On j -near concepts in rough sets with some applications

1 Introduction

Rough set theory, a mathematical tool dealing with inexact or uncertain knowledge in information systems, has originally described the in discernibility of elements by equivalence relations. In Pawlak’s original rough set theory [17, 18], an equivalence relation is a core concept, which seems to be a very stringent condition that limits the application domain of the rough set theory. To solve this problem, several authors have generalized the notion of approximation operators by using arbitrary binary relations [6, 19]. The rough set approach to an approximation of sets leads to useful forms of granular computing that are part of computational intelligence. The basic idea underlying the rough set approach and its topological generalizations to information granulation is to discover to what extent a given set of objects (which can be pixels of an image) approximates another set of objects of interest. Objects of the definite universe are compared through considering their descriptions.

Data granulation is considered a good tool for decision making in various types of real life applications. The basic ideas of data granulation have appeared in many fields, such as interval analysis, quantization, rough set theory, Dempster-Shafer theory of belief functions, divide and conquer, cluster analysis, machine learning, databases, information retrieval, and many others. Some new topological tools for data granulation using rough set approximations are initiated. Moreover, some topological measures of data granulation in topological information systems are defined. Topological generalizations using j-near open sets and their applications of information granulation are further developed. The topology induced by binary relations on the universes of information systems is used to generalize the basic rough set concepts. The suggested topological operations and structures open up the way for applying more of topological facts and methods in the process of granular computing. So, we think that the topological structure will be an important base for knowledge extraction and processing.

In 2014, Abd El-Monsef et al. [8] introduced the new concept “the j-neighborhood space” (in briefly, j- NS ) which represents a generalized type of neighborhood spaces. In fact, they introduced a framework to generalize the classical rough set theory introduced by Pawlak [31] and some of the other generalizations using different general topologies induced from binary relations. Accordingly, different rough set approximations are introduced to satisfy all properties of original rough set theory without any conditions or restrictions. Such techniques open the way for more topological applications in rough context and help in formalizing many applications from real-life data.

The standard rough set theory starts from an equivalence relation. The theory is a new mathematical tool to deal with vagueness and imperfect knowledge. It deals with the vagueness (ambiguity) of the set by using the concept of the lower and upper approximations [17, 18]. The set has the same lower and upper approximations, called crisp (exact) set, otherwise known as a rough (inexact) set. Therefore, the boundary region is defined as the difference between the upper and lower approximations, and then the accuracy of the set or its ambiguity depending on the boundary region is either empty or not respectively. Non-empty boundary region of a set means that our knowledge about the set is not sufficient to define the set precisely. The main aim of the rough set is to reduce the boundary region by increasing the lower approximation and decreasing the upper approximation. Near (or nearly) open concept had been introduced a new generalized sort of open sets to topological spaces (See: [1, 9–11], and its references). Using these concepts, we introduce the new notions “j-near approximations” as easy tools for classifying the sets and for helping in the measurement of exactness and roughness of sets. On the other hand, El-Bably [9], had applied these concepts in the case of after-set and fore-set only. In this paper, we generalize these concepts, and hence we introduce forty approaches for the approximations operators. We also use the notion of near open sets to define the lower and upper approximations (so called j-near approximations) of any set with respect to any relation. The present method reduces the boundary region by increasing the lower approximation and decreasing the upper approximation compared with other methods. Moreover, these j-near approximations have more accurate measures than that of Pawlak, Yao and other measures in [14] (See Examples 4.1, 4.2). These lower and upper approximations satisfy most of the properties of Pawlak’s spaces.

In addition, the relationships between these approximations are investigated. Hence, the best approaches given by the accuracy of approximations are superimposed. These methods are more accurate than other methods and hence it is very significant in rough set context of removing the vagueness (uncertainty) which is very useful in decision-making. Moreover, these techniques give a new connection between two important theories namely “rough set theory and the general topology”. Finally, we introduce simple practical examples illustrating the importance of the suggested techniques in rough set context.

2 j-Neighborhood space

In this section, we outline the main ideas of the j-neighborhood space cited in [8].

Definition 2.1. [8] Let R be an arbitrary binary relation on a non-empty finite set U. The j-neighborhood of x ∈ U (N _j  (x)), j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 }, can defined as:

r-neighborhood: N _r  (x) ={ y ∈ U | xRy }.

Definition 2.2. [8] Let R be an arbitrary binary relation on a non-empty finite set U and ξ _j  : U → ℘ (U) be a mapping which assigns for each x in U its j-neighborhood in ℘ (U). The triple (U,  R,  ξ _j ) is called a j-neighborhood space (in briefly, j- NS ).

The following theorem is interesting, and by using it, we can generate eight different topologies.

Theorem 2.1. [8] If (U, R, ξ _j ) is a j- NS . Then, for each j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , the collection τ _j  ={ A ⊆ U| ∀ p ∈ A, N _j  (p) ⊆ A } is a topology on U.

Definition 2.3. [8] Let (U, R, ξ _j ) be a j- NS . Then the subset A ⊆ U is called “j-open set” if A ∈ τ _j , the complement of j-open set is called “j-closed set”. The family Γ _j of all j-closed sets of a j-neighborhood space is defined by Γ j = { F ⊆ U | F c ∈ τ j } .

Definition 2.4. [8] Let (U, R, ξ _j ) be j- NS and A ⊆ U. Then the “j-lower” and “j-upper” approximations of A are defined respectively by R _ j ( A ) = ∪ { G ∈ τ j : G ⊆ A } = i nt j ( A ) and R ¯ j ( A ) = ∩ { H ∈ Γ j : A ⊆ H } = c l j ( A ) , where, int _j  (A) (resp. cl _j  (A)) represents j-interior (resp. j-closure) of A.

Definition 2.5. [8] Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Thus, ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , the subset A is called “j-exact set” if R - j ( A ) = R ¯ j ( A ) = A. Otherwise, A is called “j-rough set”.

Definition 2.6. [8] Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then, the“j-boundary”, “j-positive” and “j-negative” regions of A are defined respectively by B j ( A ) = R ¯ j ( A ) - R _ j ( A ) , POS j ( A ) = R _ j ( A ) and NEG j ( A ) = U - R ¯ j ( A ) .

Definition 2.7. [8] Let (U, R, ξ _j ) be a j- NS . Then the “j-accuracy” of approximations of A ⊆ U defined by δ j ( A ) = | R - j ( A ) | | R ¯ j ( A ) | , where | R ¯ j ( A ) | ≠ 0 .

Obviously, 0 ≤ δ _j  (A) ≤ 1 and if δ _j  (A) = 1, then A is j-exact set. Otherwise, it is j-rough.

3 j-near approximations in the j- NS

Most of real life situations need some sort of approximation to fit mathematical models. The beauty of using the topology in approximation is achieved via obtaining an approximation for qualitative concepts (i.e. subsets) without coding or using assumption. General topology is the appropriated mathematical model for every collection connected by relations. The main goal of this section is to introduce one of the most important topological concepts named “j-near open” sets in rough context. By using these concepts, we define new forty approximations as mathematical tools modifying and generalizing the j-approximations in the j- NS . Properties of the suggested approximations are investigated, and their connections are also examined. These approximations reduce the boundary region by increasing the lower approximation and decreasing the upper approximation compared to other methods. Moreover, these j-near approximations have more accurate measures than that of Pawlak [17], Yao [14] and other measures.

Definition 3.1. Let (U, R, ξ _j ) be a j- NS . Then, for each j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 }, the subset A ⊆ U is called:

j-Regular open (briefly R j *-open), if A = i nt j ( c l j ( A ) ) .

Remark 3.1.

The above sets are called “j-near open” sets and the families of j-near open sets of U denoted by K j O ( U ) , ∀ K ∈ { R * , P , S , γ , α , β } .

Remark 3.2. According to the results in [1, 9], the implications between the topologies τ _j and the above families of j-near open (resp. j-near closed) sets are given in Diagram 3.1.

The following example shows the above relations. In addition, it confirms that the reverse implications are not true in general.

Example 3.1. Let (U, R, ξ _j ) be a j- NS , where U ={ a, b, c, d } and R = {(a, a) ,   (a, b) ,   (b, a)} , (b, b) , (c, a) ,   (c, b) ,   (c, c) ,   (c, d) ,   (d, d)}. Then weget: τ _r  ={ U, ∅ , { d } , { a, b } , { a, b, d }} and Γ _r  ={ U, ∅ , { c } , { c, d } , { a, b, c }}. We will compute the j-near open sets in the case of j = r (and the other cases similarly) as follows: R r * O ( U ) = { U , ∅ , { d } , { a , b } } , S r O ( U ) = { U , ∅ , { d } , { a , b } , { c , d } , { a , b , c } , { a , b , d } } P r O ( U ) = { U , ∅ , { a } , { b } , { d } , { a , b } , { a , d } , { b , d } , { a , b , d } , { a , c , d } , { b , c , d } } , γ r O ( U ) = { U , ∅ , { a } , { b } , { d } , { a , b } , { a , d } , { b , d } , { c , d } , { a , b , c } , { a , b , d } , { a , c , d } , { b , c , d } } α r O ( U ) = { U , ∅ , { d } , { a , b } , { a , b , d } } and β r O ( U ) = { U , ∅ , { a } , { b } , { d } , { a , b } , { a , c } , { a , d } , { b , c } , { b , d } , { c , d } , { a , b , c } , { a , b , d } , { a , c , d } , { b , c , d } } .

Remark 3.3. According to the above example, we notice that P _j and S _j are not comparable for each j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } .

Remark 3.4. Let (U, R, ξ _j ) be a j- NS . Then, for each j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and K ∈ { R * , P , S , γ , α , β }, the following statements are not true in general: ( i ) K u O ( U ) ⊆ K r O ( U ) ⊆ K i O ( U ) . ( i i ) K u O ( U ) ⊆ K l O ( U ) ⊆ K i O ( U ) . ( i i i ) K 〈 u 〉 O ( U ) ⊆ K 〈 r 〉 O ( U ) ⊆ K 〈 i 〉 O ( U ) . ( i v ) K 〈 u 〉 O ( U ) ⊆ K 〈 l 〉 O ( U ) ⊆ K 〈 i 〉 O ( U ) . ( v ) K r O ( U ) isthedualof K l O ( U ) .

This means that, the relationships among j-near open sets for different types of topologies τ _j are not comparable as illustrated in the following example:

Example 3.2. Let (U, R, ξ _j ) be a j- NS , whereU ={ a, b, c, d } and R = {(a, a) ,   (a, b) ,   (b, a) , (b, b) ,   (c, a) ,   (c, b) ,   (c, c) ,   (c, d) ,   (d, d)}. Thus, we get τ r = { U , ∅ , { a } , { d } , { a , b } , { a , d } , { a , b , d } , { a , c , d } } , Γ r = { U , ∅ , { b } , { c } , { b , c } , { c , d } , { a , b , c } , { b , c , d } } , τ 〈 ℓ 〉 = { U , ∅ , { b } , { d } , { b , d } , { b , c , d } } , Γ 〈 ℓ 〉 = { U , ∅ , { a } , { a , c } , { a , b , c } , { a , c , d } } τ 〈 i 〉 = { U , ∅ , { a } , { b } , { d } , { a , b } , { a , d } , { b , d } , { c , d } , { a , b , d } , { a , c , d } , { b , c , d } , Γ 〈 i 〉 = { U , ∅ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { c , d } , { a , b , c } , { a , c , d } , { b , c , d } , τ u = { U , ∅ , { d } } and Γ 〈 u 〉 = { U , ∅ , { a , b , c } } .

We will compute the j-near open sets for each j ∈ { 〈 r 〉 , 〈 ℓ 〉 , 〈 u 〉 , 〈 i 〉 } as follows: P 〈 r 〉 O ( U ) = { U , ∅ , { a } , { d } , { a , b } , { a , d } , { a , b , d } , { a , c , d } } P 〈 ℓ 〉 O ( U ) = { U , ∅ , { b } , { d } , { b , d } , { c , d } , { a , b , d } , { b , c , d } } and P 〈 u 〉 O ( U ) = { U , ∅ , { b } , { d } , { a , d } , { c , d } , { a , b , d } , { a , c , d } , { b , c , d } } .

It is clear that P 〈 u 〉 O ( U ) ⊈ P 〈 r 〉 O ( U ) and P 〈 u 〉 O ( U ) ⊈ P 〈 ℓ 〉 O ( U ). Also, S 〈 i 〉 O ( U ) = { U , ∅ , { a } , { b } , { d } , { b , d } , { a , b } , { a , d } , { c , d } , { a , b , d } , { a , c , d } , { b , c , d } } and S 〈 r 〉 O ( U ) = { U , ∅ , { a } , { d } , { a , b } , { a , c } , { a , d } , { c , d } , { a , b , c } , { a , b , d } , { a , c , d } } .

Clearly, S 〈 r 〉 O ( U ) ⊈ S i O ( U ).

Also, one can give another example to show that S 〈 l 〉 O ( U ) ⊈ S i O ( U ).

By using the j-near open sets, we introduce new methods for approximating rough sets as illustrated in the following definitions:

Definition 3.2. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then, ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and k∈ { r^*, p, s, γ, α, β }, the j-near lower and j-near upper approximations of A are defined respectively by R - j k ( A ) = ∪ { G ∈ K j O ( U ) : G ⊆ A } = j - near interior of A , and R ¯ j k ( A ) = ∩ { H ∈ K j C ( U ) : A ⊆ H } = j - near closure of A .

Definition 3.3. Let (U, R, ξ _j ) be a j- NS . Then, ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and k∈ { r^*, p, s, γ, α, β }, the subset A ⊆ U is called j-near definable (j-near exact) set if R - j k ( A ) = R ¯ j k ( A ) = A .

Otherwise, A is called j-near rough set.

Definition 3.4. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and k∈ { r^*, p, s, γ, α, β }, the j-near boundary, j-near positive and j-near negative regions of A are defined respectively by B j k ( A ) = R ¯ j k ( A ) - R - j k ( A ) , POS j k ( A ) = R - j k ( A ) and NEG j k ( A ) = U - R ¯ j k ( A ) .

Definition 3.5. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then, ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and k∈ { r^*, p, s, γ, α, β }, the j-near accuracy of the approximations of A ⊆ U is defined by δ j k ( A ) = | R - j k ( A ) | | R ¯ j k ( A ) | , where | R ¯ j k ( A ) | ≠ 0 .

Obviously, 0 ≤ δ j k ( A ) ≤ 1 and if δ j k ( A ) = 1, then A is j-near exact set. Otherwise, it is j-near rough.

Remark 3.5. In the j- NS , we can compute thej-near approximations of any subset directly, without computing j-near sets, by using the j-lower and j-upper approximations as illustrated in the following proposition:

Proposition 3.1. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then :

R - j p ( A ) = A ∩ R - j ( R ¯ j ( A ) ) .

Proof. By using the properties of j-near closure and j-near interior operators, the proof is straightforward by using the results in [9, 11]. ■

The following proposition gives the fundamental properties of j-near approximations.

Proposition 3.2. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and k∈ { r^*, p, s, γ, α, β }:

R - j k ( A ) ⊆ A ⊆ R ¯ j k ( A ) .

Proof. By using the properties of j-near interior and j-near closure, the proof is obvious. ■

Remark 3.6.

As shown in [11] that α _j O (U) represent different topologies on U. Accordingly, in the case of k = α, the properties (5) and (8) can be replaced by the following properties, respectively: ( i ) R _ j k ( A ∩ B ) = R _ j k ( A ) ∩ R - j k ( B ) . ( ii ) R ¯ j k ( A ∪ B ) = R ¯ j k ( A ) ∪ R ¯ j k ( B ) .

The main goal of the following results is to show the importance of using j-near approximations in rough context.

Theorem 3.1. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 }, k ∈ { p, s, γ, α, β } andk ≠ r^*: R _ j ( A ) ⊆ R _ j k ( A ) ⊆ A ⊆ R ¯ j k ( A ) ⊆ R ¯ j ( A ) .

Proof. Since the families of j-near open sets k _j O (U) (resp. j-near closed sets k _j C (U)) are larger than the topologies τ _j (resp. the families of j-closed sets Γ _j ). Then we have R _ j ( A ) = ∪ { G ∈ τ j : G ⊆ A } ⊆ ∪ { G ∈ k j O ( U ) : G ⊆ A } = R _ j k ( A ) .

In the same way, we can prove R ¯ j k ( A ) ⊆ R ¯ j ( A ). ■

Corollary 3.1. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , k ∈ { p, s, γ, α, β } and k ≠ r^* :

B j k ( A ) ⊆ B j ( A ).

Corollary 3.2. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , k ∈ { p, s, γ, α, β } and k ≠ r^*, the set A is j-exact set in τ _j if it is j-near exact.

Proof. Let A is a j-exact in τ _j . Thus B j ( A ) = ∅ , foreach j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } and thisimplies, by Corollary 3.1, B j k ( A ) = ∅ , ∀ k ∈ { p , s , γ , α , β } and k ≠ r^*. Hence A is j-near exact. ■

Remark 3.7. From the above results, we notice that:

Applying the j-near concepts in rough sets theory is very useful because the use of j-near structures can help further developments in the theories and applications of rough sets. Moreover, the j-near approximations can help in the discovery of hidden information in data that were collected from real-life applications as the boundary regions are decreased or deleted by increasing the lower approximation and decreasing the upper approximation, and this is very important in decision-making. Furthermore, by using j-near structures, we have introduced different and several accurate topological measures for rough sets.

The converse of the above results is not true in general as illustrated in the following example:

Example 3.3. In Example 3.1, by using Definition 3.3, Tables 3.1 and 3.2 give comparisons between the j-approximations (resp. the j-accuracy of approximations) and the k _j -approximations (resp. the k _j -accuracy of approximations) of the all subsets of U, where j = r, k∈ { p, s, γ, α, β }:

From the two tables, Table 3.1, we notice that:

There are different methods to approximate the sets. The best of these methods is given by using β _r in constructing the approximations of sets, since the boundary regions in this case are decreased (or cancelled) by increasing the lower approximation and decreasing the upper approximation that is B r β ( A ) ⊆ B j ( A ). Accordingly, this will play an important role in removing the vagueness (uncertainty) of rough sets. For example, the shaded sets in the above tables are exact in β _r but there are rough in the othertypes.

From Table 3.2, we notice that:

Using β _j in constructing the approximations of sets is more accurate than other types, because for any subset A ⊆ U, δ r ( A ) ≤ δ r β ( A ) and δ r k ( A ) ≤ δ r β ( A ) , ∀ k ∈ { p , s , γ , α }. Thus, these approaches will help in extracting and in discovering the hidden information in the data that collected from real-life applications. Accordingly, these approaches will be very useful in decision-making and in the reduction of attributes in rough set theory.

For example, let A ={ a, c } thus, δ r ( A ) = 0 , δ r p ( A ) = 1 / 2 , δ r s ( A ) = 0 , δ r γ ( A ) = 1 / 2 and δ r α ( A ) = 0, but δ _i  (A) = 1.

The following results are given to illustrate the relationships between the j-near approximations, j-near boundary and j-near accuracy respectively. Moreover, they give the best of these approaches.

Proposition 3.3. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , the following statements are true in general.

R _ j α ( A ) ⊆ R _ j p ( A ) ⊆ R _ j γ ( A ) ⊆ R _ j β ( A ) .

Proof. By using the implications between the different families of j-near open sets k _j O (U) for each k∈ { p, s, α, γ, β }, the proof is obvious. ■

Corollary 3.3. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , the following statements are true in general.

δ j α ( A ) ≤ δ j p ( A ) ≤ δ j γ ( A ) ≤ δ j β ( A ) .

Corollary 3.4. Let (U, R, ξ _j ) be a j- NS , and A ⊆ U. Then ∀ j ∈ { r , ℓ , 〈 r 〉 , 〈 ℓ 〉 , u , i , 〈 u 〉 , 〈 i 〉 } , the following statements are true in general.

A is j-exact ⇒A is α _j -exact ⇒A is P _j -exact ⇒A is γ _j -exact ⇒A is β _j -exact.

By using Example 3.3, we can show that the converse of the above implications is not true ingeneral.

Remark 3.8. From the above results, we notice that:

There are different methods to approximate the sets. The best of these methods is there are given by using β _j in constructing the approximations of sets, since the boundary regions in this case are decreased (or cancelled) by increasing the lower approximation and decreasing the upper approximation. Moreover, the β _j -accuracy is more accurate than the othertypes.

4 Practical examples

The main goal of this section is to introduce two simple practical examples in order to illustrate the importance of applying near concept in rough context. In the first example, we use an equivalence relation generated from an information system and hence we compare between our approaches and Pawlak approach. In the second example, we use a reflexive relation generated from a multi-valued information system. However, Pawlak approach does notfit in this type and hence, we can use our approaches and compare them with some other generalizations using general binary relation.

Example 4.1. Consider the following information system as in Table 4.1. Suppose we are given data about 6 students:

From Table 4.1, we have

The set of universe: U ={ 1, 2, 3, 4, 5, 6 },

The set of attributes: AT = {Analysis, Algebra,  Statistics} = C ∪ { Decision } = D,

The sets of values:

V_Analysis ={ Bad, Good }, V_Algebra ={ Bad, Good },

V_Statistics ={ Bad, Medium, Good } and V_Decision ={ Accept, Reject }.

However, we take the set of condition attributes C ={ Analysis, Algebra, Statistics }.

Thus, we get U/C ={{ 1 } , { 2, 5 } , { 3 } , { 4 } , { 6 }} and the set of all r-pre open set is P r ( U ) = ℘ ( U ) ( set of all subsets in U ) .

For any concept A ⊆ U (collection of students), this concept is determined by the j-near lower R _ j k ( A ) and the j-near upper R ¯ j k ( A ) approximations which define its boundary. The accuracy increases by the decreases of the boundary region. Clearly, the accuracy measure of using the suggested class of j-near open sets in general is greater than the accuracy measure of using any other approaches.

For example, If we take the subset A = {1, 2, 3, 6} with decision “Accept” and B = {4,  5} with decision “reject”. Then we get the following table:

In the following example, we apply our approaches on a multi-valued information system (MVIS). This type of information system is a generalization of an information system that uses an arbitrary binary relation and thus, Pawlak approach does not fit in this type. Therefore, we will compare the accuracy of our approaches and Yao approaches [14].

Example 4.2. Consider the following multi-valued information system (MVIS) as in Table 4.3. Suppose we are given data about 5 persons, as shown below:

Where,

R₁ = Languages = { English, German, Arabic } ,

R₂ = Sports ={ Handball, Basketball, Tennis } and

R₃ = Skills = { Swimming, Running, Fishing } .

The binary relations are xR _n y such that v n ( x ) ⊆ v n ( y ) , ∀ n = 1 , 2 , 3 .

We will use the case of j = r and k = p as follows: aR₁ ={ a, b }, bR₁ ={ b }, cR₁ ={ b, c, d }, dR₁ ={ d }, eR₁ ={ d, e } and aR₂ ={ a, b, c }, bR₂ = {a, b, c}, cR₂ ={ c }, dR₂ ={ c, d }, eR₂ ={ e } and aR₃ =

In order to represent the set of all condition attributes, we generate the following relation from all above relations as follows: xR = ⋂ n = 1 3 xR n.

Thus we get

aR = { a, b } , bR = { b } , cR = { c, d } , dR = { d } and eR = { e } .

Ra ={ a } , Rb = { a, b } , Rc = { c } , Rd = { c, d } and Re = { e }

N _i  (a) = { a } , N _i  (b) = { b } , N _i  (c) = { c } ,

N _i  (d) = { d }  and N _i  (e) = { e } .

Thus, we get the topology τ _i  = ℘ (U).

The set of all i-closed sets of U is Γ _i  = ℘ (U).

Suppose that the subsets X ={ a, c, d } (Decision: Accept) and Y = { b, e } (Decision : Reject).

Then we get the Table 4.4

From the above examples, we notice that:

The class of j-near open sets used in our approaches is the largest granulation based on j-closure and j-interior operator in topological spaces τ _j that it constructed from binary relations. This made the accuracy measures higher than the use of any other type. Hence, these techniques are very useful in rough set context.

5 Conclusion and future work

In this paper, we introduced the new notions “j-near” concepts in rough set theory. In fact, we introduced j-near rough approximation operators on the j- NS as easy tools to approximate the sets and removing the vagueness of rough sets. These operators are more accurate than other approaches. Consequently, they are very significant in decision-making. In addition, we indicated some topological tools of data granulation by using new topological tools for rough set approximations. Moreover, we introduced using general binary relations as a refinement data granulation instead of the classical equivalence relations. By using the suggested structures in this paper, we have introduced different and several accurate topological measures for rough sets.

We conclude that the intermingling of topology and j-near concepts in the construction of some approximation space concepts will help in getting results with abundant logical statements, discovering hidden relationships among data and, moreover, probably helps in producing accurate programs.

The following points will be studied in the future:

The division of the boundary region can be applied to enlarge the range of rough membership. Consequently, it adds new notions in rough membership relations, rough membership functions, rough equality, rough inclusion, and rough power set.