick

implies . So, . Since is full, then by Proposition 17. This implies . Thus, , contradiction.

Hence, .

By Proposition 20,

Since

then

By Propositions 19, 20, and 21,

(2)(a) Suppose that . Then, there exists such that . Since is partition, then is an equivalence relation by Proposition 16. Thus, follows . So, . This implies . Hence,

By (1),

Suppose that . Pick

implies . implies that there exists such that and . So, . Note that is an equivalence relation. Then, . Thus, , contradiction.

Hence, .

This proves that

By (1),

(2)(b) By (2)(a),

Then,

By Propositions 19, 20, and 21,

Example 26. Let , , and let be a soft set over , defined as follows:

Obviously, is full. We have

Let . We have

Thus,

By Proposition 16 and Lemmas 23, 24, and 25, we have Theorem 27.

Theorem 27. Let be a soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1)If is full, then

(2)If is full and keeping union, then

3.4. The Relationship between Soft Rough Approximation Operators and Pawlak Rough Approximation Operators

In this section, we shall explore the relationship between soft rough approximation operators and Pawlak rough approximation operators.

Definition 28. Let be an equivalence relation on . Define a mapping by

for each , where . Then, is called the soft set induced by on .

Theorem 29 (see [10]). Let be an equivalence relation on , let be the soft set induced by on , and let be a soft approximation space. Then, for each ,

Thus, in this case, is a Pawlak rough set if and only if X is a soft -rough set.

By Proposition 16 and Lemmas 23 and 25, we have Theorem 30.

Theorem 30. Let be a partition soft set over , and let be a soft approximation space. Then, the following properties hold for any .

(1); (2),

where and are the Pawlak rough approximations of .

Corollary 31. Let be a full soft set over , and let be a soft approximation space. Then,

(1)every soft -definable set is a soft -definable set.(2)every soft -definable set is a soft -definable set.

Remark 32. Theorems 29 and 30 illustrate that Pawlak’s rough set models can be viewed as a special case of soft rough sets.

Remark 33. Example 4.6 in [10] illustrates that a soft rough approximation is a worth considering alternative to the rough approximation. Soft rough sets could provide a better approximation than rough sets do.

4. The Relationships between Soft Sets and Topologies

Let be a soft set over , and let be a soft approximation space. Denote

4.1. The First Sort of Topologies Induced by a Soft Set and Related Results

By Propositions 13 and 14, we have Theorem 34.

Theorem 34. Let be a full and keeping intersection or a partition soft set over and let be a soft approximation space. Then is a topology on .

Remark 35. Let be a full and keeping union soft set over , and let be an soft approximation space. Then, by Proposition 14(5),   is a indiscrete topology on .

The following theorem gives the structure of the first sort of topologies induced by a soft set.

Theorem 36. Let be a full and keeping intersection soft set over , let be a soft approximation space, and let be the topology induced by on . Then,

(1)

(2)

(3)if is topological, then

(4) is an interior operator of .

Proof. (1) By Proposition 13, we have

Obviously,

Let . Then, for some . By Proposition 13, . This implies . Thus,

Hence,

(2) For each , by Proposition 13,

Then, . So . Thus,

(3) Suppose that . If , by is topological, there exists such that . If , for each , , there exists such that . Then,

So, . Since is keeping union, then

This implies . Thus,  .

By (1), .

Hence,

(4) It suffices to show that

By (1), . By Proposition 13, . Thus

Conversely, for each with , we have by Proposition 13. Thus,

Hence,

Definition 37. Let be a topology on . Put , where is the set of indexes. Define a mapping by for each . Then, the soft set over is called the soft set induced by on .

Definition 38. Let be a topological space. If there exists a full and keeping intersection or a partition soft set over such that , then is called a soft approximating space.

The following proposition can easily be proved.

Proposition 39. (1) Let be a topology on , and let be the soft set induced by on . Then, is a full, keeping intersection, and keeping union soft set over .

(2) Let and be two topologies on , and let and be two soft sets induced, respectively, by and on . If , then

Theorem 40. Let be a topology on , let be the soft set induced by on , and let be the topology induced by on . Then, .

Proof. Put ; then, is a mapping, where for each . By Proposition 39, is full, keeping intersection, and keeping union.

By Theorem 36, .

Hence, .

Corollary 41. Every topological space on the initial universe is a soft approximating space.

Theorem 42. Let be a topological space. Then, there exists a full, keeping intersection, and keeping union soft set over such that

where is a soft approximation space.

Proof. Put , where is the set of indexes. Define a mapping by

By Proposition 39, is full, keeping intersection, and keeping union.

Let . For each , for some . Then, with . This implies .

Conversely, for each , there exists an open neighborhood of in such that . So, for some . This implies . Thus, .

Hence, .

Theorem 43. Let be a full and keeping intersection soft set over , let be the topology induced by on , and let be the soft set induced by on . Then,

(1)

(2)If is topological, then

Proof. (1) By Theorem 36, . Denote

Thus is a mapping given by , where for each .

Hence, .

(2) Since is topological, then by Theorem 36,  .

Hence,

4.2. The Second Sort of Topologies Induced by a Soft Set

By Proposition 19, we have Theorem 44.

Theorem 44. Let be a full soft set over , and let be a soft approximation space. Then, is a topology on .

Definition 45. Let be a topology on .

(1) is called an Alexandrov topology on , if is closed for arbitrary intersections.(2) is called an Alexandrov space, if is an Alexandrov topology on .(3) is called a pseudodiscrete topology on , if if and only if .

Obviously, every pseudodiscrete topology is an Alexandrov topology.

The following theorem gives the structure of the second sort of topologies induced by a soft set.

Theorem 46. Let be a full soft set over , and let be the topology induced by on . Then, is an Alexandrov space.

Proof. By Proposition 16, is reflexive and symmetric. Then, by Proposition 5 in [16], is a pseudodiscrete topology on .

Thus,

4.3. The Third Sort of Topologies Induced by a Soft Set

Example 47. Let be a full soft set over and let be a soft approximation space in Example 26.

Let , and . We have

Thus, is not a topology on .

By Proposition 21, we have Theorem 48.

Theorem 48. Let be a full soft set over , and let be a soft approximation space. Then, is a topology on .

4.4. The Relationships among Three Sorts of Topologies Induced by a Soft Set and Related Results

By Theorem 27, we have Theorem 49, which illustrates relationships among three sorts of topologies induced by a soft set.

Theorem 49. (1) If is a full soft set over , then

(2) If is a full and keeping intersection soft set over , then

(3) If is a partition soft set over , then

By Theorem 36, Proposition 39 and Theorem 49, we have Theorem 50.

Theorem 50. Let be a topology on , let be the soft set induced by on , and let , and be the topology induced, respectively, by on . Then

By Proposition 39 and Theorems 43 and 49, we have Theorem 51.

Theorem 51. (1) If is a full soft set over , let and be the topologies induced, respectively, by on , and let   resp., be the soft set induced by   resp., on . Then,

(2) Let be a full and keeping intersection soft set over , let , , and be the topologies induced respectively by on and let   resp., , be the soft set induced by   resp., , on . Then,

(a)

(b) If is keeping union, then

(3) Let be a partition soft set over , let , and be the topologies induced, respectively, by on and let   resp., , be the soft set induced by   resp., , on . Then,

Example 52. Let be a partition soft set over in Example 11, let be the topology induced by on , and let be the soft set induced by on . We have

Obviously,

Thus,

5. The Related Properties of Soft Rough Sets

In this section, four sorts of soft rough sets based on four pairs of soft rough approximations are investigated.

For , we denote

Lemma 53 (see [10]). Let be a soft set over and let be a soft approximation space. Then for each ,

By Corollary 31, we have the following Lemma 54.

Lemma 54. Let be a full soft set over , and let be a soft approximation space. Then,

(1)every soft -rough set is a soft -rough set.(2)every soft -rough set is a soft -rough set.

By Theorem 27, we have Lemma 55.

Lemma 55. Let be a full and keeping union soft set over , and let be a soft approximation space. If is a soft -definable set, then,

The following theorem gives the structure of soft rough sets.

Theorem 56. Let be a soft set over , and let be a soft approximation space. Denote

(1)If is full, then

(a)

(b)

(c)

(d)

(2)If is full and keeping union, then

(a); (b).

(3)If is full and keeping intersection, then

(a).

Proof. These hold by Proposition 14 and Lemmas 53, 54, and 55.

Theorem 57. Let be a soft set over and let be a soft approximation space. Then, for , one has

Proof. This is obtained from Propositions 19, 20, and 21.

By Proposition 14 and Theorem 27, we have Theorem 58.

Theorem 58. Let be a soft set over , and let be a soft approximation space. Then, for , one has the following.

(1)If be a full, then

(2)If is full and keeping union, then

(a); (b) and where .

Remark 59. Theorem 58 illustrates that soft -rough sets could provide a better approximation than soft -rough sets do and soft -rough sets could provide a better approximation than soft -rough sets do.

6. A Correspondence Relationship

In this section, we give a one-to-one correspondence relationship in order to reveal the broad application prospect of soft sets.

Definition 60 (see [17]). Let be a finite set of objects, let be a finite set of attributes, and let be a binary relation on from to . The triple is called a formal context.

Let be a formal context. For and , , which is also written as , means that the object possesses the attribute .

Denote

Definition 61. Let be a formal context. Define a mapping by

for each . Then, is called the soft set over induced by . We denote by .

Definition 62. Let be a soft set over . Define a binary relation by

for each and each . Then, is called the formal context induced by . We denote by .

Lemma 63. Let be a formal context, let be the soft set induced by , and let be the formal context induced by . Then,

Proof. Obviously, .

For each ,

For each ,

Then,

Thus, for each , . This implies .

Hence,

Lemma 64. Let be a soft set over , let be the formal context induced by , and let

be the soft set induced by . Then,

Proof. Obviously, .

Since , then

So, .

Obviously, for each .

Thus, for each , .

Hence,

Theorem 65. Let

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

Proof. Two mappings and are defined as follows:

By Lemma 63,

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

By Lemma 64,

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

Hence, and are both a one-to-one correspondence. This prove that there exists a one-to-one correspondence between and .

Remark 66. Theorem 65 illustrates that we can do formal concept analysis for soft sets or do soft analysis for formal contexts.

7. Conclusions

In this paper, we investigated soft rough approximation operators and the problems of combing soft sets with soft rough sets and topologies. Four pairs of soft rough approximation operators were considered, and their properties were given. Four sorts of soft rough sets are defined by using four pairs of soft rough approximation operators, and Pawlak’s rough set models can be viewed as a special case of soft rough sets. We researched relationships among soft sets, soft rough sets and topologies, obtained the structure of soft rough sets, and revealed that every topological space on the initial universe is a soft approximating space. We may mention that soft rough sets can be used in object evaluation and group decision making. It should be noted that the use of soft rough sets could, to some extent, automatically reduce the noise factor caused by the subjective nature of the expert's evaluation. We will investigate these problems in future papers.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11061004, 11226085), the Science Research Project of Guangxi University for Nationalities (no. 2011QD015) and the Science Foundation of Guangxi College of Finance and Economics (no. 2012ZD001).

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