A comparison of two types of rough approximations based on N j -neighborhoods

Abstract

In 1982, Pawlak proposed the concept of rough sets as a novel mathematical tool to address the issues of vagueness and uncertain knowledge. Topological concepts and results are close to the concepts and results in rough set theory; therefore, some researchers have investigated topological aspects and their applications in rough set theory. In this discussion, we study further properties of N_j-neighborhoods; especially, those are related to a topological space. Then, we define new kinds of approximation spaces and establish main properties. Finally, we make some comparisons of the approximations and accuracy measures introduced herein and their counterparts induced from interior and closure topological operators and E-neighborhoods.

Keywords

lower and upper approximations accuracy measure topological space

1 Introduction

There are several approaches to manipulate and understand imperfect knowledge, among which is the rough set theory introduced by Pawlak [18, 19] as a non-statistical approach in data analysis. Methodology of rough set theory is concerned with the classifications of objects using equivalence relations into three main areas. Since its emergence until the present, it has been successfully applied in several fields such as computer network [15], image processing [21], data mining [23], and medical science [12].

To extend the applications of rough set theory, many authors replaced a strict condition of an equivalence relation by tolerance relation [27], similarity relation [3, 4], dominance relation [42], and arbitrary binary relation [44]. The essential aim of these generalizations is to delete some objects from the negative region of decision categories and/or add new objects to the positive region.

The classical lower and upper approximations are the central idea in the rough set theory and they have been defined using different kinds of neighborhoods such as right and left neighborhoods [31, 32], minimal right neighborhoods [6] and minimal left neighborhoods [7]. Later on, Abd El-Monsef et al. [1] established new types of neighborhoods by taking the union and intersection of some previous types of neighborhoods. Abu-Donia [5] applied the right neighborhoods to study three novel types of approximations under a finite number of binary relations. In [28], the dual idea of neighborhood systems was presented under the name of remote neighborhood systems. Atef et al. [11] presented new kinds of neighborhoods, namely j-adhesion neighborhoods. Recently, Al-shami et al. [9] have defined E-neighborhood systems and explored new rough set models, and Al-shami [8] has introduced C-neighborhood systems and applied to protect from infection with the new corona-virus (COVID-19). New types of rough set models were investigated in [33 –41].

An interesting research theme in rough set theory is to study rough set theory via topology. Skowron [26] and Wiweger [29] separately investigated the role of topological aspects in rough sets. Lashin et al. [16] introduced a topology using binary relations and applied to generalize the basic rough set concepts. Salama [22] defined the concept of higher order sets as a new class of new types of nearly open and closed sets. He initiated this class by much iteration of topological closure and interior operations for a given set. El-Sharkasy [13] investigated separation axioms in the minimal structure approximation space. Singh and Tiwari [25] mentioned the important studies conducted in the interdependencies of classical rough set theory and topology during the last twenty years. Recently, Salama et al. [24] have investigated topological methods for rough continuous functions and provided some practical applications.

The main two goals of this manuscript are, first, to discuss further properties of N_j-neighborhoods from a topological view, and second, to exploit them to generate different types of approximations spaces that give higher accuracy measures than their counterparts induced from topology

After this introduction, the article is organized as follows. In Section 2, we recall some definitions and properties of rough sets and topological spaces that help the reader to well understand this article. In Section 3, we scrutinize more properties of N_j-neighborhood spaces; especially, those are related to separation axioms and algebraic structures. In Section 4, we apply N_j-neighborhoods to define and explore new approximations of a set. In Sections 5 and 6, we present a comparison between the lower and upper approximations and accuracy measures induced from different types of N_j-neighborhoods and their counterparts induced from topological operators and E_j-neighborhoods. We elucidate that our approach is better than approach given in [1] under arbitrary relation, and is better than approach given in [9] under a pre order relation. Finally, we give some conclusions and make a plan for future work in Section 7.

2 Preliminaries

This section recalls some basic properties and results of rough set theory; particularly, those are related to some types of neighborhoods systems.

Through this manuscript, U denotes a nonempty finite set.

2.1 Rough set theory and neighborhood systems

Definition 2.1. (see, [1, 2]) A binary relation Ω on U is a subset of U × U. We write xΩy if (x, y) ∈ Ω.

Definition 2.2. (see, [1, 2]) A binary relation Ω on U is said to be:

(i) reflexive if xΩx for each x ∈ U.

(ii) symmetric if xΩy ⇔ yΩx.

(iii) antisymmetric if x = y whenever xΩy and yΩx.

(iv) transitive if xΩz whenever xΩy and yΩz.

(v) equivalence if it is reflexive, symmetric and transitive.

(vi) partial order if it is reflexive, antisymmetric and transitive.

(vii) diagonal if Ω = {(x, x) : x ∈ U}.

(viii) comparable if xΩy or yΩx for each x, y ∈ U.

Definition 2.3. [18, 19] Let Ω be an equivalence relation on U. We associate every X ⊆ U with two subsets: $\underline{Ω} (X) = ⋃ {A \in U / Ω : A \subseteq X}$ , and

$\bar{Ω} (X) = ⋃ {A \in U / Ω : A ⋂ X \neq \emptyset}$

The two sets $\underline{Ω} (X)$ and $\bar{Ω} (X)$ are respectively called lower and upper approximations of X, where U/Ω is the set of all equivalence classes.

Proposition 2.4. [18, 19] Let Ω be an equivalence relation on U and G, H ⊆ U. Then the following properties hold.

(L1) $\underline{Ω} (G) \subseteq G$

(U1) $G \subseteq \bar{Ω} (G)$

(L2) $\underline{Ω} (\emptyset) = \emptyset$

(U2) $\bar{Ω} (\emptyset) = \emptyset$

(L3) $\underline{Ω} (U) = U$

(U3) $\bar{Ω} (U) = U$

(L4) If G ⊆ H, then $\underline{Ω} (G) \subseteq \underline{Ω} (H)$

(U4) If G ⊆ H, then $\bar{Ω} (G) \subseteq \bar{Ω} (H)$

(L5) $\underline{Ω} (G \cap H) = \underline{Ω} (G) \cap \underline{Ω} (H)$

(U5) $\bar{Ω} (G \cap H) \subseteq \bar{Ω} (G) \cap \bar{Ω} (H)$

(L6) $\underline{Ω} (G) \cup \underline{Ω} (H) \subseteq \underline{Ω} (G \cup H)$

(U6) $\bar{Ω} (G \cup H) = \bar{Ω} (G) \cup \bar{Ω} (H)$

(L7) $\underline{Ω} (G^{c}) = (\bar{Ω} (G))^{c}$

(U7) $\bar{Ω} (G^{c}) = (\underline{Ω} (G))^{c}$

(L8) $\underline{Ω} (\underline{Ω} (G)) = \underline{Ω} (G)$

(U8) $\bar{Ω} (\bar{Ω} (G)) = \bar{Ω} (G)$

(L9) $\underline{Ω} ((\underline{Ω} (G))^{c}) = (\bar{Ω} (G))^{c}$

(U9) $\bar{Ω} ((\bar{Ω} (G))^{c}) = (\bar{Ω} (G))^{c}$

(L10) $\forall K \in U / Ω \Rightarrow \underline{Ω} (K) = K$

(U10) $\forall K \in U / Ω \Rightarrow \bar{Ω} (K) = K$

Now, we recall the concepts of N_j-neighborhoods and E_j-neighborhoods which we apply to initiate new approximations and make some comparisons in this manuscript.

Definition 2.5. [1 , 32] Let Ω be a binary relation on U. The j-neighborhoods of x ∈ U (denoted by N_j (x)) are given for each j ∈ {r, l, 〈r〉, 〈l〉, i, u, 〈i〉, 〈u〉} as follows.

(i) N_r (x) = {y ∈ U : xΩy}.

(ii) N_l (x) = {y ∈ U : yΩx}.

(iii) $N_{〈 r 〉} (x) = {\begin{matrix} ⋂_{x \in N_{r} (y)} N_{r} (y) & there exists N_{r} (y) containing x \\ \emptyset & Otherwise \end{matrix}$

(iv) $N_{〈 l 〉} (x) = {\begin{matrix} ⋂_{x \in N_{l} (y)} N_{l} (y) & there exists N_{l} (y) containing x \\ \emptyset & Otherwise \end{matrix}$

(v) N_i (x) = N_r (x) ⋂ N_l (x).

(vi) N_u (x) = N_r (x) ⋃ N_l (x).

(vii) N_〈i〉 (x) = N_〈r〉 (x) ⋂ N_〈l〉 (x).

(viii) N_〈u〉 (x) = N_〈r〉 (x) ⋃ N_〈l〉 (x).

Henceforth, we consider j ∈ {r, l, 〈r〉, 〈l〉, i, u, 〈i〉, 〈u〉}, unless stated otherwise.

In [6, 32], the authors studied lower and upper approximations of a subset using N_〈r〉 and N_r under any arbitrary binary relation. We complete these two studies by introducing lower and upper approximations of a subset using all j-neighborhoods.

Definition 2.6. [1] Let Ω be a binary relation on U and λ_j be a map from U to 2^U which associated each x ∈ U with its j-neighborhood in 2^U. We called a triple (U, Ω, λ_j) a j-neighborhood space (briefly, j-NS)

Definition 2.7. [9] Let Ω be a binary relation on U. The E-Neighborhoods of x ∈ U (briefly, E_j (x)) are defined for each j as follows.

(i) E_r (x) = {y ∈ U : N_r (y) ⋂ N_r (x) ≠ ∅}.

(ii) E_l (x) = {y ∈ U : N_l (y) ⋂ N_l (x) ≠ ∅}.

(iii) E_i (x) = E_r (x) ⋂ E_l (x).

(iv) E_u (x) = E_r (x) ⋃ E_l (x).

(v) E_〈r〉 (x) = {y ∈ U : N_〈r〉 (y) ⋂ N_〈r〉 (x) ≠ ∅}.

(vi) E_〈l〉 (x) = {y ∈ U : N_〈l〉 (y) ⋂ N_〈l〉 (x) ≠ ∅}.

(vii) E_〈i〉 (x) = E_〈r〉 (x) ⋂ E_〈l〉 (x).

(viii) E_〈u〉 (x) = E_〈r〉 (x) ⋃ E_〈l〉 (x).

Definition 2.8. [9] Let X be a subset of a j-NS (U, Ω, λ_j). Then E_j-lower approximation $F_{E_{j}} (X)$ , E_j-upper approximations $F^{E_{j}} (X)$ and E-accuracy measure $M_{E_{j}} (X)$ of X are respectively defined as follows $F_{E_{j}} (X) = {x \in U : E_{j} (x) \subseteq X}$ and

$F^{E_{j}} (X) = {x \in U : E_{j} (x) ⋂ X \neq \emptyset}$

$M_{E_{j}} (X) = \frac{∣ F_{E_{j}} (X) ⋂ X ∣}{∣ F^{E_{j}} (X) ⋃ X ∣}$ .

2.2 Some topological concepts

We begin this subsection by mentioning the definition a topology.

Definition 2.9. A class θ of subsets of a nonempty set U is called a topology if it is closed under arbitrary union and finite intersection. We called an ordered pair (U, θ) a topological space.

A topology satisfying that every open set is also closed is called a clopen topology, and a topology satisfying that all subsets are open is called a discrete topology.

Definition 2.10. For a subset A of (X, θ), the interior of A (briefly, int_θ (A)) is the union of all open sets that are contained in A, the closure of A (briefly, cl_θ (A)) is the intersection of all closed sets containing A.

Definition 2.11. A topological space (U, θ) is called:

(i) T₀ if for every x ≠ y ∈ U there is an open set V such that x ∈ V and y ∉ V, or y ∈ V and x ∉ V.

(ii) T₁ if for every x ≠ y ∈ U there are two open sets V and W such that x ∈ V \ W and and y ∈ W \ V. If V and W are disjoint, then (U, θ) is called T₂.

(iii) regular if for every closed set H such that x ∉ H, there are two disjoint open sets; one of them containing H and the other containing x.

(iv) normal if for every two disjoint closed sets are separated by two disjoint open sets.

(v) T₃ (resp. T₄) if it is both T₁ and regular (resp. normal).

Proposition 2.12. Every T_i-space is T_i-1 for each i = 1, 2, 3, 4.

Proposition 2.13. If (U, θ) is finite T₁, then θ is the discrete topology.

The following result shows the method of generating different types of topology using N_j-neighborhood systems.

Theorem 2.14. [1] If (U, Ω, λ_j) is a j-NS, then θ_{N
_j} = {A ⊆ U : N_j (x) ⊆ A for each x ∈ A} forms a topology on U for each j.

Definition 2.15. [1] A subset A of a j-NS (U, Ω, λ_j) is said to be a N_j-open set if A belongs to θ_{N
_j}, and its complement is said to be a N_j-closed set.

A class Γ_j of all N_j-closed subsets of a j-NS (U, Ω, λ_j) is denoted by Γ_{N
_j} = {F ⊆ U : F^c ∈ θ_{N
_j}}.

The following two definitions explain the method of formulating the approximations and accuracy measures of a set using interior and closure operators.

Definition 2.16. [1] Let (U, Ω, λ_j) be a j-NS. For each j, the N_j-lower and N_j-upper approximations of a set X ⊆ U are respectively given by $F_{θ_{N_{j}}} (X) = ⋃ {G \in θ_{N_{j}} : G \subseteq X} = {int}_{j} (X)$ , where (int_j is the interior operator in (U, θ_{N
_j}))

$F^{θ_{N_{j}}} (X) = ⋃ {F \in Γ_{N_{j}} : X \subseteq F} = {cl}_{j} (X)$ , where (cl_j is the closure operator in (U, θ_{N
_j}))

Definition 2.17. [1] The θ_{N
_j}-boundary, and θ_{N
_j}-accuracy measure of a subset X of a j-NS (U, Ω, λ_j) are formulated respectively by B_{θ
_{N
_j}} (X) = cl_j (X) \ int_j (X),

$M_{θ_{N_{j}}} (X) = \frac{∣ {int}_{j} (X) ∣}{∣ {cl}_{j} (X) ∣}$ provided that X≠ ∅.

3 Further properties of j-neighborhood spaces

In this section, we establish some topological and algebraic properties of j-neighborhood spaces under different kinds of binary relations.

Theorem 3.1. Let (U, Ω, λ_j) be a j-NS. For each j ∈ {r, l, u}, the following two properties are equivalent:

(i) a topological space (U, θ_{N
_j}) is T_i for each i = 1, 2, 3, 4.

(ii) Ω is a subset of the diagonal relation.

Proof. (i)⇒ (ii): Since U is finite and (U, θ_{N
_j}) is a T_i-space for each i = 1, 2, 3, 4, then θ_{N
_j} is the discrete topology. Therefore, every singleton subset of U is open. Thus, for each j ∈ {r, l, u}, N_j (x) equals to {x} or the empty set for each x ∈ U. This implies that xΩy ⇒ x = y. Hence, Ω is a subset of the diagonal relation.

(ii)⇒ (i): If Ω is a subset of the diagonal relation, then N_j (A) equals to A or the empty set for each A ⊆ U. Therefore, A ∈ θ_{N
_j} for each A ⊆ U. Consequently, θ_{N
_j} is the discrete topology. Hence, (U, θ_{N
_j}) is T₄, as required.□

The above result fails in the cases of j ∉ {r, l, u} as the next example explains.

Example 3.2. Consider Ω = {(v, v) , (w, w) , (v, w)} is a binary relation on U = {v, w}. By calculating we find that: N_j (v) = {v} and N_j (w) = {w} for each j ∈ {〈r〉, 〈l〉, i, 〈i〉, 〈u〉}. Then θ_{N
_j} is the discrete topology. Therefore (U, θ_{N
_j}) is T_i (i = 1, 2, 3, 4) in spite of Ω is not a subset of the diagonal relation. Note that θ_{N
_r} = {∅ , U, {w}} , θ_{N
_l} = {∅ , U, {v}} and θ_{N
_u} = {∅ , U} are not T₁.

Theorem 3.3. Let (U, Ω, λ_j) be a j-NS. If Ω is a subset of the diagonal relation, then for each j a topological space (U, θ_{N
_j}) is T_i for each i = 1, 2, 3, 4.

Proof. One can prove this theorem using a similar manner of the proof of Theorem 3.1.□

Theorem 3.4. Let (U, Ω, λ_j) be a j-NS and j ∈ {r, l, i}. If Ω is comparable, anti-reflexive, anti-symmetric and transitive relation, then (U, θ_{N
_j}) is T₀.

Proof. To prove the theorem in the case of j = r, let x ≠ y ∈ U. Since Ω is comparable and anti-symmetric, then xΩy such that (y, x) ∉ Ω, or yΩx such that (x, y) ∉ Ω. Say, xΩy such that (y, x) ∉ Ω. Then y ∈ N_r (x). Since Ω is anti-reflexive, then x ∉ N_r (x). It remains to prove that N_r (x) ∈ θ_{N
_j}. Let z ∈ N_r (x). Then xΩz. For each a ∈ N_r (z), we have zΩa. Since Ω is transitive, then xΩa. Therefore a ∈ N_r (x). Thus N_r (z) ⊆ N_r (x) which means N_r (x) ∈ θ_{N
_j}. Hence, (U, θ_{N
_j}) is T₀, as required.

Following similar arguments, one can prove the theorem in the case of j = l.

To prove the theorem in the case of j = i, note that N_i (x) =∅ for each x ∈ U because Ω is anti-reflexive and anti-symmetric. Then θ_{N
_i} is the discrete topology on U; therefore, (U, θ_{N
_i}) is T₀.□

Remark 3.5. In the above proposition, we obtain the indiscrete topology in the case of j = u. This follows from the fact N_u (x) under those conditions is U \ {x} for each x ∈ U. This implies that U and ∅ are the only sets in θ_{N
_u}; therefore, it forms the indiscrete topology on U.

Proposition 3.6. Let (U, Ω, λ_j) be a j-NS such that x, y ∈ U and Ω is reflexive and anti-symmetric. Then

(i) N_i (x) = N_〈i〉 (x) = {x} for each x ∈ U.

(ii) N_j (x) ≠ N_j (y) for each x ≠ y, where j ∈ {r, l, 〈r〉, 〈l〉}.

Proof.

(i) Since Ω is reflexive, then x ∈ N_i (x). Now, let y ∈ N_i (x). Then y ∈ N_r (x) and y ∈ N_l (x). This means that xΩy and yΩx. Since Ω is anti-symmetric, then x = y. Thus N_i (x) = {x}. Since Ω is reflexive, then x ∈ N_〈i〉 (x) and N_〈i〉 (x) ⊆ N_i (x). Hence, the desired result is proved.

(ii) We give proofs of the two cases j = r, 〈r〉, and one can prove the other two cases similarly.

For j = r. Suppose that N_r (x) = N_r (y). Since Ω is reflexive, then x ∈ N_r (x) and y ∈ N_r (y). By assumption, we obtain xΩy and yΩx. Since Ω is anti-symmetric, then x = y. Therefore, the desired result is proved.

For j = 〈r〉. Suppose that N_〈r〉 (x) = N_〈r〉 (y). Since Ω is reflexive, then x ∈ N_〈r〉 (x) and y ∈ N_〈r〉 (y). By assumption, we obtain y ∈ N_〈r〉 (x) and x ∈ N_〈r〉 (y). Then y ∈ N_r (x) and x ∈ N_r (y). Therefore, xΩy and yΩx. Since Ω is anti-symmetric, then x = y. Hence, the desired result is proved.□

Proposition 3.7. Let (U, Ω, λ_j) be a j-NS. The following results hold.

(i) If Ω is comparable, then N_u (x) equals U or U \ {x} for each x ∈ U.

(ii) If Ω is symmetric, then N_r (x) = N_l (x) = N_i (x) = N_u (x) and N_〈r〉 (x) = N_〈l〉 (x) = N_〈i〉 (x) = N_〈u〉 (x).

(iii) If Ω is pre order, then N_j (x) = N_〈j〉 (x) for each j ∈ {r, l, i, u}.

Proof.

(i) It is obvious.

(ii) It follows from the fact that Ω is symmetric if and only if xΩy ⇔ yΩx.

(iii) When j = r. Let z ∈ N_r (x). Now, let x ∈ N_r (y). Then yΩx. Since Ω is transitive, then z ∈ N_r (y). This implies that z ∈ N_〈r〉 (x). Therefore, N_r (x) ⊆ N_〈r〉 (x). On the other hand, the reflexivity of Ω implies that N_〈r〉 (x) ⊆ N_r (x). Hence, we obtain the desired result.

Following similar arguments, one can prove the theorem in the case of j = l. The cases of j = i, u is a result of the equality of in the case of j = r, l.□

Proposition 3.8. Let (U, Ω, λ_j) be a j-NS such that Ω is symmetric and transitive. If N_j (x)⋂ N_j (y) ≠ ∅, then N_j (x) = N_j (y) for each j.

Proof. We only prove the proposition when j = r. The other cases can be made similarly.

Since N_r (x)⋂ N_r (y) ≠ ∅, then there exists z ∈ U such that xΩz and yΩz. Since Ω is symmetric, then zΩy, and since Ω is transitive, then xΩy. Now, let a ∈ N_r (y). Then yΩa. Since xΩy, then a ∈ N_r (x). Therefore, N_r (y) ⊆ N_r (x). Following similar argument, we obtain N_r (x) ⊆ N_r (y). Hence, the proof is complete.□

Proposition 3.9. Let (U, Ω, λ_j) be a j-NS such that Ω is reflexive and anti-symmetric. Then for j ∈ {i, 〈i〉}, a family $G = {N_{j} (x) : x \in U}$ forms a partition for U.

Proof. We prove the proposition in the case of j = i. The other case follows similar lines.

Since Ω is reflexive, then U = ⋃ _x∈UN_i (x). Let x ≠ y. Suppose that N_i (x)⋂ N_i (y) ≠ ∅. Then there exists z ∈ U such that z ∈ N_i (x) and z ∈ N_i (y). This means that zΩxΩz and zΩyΩz. Since Ω is anti-symmetric, then z = y and z = x. This contradicts our assumption. Thus, N_i (x)⋂ N_i (y) = ∅ for each x ≠ y ∈ U, as required.□

Corollary 3.10. Let (U, Ω, λ_j) be a j-NS such that Ω is reflexive and anti-symmetric. Then (U, θ_{N
_j}) is a clopen topological space for j ∈ {i, 〈i〉}.

The following two propositions give some properties of relations that are partial orders with either a maximum or a minimum.

Proposition 3.11. Let (U, Ω, λ_j) be a j-NS such that Ω is a partial order relation. If x ∈ U is the smallest element, then we have the following results.

(i) N_r (x) = N_〈r〉 (x) = N_u (x) = N_〈u〉 (x) = U.

(ii) N_l (x) = N_〈l〉 (x) = N_i (x) = N_〈i〉 (x) = {x}.

Proof. (i): Since x is the smallest element, then xΩy for each y ∈ U. This implies that N_r (x) = U, and consequently N_u (x) = U. To prove that N_r (x) = N_〈r〉 (x), let x ∈ N_r (y). Since x is the smallest element, then y = x. In other words, N_r (x) is the only right neighborhood of x. Therefore, N_〈r〉 (x) = N_r (x), as required. It directly follows that N_u (x) = N_〈u〉 (x) = U. Hence, the proof is complete.

Following similar arguments one can prove (ii).□

Proposition 3.12. Let (U, Ω, λ_j) be a j-NS such that Ω is a partial order relation. If x is the largest element, then we have the following results.

(i) N_l (x) = N_〈l〉 (x) = N_u (x) = N_〈u〉 (x) = U.

(ii) N_r (x) = N_〈r〉 (x) = N_i (x) = N_〈i〉 (x) = {x}.

Proof. The proof is similar to that of Proposition 3.13.□

Proposition 3.13. Let (U, Ω, λ_j) be a j-NS and $G = {N_{j} (x) : x \in U}$ . Then for each j ∈ {r, l, 〈r〉, 〈l〉, i, u, 〈i〉, 〈u〉}, the pair $(G, Ω^{★})$ is a partially ordered set, where N_j (x) Ω^★N_j (y) if and only if N_j (x) ⊆ N_j (y).

Proof. Since N_j (x) ⊆ N_j (x), then N_j (x) Ω^★N_j (x) (reflexive). Let N_j (x) Ω^★N_j (y) and N_j (y) Ω^★N_j (x). Then N_j (x) ⊆ N_j (y) and N_j (y) ⊆ N_j (x). Therefore, N_j (x) = N_j (y) (anti-symmetric). Finally, let N_j (x) Ω^★N_j (y) and N_j (y) Ω^★N_j (z). Then N_j (x) ⊆ N_j (y) and N_j (y) ⊆ N_j (z). Therefore, N_j (x) ⊆ N_j (z). Thus, N_j (x) Ω^★N_j (z) (transitive). Hence, Ω^★ is a partial order relation on $G$ .□

Proposition 3.14. Let (U, Ω, λ_j) be a j-NS and A ⊆ U. Then each one of the following families form a topology on U.

(i) θ₁ = {∅ , U, N_i (A) , N_r (A) , N_l (A) , N_u (A)}.

(ii) θ₂ = {∅ , U, N_〈i〉 (A) , N_〈r〉 (A) , N_〈l〉 (A) , N_〈u〉 (A)}.

Proof. We prove (i), and (ii) can be made similarly.

It is clear that N_r (A) ⋂ N_l (A) = N_i (A) ∈ θ₁ and N_r (A) ⋃ N_l (A) = N_u (A) ∈ θ₁. Since N_i (A) ⊆ N_r (A) ⊆ N_u (A) and N_i (A) ⊆ N_l (A) ⊆ N_u (A), then the intersection (union) of any sets of them is a member of θ₁. Hence, θ₁ is a topology on U.□

Proposition 3.15. Let (U, Ω, λ_j) be a j-NS such that Ω is transitive. Then N_j (x) ∈ θ_{N
_j} for each x ∈ X and j ∈ {r, l, i}.

Proof. The proof is trivial if N_r (x) =∅. Suppose that a ∈ N_r (x). Let b ∈ N_r (a), i.e. aΩb. By the transitivity of Ω, xΩb. Therefore, b ∈ N_r (x). Thus, N_r (a) ⊆ N_r (x). Hence, N_r (x) ∈ θ_{N
_r}.

Similarly, one can prove the remaining two cases l and i.□

Corollary 3.16. Let (U, Ω, λ_j) be a j-NS such that Ω is pre order (i.e.,reflexive and transitive). Then N_j (x) ∈ θ_{N
_j} for each x ∈ X and j ∈ {〈r〉, 〈l〉, 〈i〉}.

Proof. The proof is trivial if N_〈r〉 (x) =∅. Suppose that N_〈r〉 (x)≠ ∅. From (iii) of Definition 2.5, N_〈r〉 (x) = ⋂ _{x∈N_r(y)}N_r (y). According to the above proposition, N_r (y) ∈ θ_{N
_r} for each y ∈ X. Since θ_{N
_r} is a topology on a finite set U, ⋂_{x∈N_r(y)}N_r (y) ∈ θ_{N
_r}

Similarly, one can prove the remaining two cases 〈l〉 and 〈i〉.□

Lemma 3.17. Let (U, Ω, λ_i) be an i-NS. Then y ∈ N_i (x) iff x ∈ N_i (y).

Proposition 3.18. Let (U, Ω, λ_i) be an i-NS. Then θ_{N
_i} is a clopen topology on U.

Proof. For each X ∈ θ_{N
_i}, we need to prove N_j (v) ⊆ X^c for each v ∈ X^c. Without loss of generality, we assume N_j (v)¬ = ∅. Suppose y ∈ N_j (v) ∩ X. Then, by the above lemma, v ∈ N_j (y) and N_j (y) ⊆ X (because X ∈ θ_{N
_i}). Thus v ∈ X. This is a contradiction. Therefore, N_j (x)∩ X = ∅, i.e., N_j (v) ⊆ X^c. This ends the proof that θ_{N
_i} is a clopen topology.□

4 Generating different types of approximations space using N_j-neighborhoods

We devote this section to formulate the concepts of N_j-lower and N_j-upper approximations, N_j-boundary region, N_j-positive region, N_j-negative region, and N_j-accuracy measure of a subset X. We illustrate the relationships between them and reveal main properties with the help of examples.

Definition 4.1. Let X be a subset of a j-NS (U, Ω, λ_j). We associate X with two sets using the concept of N_j-neighborhoods as follows. $F_{N_{j}} (X) = {x \in U : N_{j} (x) \subseteq X}$ and

$F^{N_{j}} (X) = {x \in U : N_{j} (x) ⋂ X \neq \emptyset}$

The two sets $F_{N_{j}} (X)$ and $F^{N_{j}} (X)$ are respectively called N_j-lower and N_j-upper approximations of X.

Definition 4.2. Let X be a subset of a j-NS (U, Ω, λ_j). We associate X with three regions using the concept of N_j-neighborhoods as follows. $B_{N_{j}} (X) = F^{N_{j}} (X) \ F_{N_{j}} (X)$

${POS}_{N_{j}} (X) = F_{N_{j}} (X)$

${NEG}_{N_{j}} (X) = U \ F^{N_{j}} (X)$

The three regions $B_{N_{j}} (X)$ , POS_{N
_j} (X) and NEG_{N
_j} (X) are respectively called N_j-boundary, N_j-positive and N_j-negative regions of X.

In the following definition, we associate each subset of a j-NS (U, Ω, λ_j) with a number shows the size of the boundary region without information of its structure

Definition 4.3. The N_j-accuracy measure of a subset X of a j-NS (U, Ω, λ_j) is a rational number lies in the closed interval [0, 1] and calculated by the following formula. $M_{N_{j}} (X) = \frac{∣ F_{N_{j}} (X) ⋂ X ∣}{∣ F^{N_{j}} (X) ⋃ X ∣}$ .

The following example exhibits how to calculate the approximations, regions, and accuracy measure given in the above three definitions for each j.

Example 4.4. Consider Ω = {(v, v) , (v, x) , (v, y), (w, x), (y, x)} is a binary relation on U = {v, w, x, y}. Then we calculate N_j-neighborhoods as given in Tables 1.

Now, let X = {v, x} ⊆ U. By calculating, we obtain the following for each j.

(i) $F_{N_{r}} (X) = {w, x, y}$ and $F^{N_{r}} (X) = {v, w, y}$ ; hence, $B_{N_{r}} = {v}$ and $M_{N_{r}} (X) = \frac{1}{4}$ .

(ii) $F_{N_{l}} (X) = {v, w, y}$ and $F^{N_{l}} (X) = {v, x, y}$ ; hence, $B_{N_{l}} = {x}$ and $M_{N_{l}} (X) = \frac{1}{3}$ .

(iii) $F_{N_{i}} (X) = U$ and $F^{N_{i}} (X) = {v}$ ; hence, $B_{N_{i}} = \emptyset$ and $M_{N_{i}} (X) = 1$ .

(iv) $F_{N_{u}} (X) = {w, y}$ and $F^{N_{u}} (X) = U$ ; hence, $B_{N_{u}} = {v, x}$ and $M_{N_{u}} (X) = 0$ .

(v) $F_{N_{〈 r 〉}} (X) = {w, x}$ and $F^{N_{〈 r 〉}} (X) = {v, x, y}$ ; hence, $B_{N_{〈 r 〉}} = {v, y}$ and $M_{N_{〈 r 〉}} (X) = \frac{1}{3}$ .

(vi) $F_{N_{〈 l 〉}} (X) = {v, x}$ and $F^{N_{〈 l 〉}} (X) = {v, w, y}$ ; hence, $B_{N_{〈 l 〉}} = {w, y}$ and $M_{N_{〈 l 〉}} (X) = \frac{2}{4} = \frac{1}{2}$ .

(vii) $F_{N_{〈 i 〉}} (X) = {v, w, x}$ and $F^{N_{〈 i 〉}} (X) = {v, y}$ ; hence, $B_{N_{〈 i 〉}} = {y}$ and $M_{N_{〈 i 〉}} (X) = \frac{2}{3}$ .

(viii) $F_{N_{〈 u 〉}} (X) = {x}$ and $F^{N_{〈 u 〉}} (X) = U$ ; hence, $B_{N_{〈 u 〉}} = {v, w, y}$ and $M_{N_{〈 u 〉}} (X) = \frac{1}{4}$ .

Proposition 4.5. If Ω is reflexive, then $M_{N_{j}} (X) = \frac{∣ F_{N_{j}} (X) ∣}{∣ F^{N_{j}} (X) ∣}$ .

Proof. Since Ω is reflexive, then x ∈ N_j (x) for each x. Therefore, $F_{N_{j}} (U) \subseteq X$ and $X \subseteq F^{N_{j}} (X)$ . Hence, $M_{N_{j}} (U) = \frac{∣ F_{N_{j}} (U) ⋂ U ∣}{∣ F^{N_{j}} (U) ⋃ U ∣} = \frac{∣ F_{N_{j}} (U) ∣}{∣ F^{N_{j}} (U) ∣}$ , as required.□

Theorem 4.6. Let (U, Ω, λ_j) be a j-NS and X, Y ⊆ U. Then the following properties hold for each j.

(i) $\emptyset \subseteq F_{N_{j}} (\emptyset)$ and $F_{N_{j}} (U) = U$ .

(ii) If X ⊆ Y, then $F_{N_{j}} (X) \subseteq F_{N_{j}} (Y)$ .

(iii) $F_{N_{j}} (X \cap Y) = F_{N_{j}} (X) \cap F_{N_{j}} (Y)$ .

(iv) $F_{N_{j}} (X^{c}) = (F^{N_{j}} (X))^{c}$ .

Proof.

(i) It is clear

(ii) Since X ⊆ Y, then $F_{N_{j}} (X) = {x \in U : N_{j} (x) \subseteq X} \subseteq {x \in U : N_{j} (x) \subseteq Y} = F_{N_{j}} (Y)$ .

(iii) It follows from (ii) that $F_{N_{j}} (X \cap Y) \subseteq F_{N_{j}} (X) \cap F_{N_{j}} (Y)$ . Conversely, let $x \in F_{N_{j}} (X) \cap F_{N_{j}} (Y)$ . Then $x \in F_{N_{j}} (X)$ and $x \in F_{N_{j}} (Y)$ . Therefore, N_j (x) ⊆ X and N_j (x) ⊆ Y. Thus, N_j (x) ⊆ X ∩ Y. Hence, $x \in F_{N_{j}} (X \cap Y)$ , as required.

(iv) $x \in F_{N_{j}} (X^{c}) \Leftrightarrow N_{j} (x) \subseteq X^{c} \Leftrightarrow N_{j} (x) ⋂ X = \emptyset \Leftrightarrow x \notin F^{N_{j}} (X) \Leftrightarrow x \in (F^{N_{j}} (X))^{c}$ .□

Corollary 4.7. Let (U, Ω, λ_j) be a j-NS and X, Y ⊆ Ω. Then $F_{N_{j}} (X) \cup F_{N_{j}} (Y) \subseteq F_{N_{j}} (X \cup Y)$ .

To elucidate that the converses of the items (i) and (ii) of Theorem 4.6, and Corollary 4.7 are not always true, we present the next example.

Example 4.8. Let a j-NS (U, Ω, λ_j) be the same as given in Example 4.4. Note the following.

(i) $F_{N_{r}} (\emptyset) = F_{N_{〈 l 〉}} (\emptyset) = {x}$ , $F_{N_{l}} (\emptyset) = F_{N_{〈 r 〉}} (\emptyset) = {w}$ , $F_{N_{〈 i 〉}} (\emptyset) = {w, x}$ and $F_{N_{i}} (\emptyset) = {w, x, y}$ . Then $F_{N_{j}} (\emptyset) ⊊ \emptyset$ in general.

(ii) For X = {v, x} and Y = {x, y}, we have $F_{N_{r}} (X) = F_{N_{r}} (Y) = {w, x, y}$ . But neither X ⊊ Y nor Y ⊊ X.

(iii) For the same sets X and Y given in (ii), we have $F_{N_{r}} (X) ⋃ F_{N_{r}} (Y) = {w, x, y}$ . But $F_{N_{r}} (X ⋃ Y) = U$ . Therefore, $F_{N_{j}} (X ⋃ Y) ⊊ F_{N_{j}} (X) ⋃ F_{N_{j}} (Y)$ in general.

It can be noted that some basic properties of Pawlak’s rough sets with respect to lower approximation may evaporate. In what follows, we mention those missing properties.

(i) $F_{N_{j}} (X) \subseteq X$ .

(ii) $F_{N_{j}} (\emptyset) = \emptyset$ .

(iii) $F_{N_{j}} (F_{N_{j}} (X)) = F_{N_{j}} (X)$ .

The next example supports assertion (iii) above.

Example 4.9. Let X = {v, w} be a j-NS (U, Ω, λ_j) given in Example 4.4. Then $F_{N_{l}} ({v, w}) = {v, w, y}$ and $F_{N_{l}} (F_{N_{l}} ({v, w, y})) = U$ . Hence, $F_{N_{j}} (F_{N_{j}} (X)) \neq F_{N_{j}} (X)$ .

The next result determines the conditions under which the properties mentioned above hold.

Proposition 4.10. Let (U, Ω, λ_j) be a j-NS and X ⊆ U such that Ω is reflexive. Then the following properties hold for each j.

(i) $F_{N_{j}} (\emptyset) = \emptyset$ .

(ii) $F_{N_{j}} (X) \subseteq X$ .

(iii) $F_{N_{j}} (F_{N_{j}} (X)) \subseteq F_{N_{j}} (X)$ .

Proof.

(i) Since Ω is reflexive, then x ∈ N_j (x) for each x ∈ U. Therefore, $F_{N_{j}} (\emptyset) = \emptyset$ .

(ii) Let $x \in F_{N_{j}} (X)$ . Then N_j (x) ⊆ X. Since Ω is reflexive, then x ∈ X. Therefore, $F_{N_{j}} (X) \subseteq X$ .

(iii) It directly follows from (ii) above.□

Theorem 4.11. Let (U, Ω, λ_j) be a j-NS and X, Y ⊆ U. Then the following properties hold for each j.

(i) $F^{N_{j}} (\emptyset) = \emptyset$ and $F^{N_{j}} (U) \subseteq U$ .

(ii) If X ⊆ Y, then $F^{N_{j}} (X) \subseteq F^{N_{j}} (Y)$ .

(iii) $F^{N_{j}} (X \cup Y) = F^{N_{j}} (X) \cup F^{N_{j}} (Y)$ .

(iv) $F^{N_{j}} (X^{c}) = (F_{N_{j}} (X))^{c}$ .

Proof. The proof is similar to that of Theorem 4.6.□

Corollary 4.12. Let (U, Ω, λ_j) be a j-NS and X, Y ⊆ Ω. Then $F^{N_{j}} (X \cap Y) \subseteq F^{N_{j}} (X) \cap F^{N_{j}} (Y)$ .

To elucidate that the converses of the items (i) and (ii) of Theorem 4.11, and Corollary 4.12 are not always true, we present the next example.

Example 4.13. Let a j-NS (U, Ω, λ_j) be the same as given in Example 4.4. Note the following.

(i) $F^{N_{r}} (U) = F^{N_{〈 l 〉}} (U) = {v, w, y} ⊉ U$ . Then $U ⊊ F^{N_{j}} (U)$ in general.

(ii) Let X = {v} and Y = {x, y}. Then $F^{N_{u}} (X) = {v, x, y}$ and $F^{N_{u}} (Y) = U$ . But X ⊊ Y.

(iii) For the same sets X and Y given in (ii), we have $F^{N_{u}} (X ⋂ Y) = F^{N_{u}} (\emptyset) = \emptyset$ . Therefore, $F^{N_{u}} (X) ⋂ F^{N_{u}} (Y) = {v, x, y} ⊊ F^{N_{u}} (X ⋂ Y)$ in general.

It can be noted that some basic properties of Pawlak’s rough sets with respect to upper approximation may evaporate. In what follows, we mention those missing properties.

(i) $F^{N_{j}} (U) = U$ .

(ii) $X \subseteq F^{N_{j}} (X)$ .

(iii) $F^{N_{j}} (F^{N_{j}} (X)) = F^{N_{j}} (X)$ .

The next example supports assertion (iii) above.

Example 4.14. Let X = {w} be a j-NS (U, Ω, λ_j) given in Example 4.4. Then $F^{N_{〈 u 〉}} ({x}) = {v, x}$ and $F^{N_{〈 u 〉}} (F^{N_{〈 u 〉}} ({v, x})) = U$ . Hence, $F^{N_{〈 u 〉}} (F^{N_{〈 u 〉}} (X)) \neq F^{N_{〈 u 〉}} (X)$ .

The next result determines the conditions under which the properties mentioned above hold.

Proposition 4.15. Let (U, Ω, λ_j) be a j-NS and X ⊆ U such that Ω is reflexive. Then the following properties hold for each j.

(i) $F^{N_{j}} (U) = U$ .

(ii) $X \subseteq F^{N_{j}} (X)$ .

(iii) $F^{N_{j}} (X) \subseteq F^{N_{j}} (F^{N_{j}} (X))$ .

Proof. The proof is similar to that of Proposition 4.10.□

5 Comparison $F_{N_{j}}$ and $F^{N_{j}}$ approximations with the approximations induced by N-topologies

In this section, we compare between the approximations given in the previous section and those were defined using interior and closure topological operators in [1]. We prove that the accuracy measures induced from the approximations given herein are higher than their counterparts induced from topological operators. We support the obtained results by an elucidative example.

We begin this section by proving that the accuracy measures obtained from our approximations $M_{N_{j}} (X)$ (which were introduced in the previous section) are better than those $M_{θ_{j}} (X)$ given in [1] (which are based on N_j-topologies).

Theorem 5.1. $M_{θ_{N_{j}}} (X) \leq M_{N_{j}} (X)$ for every nonempty set X ∈ 2^X.

Proof. First, let z ∈ int (X). Then there is A ∈ θ_j (given in Theorem 2.14) such that z ∈ A ⊆ X; so N_j (z) ⊆ A ⊆ X. Thus, $z \in F_{N_{j}} (X)$ . Hence, $int (X) \subseteq F_{N_{j}} (X)$ . Since int (X) ⊆ X, then $int (X) \subseteq F_{N_{j}} (X) ⋂ X$ . Consequently, we obtain the following inequality $| int (X) | \leq | F_{N_{j}} (X) \cap X |$ (1)

Second, let $z \in F^{N_{j}} (X) \cup X$ . Then we have two cases:

Case 1: z ∈ X. This directly leads to that z ∈ cl (X).

Case 2: z ∉ X. Then $z \in F^{N_{j}} (X)$ ; therefore, N_j (z)⋂ X ≠ ∅. This means there is v ≠ z ∈ U such that v ∈ N_j (z) and v ∈ X. Now, for any A ∈ θ containing z we get v ∈ A. Therefore, A∩ X ≠ ∅; consequently z ∈ cl (X).

From the above two cases, we obtain $F^{N_{j}} (X) \cup X \subseteq cl (X)$ . Thus, we obtain the following inequality $\frac{1}{| cl (X) |} \leq \frac{1}{| F^{N_{j}} (X) \cup X |}$ (2)

From equalities 1 and 2, we find $\frac{| int (X) |}{| cl (X) |} \leq \frac{| F_{N_{j}} (X) \cap X |}{| F^{N_{j}} (X) \cup X |}$ .

Hence, the desired result is proved.□

To support the above result, we calculate the approximations of a set using the idea of N_j-topologies given in [1], and compare them with their counterparts given in the previous section. To this end, we present the following example.

Example 5.2. Consider Ω = {(v, v) , (v, x) , (v, y) , (y, v), (w, v) , (w, x) , (y, x)} is a binary relation on U = {v, w, x, y}. Then we calculate N_j-neighborhoods as given in Tables 2.

Now, we calculate the approximations of each subset of U using a topological method given in [1]. For the sake of succinctness, we suffice by the cases of j ∈ {u, r, l, i}. First, we determine the 4 different topologies induced by N_j neighborhoods as follows.

(i) θ_{N
_u} = {∅ , U}.

(ii) θ_{N
_r} = {∅ , U, {x} , {v, x, y}}.

(iii) θ_{N
_l} = {∅ , U, {w} , {v, w, y}}.

(iv) θ_{N
_i} = {∅ , U, {w} , {x} , {w, x} , {v, y} , {v, w, y} , . {v, x, y}}.

Second, we calculate the topological approximations of each subset of U for j ∈ {u, r, l, i} in the following table.

Third, we calculate the approximations of each subset of U using our method given in the previous section in the following table.

Finally, we calculate the accuracy measure values of each subset of U for the approximations given in Tables 3 and 4.

Table 1

j-neighborhoods of each element in U

N_j\U	v	w	x	y
N _r	{v, x, y}	{x}	∅	{x}
N _l	{v}	∅	{v, w, y}	{v}
N _i	{v}	∅	∅	∅
N _u	{v, x, y}	{x}	{v, w, y}	{v, x}
N _〈r〉	{v, x, y}	∅	{x}	{v, x, y}
N _〈l〉	{v}	{v, w, y}	∅	{v, w, y}
N _〈i〉	{v}	∅	∅	{v, y}
N _〈u〉	{v, x, y}	{v, w, y}	{x}	U

Table 2

j-neighborhoods of each element in U

N_j\U	v	w	x	y
N _r	{v, x, y}	{v, x}	∅	{v, x}
N _l	{v, w, y}	∅	{v, w, y}	{v}
N _i	{v, y}	∅	∅	{v}
N _u	U	{v, x}	{v, w, y}	{v, x}
N _〈r〉	{v, x}	∅	{v, x}	{v, x, y}
N _〈l〉	{v}	{v, w, y}	∅	{v, w, y}
N _〈i〉	{v}	∅	∅	{v, y}
N _〈u〉	{v, x}	{v, w, y}	{v, x}	U

Table 3

The topological approximations $F_{θ_{N_{j}}}$ and $F^{θ_{N_{j}}}$ for j ∈ {r, l, i, u}

X	$F_{θ_{N_{u}}} (X)$	$F^{θ_{N_{u}}} (X)$	$F_{θ_{N_{r}}} (X)$	$F^{θ_{N_{r}}} (X)$	$F_{θ_{N_{l}}} (X)$	$F^{θ_{N_{l}}} (X)$	$F_{θ_{N_{i}}} (X)$	$F^{θ_{N_{i}}} (X)$
{v}	∅	U	∅	{v, w, y}	∅	{v, x, y}	∅	{v, y}
{w}	∅	U	∅	{w}	{w}	U	{w}	{w}
{x}	∅	U	{x}	U	∅	{x}	{x}	{x}
{y}	∅	U	∅	{v, w, y}	∅	{v, x, y}	∅	{v, y}
{v, w}	∅	U	∅	{v, w, y}	{w}	U	{w}	{v, w, y}
{v, x}	∅	U	{x}	U	∅	{v, x, y}	{x}	{v, x, y}
{v, y}	∅	U	∅	{v, w, y}	∅	{v, x, y}	{v, y}	{v, y}
{w, x}	∅	U	{x}	U	{w}	{w, x}	{w, x}	{w, x}
{w, y}	∅	U	∅	{v, w, y}	{w}	U	{w}	{v, w, y}
{x, y}	∅	U	{x}	U	∅	{v, x, y}	{x}	{v, x, y}
{v, w, x}	∅	U	{x}	U	{w}	U	{w, x}	U
{v, w, y}	∅	U	∅	{v, w, y}	{v, w, y}	U	{v, w, y}	{v, w, y}
{v, x, y}	∅	U	{v, x, y}	U	∅	{v, x, y}	{v, x, y}	{v, x, y}
{w, x, y}	∅	U	{x}	U	{w}	U	{w, x}	U
U	U	U	U	U	U	U	U	U
∅	∅	∅	∅	∅	∅	∅	∅	∅

Table 4

Our approximations $F_{j}$ and $F^{j}$ for j ∈ {r, l, i, u}

X	$F_{N_{u}} (X)$	$F^{N_{u}} (X)$	$F_{N_{r}} (X)$	$F^{N_{r}} (X)$	$F_{N_{l}} (X)$	$F^{N_{l}} (X)$	$F_{N_{i}} (X)$	$F^{N_{i}} (X)$
{v}	∅	U	{x}	{v, w, y}	{w, y}	{v, x, y}	{w, x, y}	{v, y}
{w}	∅	{v, x}	{x}	∅	{w}	{v, x}	{w, x}	∅
{x}	∅	{w, x}	{x}	{v, w, y}	{w}	∅	{w, x, y}	∅
{y}	∅	{v, x, y}	{x}	{v}	{w}	{v, x}	{w, x}	{v}
{v, w}	∅	U	{x}	{v, w, y}	{w, y}	{v, x, y}	{w, x, y}	{v, y}
{v, x}	{w, y}	U	{w, x, y}	{v, w, y}	{w, y}	{v, x, y}	{w, x, y}	{v, y}
{v, y}	∅	U	{x}	{v, w, y}	{w, y}	{v, x, y}	U	{v, y}
{w, x}	∅	U	{x}	{v, w, y}	{w}	{x}	{w, x}	∅
{w, y}	∅	{v, x}	{x}	{v}	{w}	{x}	{w, x}	{v}
{x, y}	∅	U	{x}	{v, w, y}	{w}	{x}	{w, x}	{v}
{v, w, x}	{w, y}	U	{w, x, y}	{v, w, y}	{w, y}	{v, x, y}	{w, x, y}	{v, y}
{v, w, y}	{x}	U	{x}	{v, w, y}	U	{v, x, y}	U	{v, y}
{v, x, y}	{w, y}	U	U	{v, w, y}	{w, y}	{v, x, y}	U	{v, y}
{w, x, y}	∅	U	{x}	{v, w, y}	{w}	{v, x}	{w, x}	{v}
U	U	U	U	{v, w, y}	U	{v, x, y}	U	{v, y}
∅	∅	∅	{x}	∅	{w}	∅	{w, x, y}	∅

It can be seen from Table 5 the following:

Table 5

Comparison between N_j-accuracy measure forj ∈ {r, l, i, u}

(i) the boxes with red colore show that $M_{θ_{N_{u}}} (X) < M_{N_{u}} (X)$ .

(ii) the box with yellow colore shows that $M_{θ_{N_{r}}} (X) < M_{N_{r}} (X)$ .

(iii) the boxes with green colore show that $M_{θ_{N_{l}}} (X) < M_{N_{l}} (X)$ .

(iii) The accuracy measures are equal in case of j = i.

In the rest of this section, we determine under what conditions the accuracy measures obtained from N_j-neighborhoods and N_j-topologies are identical.

Theorem 5.3. Let (U, Ω, λ_j) be a j-NS such that Ω is pre order (i.e., reflexive and transitive). Then $M_{θ_{N_{j}}} (X) = M_{N_{j}} (X)$ for every nonempty set X ∈ 2^X and j ∈ {r, l, i, 〈r〉, 〈l〉, 〈i〉}.

Proof. Let $z \in F_{N_{j}} (X) ⋂ X = F_{N_{j}} (X)$ (because Ω is reflexive). Then z ∈ N_j (z) ⊆ X. It follows from Proposition 3.15 and Corollary 3.16 that N_j (z) ∈ θ_{N
_j}. Therefore, z ∈ int (X). Thus $F_{N_{j}} (X) ⋂ X \subseteq int (X)$ . Consequently, we obtain the following inequality $| F_{N_{j}} (X) \cap X | \leq | int (X) |$ (3)

Second, let z ∈ cl (X). Since z ∈ N_j (z) ∈ θ_{N
_j}, N_j (z)⋂ X ≠ ∅. So that, $z \in F^{N_{j}} (X) \cup X$ . Thus, $cl (X) \subseteq F^{N_{j}} (X) \cup X$ . We obtain the following inequality $\frac{1}{| F^{N_{j}} (X) \cup X |} \leq \frac{1}{| cl (X) |}$ (4)

From equalities 3 and 4, we find $M_{N_{j}} (X) = \frac{| F_{N_{j}} (X) \cap X |}{| F^{N_{j}} (X) \cup X |} \leq \frac{| int (X) |}{| cl (X) |} = M_{θ_{j}} (X)$ . On the other hand, it follows from Theorem 5.1 that $M_{θ_{j}} (X) \leq M_{N_{j}} (X)$ . Hence, the desired result is proved.□

6 Comparison

F_{N_{j}}

and

F^{N_{j}}

approximations with the approximations induced by E-neighborhoods

In this section, we compare between the approximations given in Section 3 and those were introduced in [9] with respect to a reflexive relation. We elucidate that our approach is better than approach given in [9] in terms of the approximations and accuracy measures.

Lemma 6.1. Let (U, Ω, λ_j) be a j-NS and X ⊆ U. If Ω is reflexive, then N_j (x) ⊆ E_j (x).

Proof. Let a ∈ N_j (x). By hypothesis, a ∈ N_j (a); therefore, N_j (a)⋂ N_j (x) ≠ ∅. Hence, a ∈ E_j (x), as required.□

Proposition 6.2. Let (U, Ω, λ_j) be a j-NS and X ⊆ U. If Ω is reflexive, then the following results hold for each j.

(i) $F_{E_{j}} (X) \subseteq F_{N_{j}} (X)$ .

(ii) $F^{N_{j}} (X) \subseteq F^{E_{j}} (X)$ .

Proof. To prove (i), let $a \in F_{E_{j}} (X)$ . Then E_j (a) ⊆ X. Since Ω is reflexive, then it follows from the Lemma 6.1 that a ∈ N_j (a) ⊆ E_j (a). Hence, $a \in F_{N_{j}} (X)$ , as required.

Similarly, one can prove (ii)□

Corollary 6.3. Let X be a nonempty subset of a j-NS (U, Ω, λ_j). If Ω is reflexive, then $M_{E_{j}} (X) \leq M_{N_{j}} (X)$ for each j.

Proof. Since Ω is reflexive, then it follows from the above proposition that $F_{E_{j}} (X) \subseteq F_{N_{j}} (X)$ , and $F^{N_{j}} (X) \subseteq F^{E_{j}} (X)$ . Therefore, $∣ F_{E_{j}} (X) ∣ \leq ∣ F_{N_{j}} (X) ∣$ , and $∣ F^{N_{j}} (X) ∣ \leq ∣ F^{E_{j}} (X) ∣$ . By hypothesis, X is nonempty, then $F^{E_{j}} (X)$ and $F^{N_{j}} (X)$ are nonempty sets. Therefore, $\frac{1}{∣ F^{E_{j}} (X) ∣} \leq \frac{1}{∣ F^{N_{j}} (X) ∣}$ . Thus, $\frac{∣ F_{E_{j}} (X) ∣}{∣ F^{E_{j}} (X) ∣} \leq \frac{∣ F_{N_{j}} (X) ∣}{∣ F^{N_{j}} (X) ∣}$ . Since Ω is reflexive, then $F_{E_{j}} (X) \subseteq X \subseteq F^{E_{j}} (X)$ and $F_{N_{j}} (X) \subseteq X \subseteq F^{N_{j}} (X)$ . Hence, $\frac{∣ F_{E_{j}} (X) ⋂ X ∣}{∣ F^{E_{j}} (X) ⋃ X ∣} \leq \frac{∣ F_{N_{j}} (X) ⋂ X ∣}{∣ F^{N_{j}} (X) ⋃ X ∣}$ , as required.□

To support the obtained results, we construct the following example.

Example 6.4. Consider Ω = {(v, v) , (w, w) , (x, x), (v, w), (w, v) , (w, x)} is a binary relation on U = {v, w, x}. Then we calculate N_j-neighborhoods and E_j-neighborhoods in the following table. For the sake of the brevity, we suffice by j ∈ {r, l, i, u} Table 6.

Now, we calculate the approximations and accuracy measure of each subset of U with respect to E_j-neighborhoods and N_j-neighborhoods.

It can be seen from Table 6, Table 7, Table 8, Table 9 and Table 10 that $M_{Ej} (X) < M_{Nj} (X)$ for j ∈ {r, l, i, u}.

Table 6
N_j-neighborhoods and E_j-neighborhoods of each element in U

v w x

N _r {v, w} U {x}

N _l {v, w} {v, w} {w, x}

N _i {v, w} {v, w} {x}

N _u {v, w} U {w, x}

E _r {v, w} U {w, x}

E _l U U U

E _i {v, w} U {w, x}

E _u U U U

	v	w	x
N _r	{v, w}	U	{x}
N _l	{v, w}	{v, w}	{w, x}
N _i	{v, w}	{v, w}	{x}
N _u	{v, w}	U	{w, x}
E _r	{v, w}	U	{w, x}
E _l	U	U	U
E _i	{v, w}	U	{w, x}
E _u	U	U	U

Table 7

The approximations and accuracy measure in case of j = r

X	$F_{E_{r}} (X)$	$F^{E_{r}} (X)$	$M_{E_{r}} (X)$	$F_{N_{r}} (X)$	$F^{N_{r}} (X)$	$M_{N_{r}} (X)$
{v}	∅	{v, w}	0	∅	{v, w}	0
{w}	∅	U	0	∅	{v, w}	0
{x}	∅	{w, x}	0	{x}	{w, x}	$\frac{1}{2}$
{v, w}	{v}	U	$\frac{1}{3}$	{v}	{v, w}	$\frac{1}{2}$
{v, x}	∅	U	0	{x}	U	$\frac{1}{3}$
{w, x}	{x}	U	$\frac{1}{3}$	{x}	U	$\frac{1}{3}$
U	U	U	1	U	U	1

Table 8

The approximations and accuracy measure in case of j = l

X	$F_{E_{l}} (X)$	$F^{E_{l}} (X)$	$M_{E_{l}} (X)$	$F_{N_{l}} (X)$	$F^{N_{l}} (X)$	$M_{N_{l}} (X)$
{v}	∅	U	0	∅	{v, w}	0
{w}	∅	U	0	∅	U	0
{x}	∅	U	0	∅	{x}	0
{v, w}	∅	U	0	{v, w}	U	$\frac{2}{3}$
{v, x}	∅	U	0	∅	U	0
{w, x}	∅	U	0	{x}	U	$\frac{1}{3}$
U	U	U	1	U	U	1

Table 9

The approximations and accuracy measure in case of j = i

X	$F_{E_{i}} (X)$	$F^{E_{i}} (X)$	$M_{E_{i}} (X)$	$F_{N_{i}} (X)$	$F^{N_{i}} (X)$	$M_{N_{i}} (X)$
{v}	∅	{v, w}	0	∅	{v, w}	0
{w}	∅	U	0	∅	{v, w}	0
{x}	∅	{w, x}	0	{x}	{x}	1
{v, w}	{v}	U	$\frac{1}{3}$	{v, w}	{v, w}	1
{v, x}	∅	U	0	{x}	U	$\frac{1}{3}$
{w, x}	{x}	U	$\frac{1}{3}$	{x}	U	$\frac{1}{3}$
U	U	U	1	U	U	1

Table 10

The approximations and accuracy measure in case of j = u

X	$F_{E_{u}} (X)$	$F^{E_{u}} (X)$	$M_{E_{u}} (X)$	$F_{N_{u}} (X)$	$F^{N_{u}} (X)$	$M_{N_{u}} (X)$
{v}	∅	U	0	∅	{v, w}	0
{w}	∅	U	0	∅	U	0
{x}	∅	U	0	∅	{w, x}	0
{v, w}	∅	U	0	{v}	U	$\frac{1}{3}$
{v, x}	∅	U	0	∅	U	0
{w, x}	∅	U	0	{x}	U	$\frac{1}{3}$
U	U	U	1	U	U	1

7 Conclusion and future work

Pawlak [18, 19] proposed a powerful mathematical tool for data representation called rough set theory. Classifications of sets in rough set content are based on the approximation operators which have topological properties similar to all/some properties of the interior and closure operators. Therefore, investigation of rough sets using topological concepts is fruitful to model real-life problems such as image processing, machine learning, data mining, pattern recognition and medical events.

One of the hot topics on rough sets is reducing the boundary region to increase the accuracy measure of decision-making. Neighborhood systems is one of the followed techniques to achieve this goal. Therefore, this article, along with its study more interesting properties of j-neighborhoods under different types of binary relations, applied j-neighborhoods to define new approximations of a set. In addition, we have compared between the approximations and accuracy measure introduced in this work and those induced from topological concepts and E_j-neighborhoods. We have illustrated the advantages of our approaches to obtain higher accuracy measures than those proposed in [1] and [9].

In the upcoming works, we will study new types of neighborhoods in rough set theory and use to generate topological structures. Also, we will investigate them in fuzzy rough set and soft rough set contents.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Footnotes

Acknowledgment

The authors are grateful to the editor and referees for their valuable comments and suggestions.

References

Abd El-Monsef

M.E.

, Embaby

O.A.

and El-Bably

M.K.

, Comparison between rough set approximations based on different topologies, International Journal of Granular Computing, Rough Sets andIntelligent Systems 3(4) (2014), 292–305.

Abd El-Monsef

M.E.

, Kozae

A.M.

and El-Bably

M.K.

, On generalizingcovering approximation space, Journal of the EgyptianMathematical Society 23(3) (2015), 535–545.

Abo-Tabl

E.A.

, Rough sets and topological spaces based onsimilarity, International Journal of Machine Learning andCybernetics 4 (2013), 451–458.

Abo-Tabl

E.A.

, A comparison of two kinds of definitions of roughapproximations based on a similarity relation, Inform. Sci. 181 (2011), 2587–2596.

Abu-Donia

H.M.

, Comparison between different kinds of approximationsby using a family of binary relations, Knowledge-Based Systems 21 (2008), 911–919.

Allam

A.A.

, Bakeir

M.Y.

and Abo-Tabl

E.A.

, Newapproach for basic rough set concepts. In: International workshop on rough sets, fuzzy sets, data mining, and granular computing. Lecture Notes in Artificial Intelligence, 3641, Springer, Regina, (2005), 64–73.

Allam

A.A.

, Bakeir

M.Y.

and Abo-Tabl

E.A.

, New approach for closurespaces by relations.ziensis}, Acta Mathematica Academiae PaedagogicaeNyiregyh’{a 22 (2006), 285–304.

Al-shami

T.M.

, An improvement of rough sets’ accuracy measure using containment neighborhoods with a medical application 569 (2021) 110–124.

Al-shami

T.M.

, Fu

W.Q.

and Abo-Tabl

E.A.

, New rough approximations based on E-neighborhoods, Complexity, Volume 2021, Article ID 6666853, 6 pages.

10.

Amer

W.S.

, Abbas

M.I.

and El-Bably

M.K.

, On j-near concepts inrough sets with some applications, Journal of Intelligent andFuzzy Systems 32(1) (2017), 1089–1099.

11.

Atef

, Khalil

A.M.

, Li

S.G.

, Azzam

and El Atik

, Comparison of six types of rough approximations based on j-neighborhood spaceand j-adhesion neighborhood space, Journal of Intelligent andFuzzy Systems 39(3) (2020), 4515–4531.

12.

Azzam

, Khalil

A.M.

and Li

S.-G.

, Medical applications via minimaltopological structure, Journal of Intelligent and FuzzySystems 39(3) (2020), 4723–4730.

13.

El-Sharkasy

M.M.

, Minimal structure approximation space and some ofits application, Journal of Intelligent and Fuzzy Systems 40(1) (2021), 973–982.

14.

Kozae

A.M.

, Abo Khadra

and Medhat

, Topological approach for approximation space (TAS), Proceeding of the 5th International Conference INFOS 2007 on Informatics and Systems, Faculty of Computers, Information Cairo University, Cairo, Egypt, 2007, pp. 289–302.

15.

Kozae

A.M.

, El-Sheikh

S.A.

, Aly

E.A.

and Hosny

, Rough sets andits applications in a computer network, Annals of FuzzyMathematics and Informatics 6(3) (2013), 605–624.

16.

Lashin

E.F.

, Kozae

A.M.

, Abo Khadra

A.A.

and Medhat

, Rough settheory for topological spaces, Int. J. Approx. Reason. 40 (2005), 35–43.

17.

, Xie

and Li

, Topological structure of generalized roughsets, Comput. Math. Appl. 63 (2012), 1066–1071.

18.

Pawlak

, Rough sets, Int. J. of Information and ComputerSciences 11(5) (1982), 341–356.

19.

Pawlak

, Rough sets, Theoretical Aspects of Reasoning About Data, Kluwer Acadmic Publishers Dordrecht, (1991).

20.

Pei

, Pei

and Zheng

, Topology vs generalized rough sets, International Journal of Approximate Reasoning 52(2) (2011), 231–239.

21.

Peters

J.F.

, Skowron

and Stepaniuk

, Nearness of visualobjects, Application of rough sets in proximity spaces, Fund.Inform. 128 (2013), 159–176.

22.

Salama

A.S.

, Sequences of topological near open and near closed setswith rough applications, Filomat 34(1) (2020), 51–58.

23.

Salama

A.S.

, Topological solution for missing attribute values inincomplete information tables, Information Sciences 180(2010), 631–639.

24.

Salama

A.S.

, Mhemdi

, Elbarbary

O.G.

and Al-shamiTopological

T.M.

, A new approach to rough set based on remote neighborhood systems, Mathematical Problems in Engineering, Volume 2019, Article ID 8712010, 8 pages.

25.

Singh

P.K.

and Tiwari

, Topological structures in rough settheory: A survey, Hacet. J. Math. Stat. 49(4) (2020), 1270–1294.

26.

Skowron

, On topology in information system, Bulletin ofPolish Academic Science and Mathematics 36 (1988), 477–480.

27.

Skowron

and Stepaniuk

, Tolerance approximation spaces, Fundamenta Informaticae 27 (1996), 245–253.

28.

Sun

, Li

and Hu

, A new approach to rough set based on remote neighborhood systems, Mathematical Problems in Engineering, Volume 2019, Article ID 8712010, 8 pages.

29.

Wiweger

, On topological rough sets, Bull Pol Acad Sci Math 37 (1989), 89–93.

30.

Q.E.

, Wang

, Huang

Y.X.

and Li

J.S.

, Topology theory on roughsets, IEEE Transactions on Systems, man and Cybernetics, Part B(Cybernetics) 38(1) (2008), 68–77.

31.

Yao

Y.Y.

, Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning 15(1996), 291–317.

32.

Yao

Y.Y.

, Generalized rough set models, Rough sets in knowledge Discovery 1, L. Polkowski, A. Skowron (Eds.), Physica Verlag, Heidelberg, (1998), 286–318.

33.

, Zhan

, Ding

and Fujita

, A novel fuzzy rough set modelwith fuzzy neighborhood operators, Information Sciences 544 (2021), 266–297.

34.

Zhan

, Jiang

and Yao

, “Three-way multi-attribute decision-making based on outranking relations,” in IEEE Transactions on Fuzzy Systems, doi: 10.1109/TFUZZ.2020.3007423.

35.

Zhan

and Sun

, Covering-based intuitionistic fuzzy rough setsand applications in multi-attribute decision-making, ArtificialIntelligence Review 53 (2020), 671–701.

36.

Zhan

and Wang

, Certain types of soft coverings based roughsets with applications, International Journal of MachineLearning and Cybernetics 10(5) (2019), 1065–1076.

37.

Zhan

and Xu

, Two types of coverings based multigranulationrough fuzzy sets and applications to decision making, Artificial Intelligence Review 53(1) (2020), 167–198.

38.

Zhan

and Zhu

, Reviews on decision making methods based on(fuzzy) soft sets and rough soft sets, Journal of Intelligent& Fuzzy Systems 29(3) (2015), 1169–1176.

39.

Zhang

, Zhan

and Wu

W.Z.

, Novel fuzzy rough set models andcorresponding applications to multi-criteria decision-making, Fuzzy Sets and Systems 383 (2020), 92–126.

40.

Zhang

, Zhan

and Wu

, “On multi-criteria decision-making method based on a fuzzy rough set model with fuzzy α-neighborhoods,” in IEEE Transactions on Fuzzy Systems, doi: 10.1109/TFUZZ.2020.3001670.

41.

Zhang

, Zhan

and Xu

, Covering-based generalized IF roughsets with applications to multi-attribute decision-making, Information Sciences 478 (2019), 275–302.

42.

Zhang

, Ouyang

and Wangc

, Note on generalized rough setsbased on reflexive and transitive relations, InformationSciences 179 (2009), 471–473.

43.

Zhang

Y.L.

, Li

and Li

, Topological structure ofrelational-based generalized rough sets, Fund. Inform. 147(4) (2016), 477–491.

44.

Zhu

, Relationship between generalized rough sets based on binary relation and covering, Information Sciences 179 (2009), 210–225.

A comparison of two types of rough approximations based on N j -neighborhoods

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Rough set theory and neighborhood systems

2.2 Some topological concepts

3 Further properties of j-neighborhood spaces

4 Generating different types of approximations space using N j -neighborhoods

5 Comparison F N j and F N j approximations with the approximations induced by N-topologies

Table 6 N j -neighborhoods and E j -neighborhoods of each element in U v w x N r {v, w} U {x} N l {v, w} {v, w} {w, x} N i {v, w} {v, w} {x} N u {v, w} U {w, x} E r {v, w} U {w, x} E l U U U E i {v, w} U {w, x} E u U U U

Conflict of interests

Footnotes

Acknowledgment

References

4 Generating different types of approximations space using N_j-neighborhoods

5 Comparison $F_{N_{j}}$ and $F^{N_{j}}$ approximations with the approximations induced by N-topologies

Table 6
N_j-neighborhoods and E_j-neighborhoods of each element in U

v w x

N _r {v, w} U {x}

N _l {v, w} {v, w} {w, x}

N _i {v, w} {v, w} {x}

N _u {v, w} U {w, x}

E _r {v, w} U {w, x}

E _l U U U

E _i {v, w} U {w, x}

E _u U U U