Axiomatic Characterizations of IVF Rough Approximation Operators

Abstract

This paper is devoted to the study of axiomatic characterizations of IVF rough approximation operators. IVF approximation spaces are investigated. The fact that different IVF operators satisfy some axioms to guarantee the existence of different types of IVF relations which produce the same operators is proved and then IVF rough approximation operators are characterized by axioms.

1. Introduction

Rough set theory was proposed by Pawlak [1, 2] as a mathematical tool for data reasoning. It may be seen as an extension of classical set theory; it has been proved to be an effective approach to deal with intelligent systems characterized by insufficient and incomplete information 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.

There are mainly two approaches to the development of rough set theory. One is the constructive approach in which rough approximation operators are constructed by means of relations, partitions, coverings, neighborhood systems, and so on. The constructive approach is suitable for practical applications of rough sets. The other one is the axiomatic approach. In this approach, axioms sets are used to characterize rough approximation operators that guarantee the existence of certain types of relations which produce the same operators. This approach is appropriate for studying algebra structures of rough sets. Under this point of view, rough set theory may be interpreted as an extension of set theory with two additional unary operators.

As a generalization of Zadeh’s fuzzy set, interval-valued fuzzy (IVF, for short) sets were introduced by Gorzalczany [3] and Türksen [4], and they were applied to the fields of approximate inference, signal transmission, and controller.

By integrating Pawlak rough set theory with IVF set theory, Sun et al. [5] introduced IVF rough sets based on an IVF approximation space, defined IVF information systems, and discussed their attribute reduction. Gong et al. [6] studied the knowledge discovery in IVF information systems. Zhang et al. [7] discussed -IVF rough sets based on an IVF approximation space on two universes of discourse.

The purpose of this paper is to investigate IVF rough sets by using axiomatic approaches and to give axiomatic characterizations of IVF rough approximation operators.

2. Preliminaries

Throughout this paper, “interval-valued fuzzy” denotes briefly by “IVF.” denotes a nonempty finite set called the universe of discourse. denotes and denotes . denotes the family of all subsets of . denotes for each .

For any , we define the following:

Obviously, for each .

Definition 1 (see [3, 4]). For any , we define the following: where .

Definition 2 (see [3, 4]). An IVF set in is defined by a mapping . Denote that Then (resp., ) is called the lower (resp., upper) degree to 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 ([3, 4]): for any and , consider the following.(1) for each .(2) for each .(3) for each .(4) for each .(5) for each .
Moreover, where .(6) for each .

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

Remark 4. Consider  

3. IVF Rough Approximation Operators and IVF Rough Sets

Recall that is called an IVF relation on if .

Definition 5 (see [5]). Let be an IVF relation on . Then is called the following.(1)Serial, if for each .(2)Reflexive, if for each .(3)Symmetric, if for any .(4)Transitive, if for any .

Let be an IVF relation on . is called preorder if is reflexive and transitive ([9]).

Definition 6 (see [5]). Let be an IVF relation on . The pair is called an IVF approximation space. The IVF lower and the IVF upper approximation of with respect to , denoted by and , are, respectively, defined as follows:
The pair is called the IVF rough set of with respect to .

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

Remark 7. Let be an IVF approximation space. Then we get the following.(1)For any , (2)For each ,  .

Proposition 8 (see [5]). Let be an IVF approximation space. Then for each ,

Theorem 9. Let be an IVF approximation space. Then for any , , and , one has the following.(1), .(2), .(3), .(4), .(5), .

Proof. (1) and (2) are obvious.
(3) ,
Then .
Pick . Since , we have (4)   ,
Then .
By (3) and , we have
Thus .
(5) ,
Then .
Similarly,

Theorem 10. Let be an IVF approximation space. Then one has the following.(1) is serial       .     .     .(2) is reflexive .       .(3) is symmetric , .           , . (4) is transitive , .              , .

Proof. (1) By Theorem 9(3), and are equivalent and and are equivalent. We only need to prove that the serialisation of is equivalent to or .
For any and , we have Assume that is serial. , . By ,
Thus .
Conversely, assume for each . ,
By , .
Put . Then .
This implies that .
Thus is serial.
Assume that is serial. , . By ,
Thus .
Conversely, assume that . By , , .
Thus is serial.
By Theorem 9, (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. , pick . By (IUR), . By Remark 7,
This implies that .
Thus is reflexive.
(3) By Theorem 9, (ILS) and (IUS) are equivalent. We only need to prove that the symmetry of is equivalent to (IUS).
, by Remark 7,
Thus the symmetry of is equivalent to (IUS).
(4) By Theorem 9, and are equivalent. We only need to prove that the transitivity of is equivalent to .
Assume that is transitive. ,
Denote that . Then , ,
Thus .
Conversely, assume that (IUT) holds. , by (IUT),
By Remark 7,
Thus is transitive.

Corollary 11. Let be an IVF approximation space. If is preorder, then

Proof. This holds by Theorem 10.

4. Axiomatic Characterizations of IVF Rough Approximation Operators

In this section we show that rough approximation operators in IVF environment can be characterized by axioms; that is, we can find axiom sets of different IVF operators that guarantee the existence of different types of IVF relations which produce the same operators.

For any , , we define the following:

Denote that . We have

Let be a mapping. Denote that

Definition 12. Let be two IVF operators. If then and are called two dual IVF operators.

Remark 13. are two dual IVF operators if and only if for each .

Theorem 14. Let be two dual IVF operators. Then there exists an IVF relation on such that and if and only if satisfies the axioms and or, equivalently, satisfies the axioms and :, , ;, ; ;, .

Proof. Note that are two dual IVF operators. Then and are equivalent to and . We only need to prove that and if and only if satisfies the axioms and .
Necessity. This holds by Theorem 9.
Sufficiency. Assume that the operator satisfies the axioms and . Define the following:
Let . By Remark 4, ,
Then . By Theorem 9(3),
Thus , .

Corollary 15. Let be two dual IVF operators. If satisfies the axioms and or, equivalently, satisfies the axioms and , then satisfies the axiom and satisfies the axiom :, ;,  .

Proof. This holds by Theorem 9   and Theorem 14.

Theorem 16. Let be two dual IVF operators. Then there exists an IVF relation on such that and , if and only if satisfies the axiom or, equivalently, satisfies the axiom , where(AL3); (AU3),   .

Proof. Note that are two dual IVF operators. Then is equivalent to . We only need to prove that and if and only if satisfies the axiom .
Necessity. Let , ,  . By Theorem 9,
Sufficiency. Assume that the operator satisfies the axiom . Define the following:
Let . Pick and . By Remark 4, then for each . Then .
Thus . By Theorem 9, .
Hence .

Corollary 17. Let be two dual IVF operators. If satisfies the axiom or, equivalently, satisfies the axiom , then satisfies the axiom and satisfies the axiom : ; .

Proof. This holds by Theorems 9 and 16.

The following results illustrate that IVF rough approximation operators generated by special IVF relations can be characterized by some axioms.

Theorem 18. Let be two dual IVF operators. Then there exists a serial IVF relation on such that and if and only if satisfies the axioms , , and or, equivalently, satisfies the axioms , , and : , .

Proof. This holds by Theorems 10 and 14.

Corollary 19. Let be dual operators. If satisfies the axioms , , and or, equivalently, satisfies the axioms , , and , then satisfies the axiom and satisfies the axiom : ; .

Proof. This holds by Theorems 10 and 18.

Theorem 20. Let be two dual IVF operators. Then there exists a reflexive IVF relation on such that and if and only if satisfies the axioms , , and or, equivalently, satisfies the axioms , , and : ; .

Proof. This holds by Theorems 10 and 14.

Theorem 21. Let be two dual IVF operators. Then there exists a reflexive IVF relation on such that and if and only if satisfies the axiom or, equivalently, satisfies the axiom :,  ; ,  .

Proof. Note that are two dual IVF operators. Then is equivalent to . We only need to prove that there exists a reflexive IVF relation on such that and if and only if satisfies the axiom .
Necessity. By Theorems 9 and 10, Sufficiency. Let . Pick , . By Remark 4,
Then for any and with ,
So By Theorem 16, there exists an IVF relation on such that and . Then . By Theorem 10(2), is reflexive.

Theorem 22. Let be two dual IVF operators. Then there exists a symmetric IVF relation on such that and if and only if satisfies the axioms , , and or, equivalently, satisfies the axioms , , and : ; .

Proof. This holds by Theorems 10(3) and 14.

Theorem 23. Let be two dual IVF operators. Then there exists a symmetric IVF relation on such that and if and only if satisfies the axiom or, equivalently, satisfies the axiom :, ,

Proof. Note that   are two dual IVF operators. Then is equivalent to . We only need to prove that there exists a symmetric IVF relation on such that and if and only if satisfies the axiom .
Necessity. This holds by Theorems 9 and 10(3).
Sufficiency. Assume that satisfies the axiom . Define the following:
,
Then is symmetric.
Let .   ,
This implies that .
Then . By Theorem 9(3), .
Thus

Theorem 24. Let be two dual IVF operators. Then there exists a transitive IVF relation on such that and if and only if satisfies the axioms , , and or, equivalently, satisfies the axioms , , and : ; .

Proof. This holds by Theorems 10(4) and 14.

Theorem 25. Let be two dual IVF operators. Then there exists a transitive IVF relation on such that and if and only if satisfies the axiom or, equivalently, satisfies the axiom :, ,

Proof. Note that are two dual IVF operators. Then is equivalent to . We only need to prove that there exists a transitive IVF relation on such that and if and only if satisfies the axiom .
Necessity. By Theorems 9 and 10(4),
Sufficiency. Let . Pick
Then
For any and with ,
So
By Theorem 16, there exists an IVF relation on such that and . So
By Theorem 10(4), is transitive.

Theorem 26. Let be two dual IVF operators. Then there exists a preorder IVF relation on such that and if and only if satisfies the axiom or, equivalently, satisfies the axiom :(ALO), (AUO)

Proof. Note that are two dual IVF operators. Then is equivalent to . We only need to prove that there exists a preorder IVF relation on such that and if and only if satisfies the axiom .
Necessity. By Theorems 9 and 10, we have
Sufficiency. Let . Pick
By Remark 4,
For any and with ,
Pick
Then
So
Thus
By Theorem 16, there exists an IVF relation on such that and . Then
By Theorem 10, is preorder.

5. Conclusions

In this paper, we have obtained axiomatic characterizations of IVF rough approximation operators; we can find axiom sets of different IVF operators that guarantee the existence of different types of IVF relations which produce the same operators. We can consider some engineering applications of our proposed notions and theories. In the future, we will study these problems.

Conflict of Interests

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

Acknowledgment

This work is supported by the Natural Science Foundation of Guangxi (no. 2012GXNSFDA276040).