Comparison of six types of rough approximations based on j -neighborhood space and j -adhesion neighborhood space

Abstract

In this paper, we generalize three types of rough set models based on j-neighborhood space (i.e, type 1 j-neighborhood rough set, type 2 j-neighborhood rough set, and type 3 j-neighborhood rough set), and investigate some of their basic properties. Also, we present another three types of rough set models based on j-adhesion neighborhood space (i.e, type 4 j-adhesion neighborhood rough set, type 5 j-adhesion neighborhood rough set, and type 6 j-adhesion neighborhood rough set). The fundamental properties of approximation operators based on j-adhesion neighborhood space are established. The relationship between the properties of these types is explained. Finally, according to j-adhesion neighborhood space, we give a comparison between the Yao’s approach and our approach.

Keywords

Rough sets lower and upper approximations accuracy measure

1 Introduction

The original concept of rough set is proposed by Pawlak [1, 2]. The rough set theory has been applied to many different fields, such as economics, engineering, medicine, biology, chemistry, banking, market research, speech recognition, information analysis, data analysis, material science, data mining, networking, linguistics and other fields (see, e.g., [3 –15]). A framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations (lower / upper) using the more familiar notion of binary relations are proposed by Yao [16]. He also introduced a special class of neighborhood systems, it is called 1-neighborhood systems and analyzed three extensions of Pawlak’s lower and upper approximations. Later, many researches presented some concepts based on a rough set, for example, similarity relation [17], the tolerance relation [18], and the arbitrary binary relation [16 , 19–22]. Abu-Donia [23] discussed three types of lower and upper approximations of any set with respect to any relation based on right neighborhood and generalized these three types of approximations into two ways by using a finite number of any binary relations. Abo-Tabl [24] gave a comparison of two kinds of definitions of the rough approximations based on a similarity relation.

In 2014, Abd El-Monsef et al. [25] presented the main ideas about the concept of “ j-neighborhood space ” and studied eight approaches for approximating rough sets. Abbas et al. [26] introduced the new concepts of j-near closure operators induced from binary relation and its applications. Hosny [27] gave a generalization of rough sets by using two different methods. Abd El-Monsef [28] presented the cover rough sets based on j-neighborhoods by approximation operations as a new type of extended covering rough set models. Amer [29] proposed the concepts of “ j-near ” in rough sets with some applications. Recently, research works on neighborhood rough sets and its applications are very active and progressing rapidly [30 –38].

To difference between the three of types of approximations definitions, the eight types of j-neighborhoods systems, and our proposed due the main notions, where in Pawlak’s original rough set theory, equivalence relation and arbitrary binary relation (see Definitions 2.2–2.4) are 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, Lin [39] and Yao [40] studied rough sets using by neighbourhood systems for the interpretation of granules and then Abd El-Monsef et al. [25, 41] introduced mixed neighbourhood systems to approximate rough sets and eight types of j-neighborhoods systems. In our proposed, we defined new types (see Definitions 4.3, 4.11, 4.19) based on j-adhesion neighborhood becouse there there are some insufficiency in eight types of j-neighborhoods systems in Definition 2.5 (see Example 3.2, where in type 1 j-neighborhood rough set, ${^{1} N}_{j R} (X) ⊈ {^{1} N}_{j}^{R} (X)) .$ Therefore, in this paper we present a new approach by using j-adhesion neighborhood space to satisfy the type 1 j-neighborhood rough set.

Regarding these developments, as the motivation of this paper, we will generalize the three types of Yao’s lower and upper approximations of any set with respect to j-neighborhood space. Further, we will define another three types of approximations based on j-adhesion neighborhood space. Properties of approximation operators based on j-adhesion neighborhood space are explained and discussed. The relationship between these types of j-adhesion neighborhood space is introduced. Finally, we will give a comparison between the Yao’s approach [16] and our approach by using j-adhesion neighborhood space.

The rest of this paper is organized as follows. In Section 2, we review some basic concepts, such as types of rough set approximations based on binary relation, j-neighborhood of element x, and j-neighborhood space. In Section 3, we generalize three types of rough sets based on j-neighborhood space. Several of the existing properties of three types of rough sets are explained. A new three types of rough set based on j-adhesion neighborhood space are defined and investigated their properties are shown in Section 4. A comparison between the Yao’s approach [16] and our approach is discussed in Section 5. Finally, conclusions are given in Section 6.

2 Preliminaries

In this section, we recall several basic notions and results which will be used in this paper as indicated below.

2.1 Types of rough set approximations based on binary relation

The concept of a rough set to any relation $R$ extended by Yao [16] as follows:

Definition 2.1. (cf. [16]). Let $R$ be an arbitrary binary relation on an universe set Ω and $(Ω, R)$ be an approximation space. Then $R (x) = {y \in Ω : x R y} .$

Definition 2.2. (cf. [16]). Let $(Ω, R)$ be an approximation space and $R \subseteq Ω \times Ω .$ The first definition for lower and upper approximations of X ⊆ Ω in $(Ω, R)$ or with respect to $R$ is a pair $(R^{-} (X), R^{+} (X))$ , defined by $R^{-} (X) = {x \in Ω : R (x) \subseteq X}$ (called the first lower approximation of X)

and $R^{+} (X) = {x \in Ω : R (x) \cap X \neq \emptyset}$ (called the first upper approximation of X).

We list the properties that are of interest in the theory of rough sets, let X, X₁, X₂ ⊆ Ω :

(1) $R^{-} (X) = (R^{+} (X^{c}))^{c}$ .

(2) $R^{-} (Ω) = Ω$ .

(3) $R^{-} (X_{1} \cap X_{2}) = R^{-} (X_{1}) \cap R^{-} (X_{2})$ .

(4) $R^{-} (X_{1}) \cup R^{-} (X_{2}) \subseteq R^{-} (X_{1} \cup X_{2})$ .

(5) If X₁ ⊆ X₂, then $R^{-} (X_{1}) \subseteq R^{-} (X_{2})$ .

(6) $R^{-} (\emptyset) = \emptyset$ .

(7) $R^{-} (X) \subseteq X$ .

(8) $X \subseteq R^{-} (R^{+} (X))$ .

(9) $R^{-} (R^{-} (X)) = R^{-} (X)$ .

(10) $R^{+} (X) = R^{-} (R^{+} (X))$ .

(11) $R^{+} (X) = (R^{-} (X^{c}))^{c}$ .

(12) $R^{+} (\emptyset) = \emptyset$ .

(13) $R^{+} (X_{1} \cap X_{2}) \subseteq R^{+} (X_{1}) \cap R^{+} (X_{2})$ .

(14) $R^{+} (X_{1} \cup X_{2}) = R^{+} (X_{1}) \cup R^{+} (X_{2})$ .

(15) If X₁ ⊆ X₂, then $R^{+} (X_{1}) \subseteq R^{+} (X_{2})$ .

(16) $R^{+} (Ω) = Ω$ .

(17) $X \subseteq R^{+} (X)$ .

(18) $R^{+} (R^{-} (X)) \subseteq X$ .

(19) $R^{+} (R^{+} (X)) = R^{+} (X)$ .

(20) $R^{-} (X) = R^{+} (R^{-} (X))$ .

(21) $R^{-} ({X_{1}}^{c} \cup X_{2}) \subseteq (R^{-} (X_{1}))^{c} \cup R^{-} (X_{2})$ .

(22) $R^{-} (X) \subseteq R^{+} (X)$ .

Definition 2.3. (cf. [16]). Let $(Ω, R)$ be an approximation space. The second definition for lower and upper approximations of X ⊆ Ω in $(Ω, R)$ or with respect to $R$ is a pair $(R^{\circ} (X), R^{•} (X))$ , defined by $R^{\circ} (X) = ⋃ {R (x) : R (x) \subseteq X}$ (called the second lower approximation of X)

and $R^{•} (X) = [R^{\circ} (X^{c})]^{c}$ (called the second upper approximation of X).

Definition 2.4. (cf. [16]). Let $(Ω, R)$ be an approximation space. The third definition for lower and upper approximations of X ⊆ Ω in $(Ω, R)$ or with respect to $R$ is a pair $(R^{⊖} (X), R^{\oplus} (X))$ , defined by $R^{⊖} (X) = [R^{\oplus} (X^{c})]^{c}$ (called the third lower approximation of X)

and $R^{\oplus} (X) = ⋃ {R (x) : R (x) \cap X \neq \emptyset}$ (called the third upper approximation of X).

2.2 j-Neighborhood of element x and j-neighborhood space

Abd El-Monsef et al. [25] proposed the concepts of j-neighborhood of element x and j-neighborhood space as indicated below.

Definition 2.5. (cf. [25]). Let $R$ be an arbitrary binary relation on Ω. The j-neighborhood of element x ∈ Ω are defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $N_{j_{1}} (x) = {y \in Ω : x R y}$ .

(2) $N_{j_{2}} (x) = {y \in Ω : y R x}$ .

(3) $N_{j_{3}} (x) = {y \in Ω : ⋂_{x \in N_{j_{1}} (y)} N_{j_{1}} (y)}$ .

(4) $N_{j_{4}} (x) = {y \in Ω : ⋂_{x \in N_{j_{2}} (y)} N_{j_{2}} (y)}$ .

(5) $N_{j_{5}} (x) = N_{j_{1}} (x) \cup N_{j_{2}} (x)$ .

(6) $N_{j_{6}} (x) = N_{j_{1}} (x) \cap N_{j_{2}} (x)$ .

(7) $N_{j_{7}} (x) = N_{j_{3}} (x) \cup N_{j_{4}} (x)$ .

(8) $N_{j_{8}} (x) = N_{j_{3}} (x) \cap N_{j_{4}} (x)$ .

By the following example, we explain the above definition as:

Example 2.6. Let Ω = {a, b, c, d} and $R \subseteq Ω \times Ω$ defined by $R = {(a, a), (b, b), (c, b), (c, c), (d, a)} .$ Then, the j-neighborhood are computed by

(1) As j = j₁, we have $N_{j_{1}} (a) = {a}, N_{j_{1}} (b) = {b},$ $N_{j_{1}} (c) = {b, c}, N_{j_{1}} (d) = {a} .$

(2) As j = j₂, we have $N_{j_{2}} (a) = {a, d}, N_{j_{2}} (b) = {b, c},$ $N_{j_{2}} (c) = {c}, N_{j_{2}} (d) = \emptyset .$

(3) As j = j₃, we have $N_{j_{3}} (a) = {a}, N_{j_{3}} (b) = {b},$ $N_{j_{3}} (c) = {b, c}, N_{j_{3}} (d) = \emptyset .$

(4) As j = j₄, we have $N_{j_{4}} (a) = {a, d}, N_{j_{4}} (b) = {b, c},$ $N_{j_{4}} (c) = {c}, N_{j_{4}} (d) = {a, d} .$

(5) As j = j₅, we have $N_{j_{5}} (a) = {a, d}, N_{j_{5}} (b) = {b, c},$ $N_{j_{5}} (c) = {b, c}, N_{j_{5}} (d) = {a} .$

(6) As j = j₆, we have $N_{j_{6}} (a) = {a}, N_{j_{6}} (b) = {b},$ $N_{j_{6}} (c) = {c}, N_{j_{6}} (d) = \emptyset .$

(7) As j = j₇, we have $N_{j_{7}} (a) = {a, d}, N_{j_{7}} (b) = {b, c},$ $N_{j_{7}} (c) = {b, c}, N_{j_{7}} (d) = {a, d} .$

(8) As j = j₈, we have $N_{j_{8}} (a) = {a}, N_{j_{8}} (b) = {b},$ $N_{j_{8}} (c) = {c}, N_{j_{8}} (d) = \emptyset .$

Definition 2.6 (cf. [25]). Let $R$ be an arbitrary binary relation on Ω, 2^Ω the set of all subsets of Ω, and ζ_j : Ω ⟶ 2^Ω be a mapping which assigns for each x in Ω its j-neighborhood in 2^Ω. The triple $(Ω, R, ζ_{j})$ is called a j-neighborhood space.

3 Generalized three types of rough set models based on j-neighborhood space

In this section, we will extent the Definitions 2.2–2.4 to j-neighborhood space as indicated below.

Definition 3.1. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. A type 1 of rough set based on j-neighborhood (briefly, type 1 j-neighborhood rough set) of X ⊆Ω in $(Ω, R, ζ_{j})$ or with respect to ζ_j is a pair $({^{1} N}_{j R} (X), {^{1} N}_{j}^{R} (X))$ , defined by ${^{1} N}_{j R} (X) = {x \in Ω : N_{j} (x) \subseteq X}$ (called the type 1 lower approximation of X) and ${^{1} N}_{j}^{R} (X) = {x \in Ω : N_{j} (x) \cap X \neq \emptyset}$ (called the type 1 upper approximation of X).

Example 3.2. (Continued from Example 2.6). If X = {a, b}. Then we have

(1) As j = j₁, we have ${^{1} N}_{j_{1} R} (X) = {a, b, d}, {^{1} N}_{j_{1}}^{R} (X) = Ω .$

(2) As j = j₂, we have ${^{1} N}_{j_{2} R} (X) = {d}, {^{1} N}_{j_{2}}^{R} (X) = {a, b} .$

(3) As j = j₃, we have ${^{1} N}_{j_{3} R} (X) = {a, b, d}, {^{1} N}_{j_{3}}^{R} (X) = {a, b, c} .$

(4) As j = j₄, we have ${^{1} N}_{j_{4} R} (X) = \emptyset, {^{1} N}_{j_{4}}^{R} (X) = {a, b, d} .$

(5) As j = j₅, we have ${^{1} N}_{j_{5} R} (X) = {d}, {^{1} N}_{j_{5}}^{R} (X) = Ω .$

(6) As j = j₆, we have ${^{1} N}_{j_{6} R} (X) = {a, b, d}, {^{1} N}_{j_{6}}^{R} (X) = {a, b} .$

(7) As j = j₇, we have ${^{1} N}_{j_{7} R} (X) = \emptyset, {^{1} N}_{j_{7}}^{R} (X) = Ω .$

(8) As j = j₈, we have ${^{1} N}_{j_{8} R} (X) = {a, b, d}, {^{1} N}_{j_{8}}^{R} (X) = {a, b} .$

Remark 3.3. If we consider j = j₁ in Definition 3.1, then we obtain on Definition 2.2.

Definition 3.4. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space and X ⊆ Ω. A type 1 j-neighborhood boundary region and a type 1 j-neighborhood accuracy measure, denoted by $^{1} B_{j}$ and $^{1} A_{j}$ , respectively, defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $^{1} B_{j} (X) = {^{1} N}_{j}^{R} (X) - {^{1} N}_{j R} (X) .$

(2) $^{1} A_{j} (X) = \frac{| {^{1} N}_{j R} (X) |}{| {^{1} N}_{j}^{R} (X) |} .$

Example 3.5. (Continued from Examples 2.6 and 3.2). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). The type 1 j₂-boundary and the type 1 j₂-accuracy, respectively, are computed as $^{1} B_{j_{2}} (X) = {a, b}$ and $^{1} A_{j_{2}} (X) = \frac{1}{2}$ .

Proposition 3.6. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{1} N}_{j}^{R} (\emptyset) = \emptyset$ .

(2) If X₁ ⊆ X₂, then ${^{1} N}_{j}^{R} (X_{1}) \subseteq {^{1} N}_{j}^{R} (X_{2})$ .

(3) ${^{1} N}_{j}^{R} (X_{1} \cap X_{2}) \subseteq {^{1} N}_{j}^{R} (X_{1}) \cap {^{1} N}_{j}^{R} (X_{2})$ .

(4) ${^{1} N}_{j}^{R} (X_{1} \cup X_{2}) = {^{1} N}_{j}^{R} (X_{1}) \cup {^{1} N}_{j}^{R} (X_{2})$ .

(5) ${^{1} N}_{j}^{R} (X) = ({^{1} N}_{j R} (X^{c}))^{c}$ .

(6) ${^{1} N}_{j}^{R} (X_{1}^{c} \cup X_{2}) \subseteq ({^{1} N}_{j R} (X_{1}))^{c} \cup {^{1} N}_{j R} (X_{2})$ .

Proof. (1) Follows from Definition 3.1.

(2) If X₁ ⊆ X₂, then ${^{1} N}_{j}^{R} (X_{1}) = {x \in Ω : N_{j} (x) \cap X_{1} \neq \emptyset} \subseteq {x \in Ω : N_{j} (x) \cap X_{2} \neq \emptyset} = {^{1} N}_{j}^{R} (X_{2}) .$

(3) and (4) Follows from Definition 3.1 and from (2) above.

(5) $({^{1} N}_{j R} (X^{c}))^{c} = ⋂ {x \in Ω : N_{j} (x) \subseteq X^{c}}^{c} = ⋂ {x \in Ω : N_{j}^{c} (x) \supseteq X} = {x \in Ω : N_{j} (x) \cap X$ $\neq \emptyset} = {^{1} N}_{j}^{R} (X) .$

(6) By contradiction, let $x \notin ({^{1} N}_{j R} (X_{1}))^{c} \cup {^{1} N}_{j R} (X_{2})$ . Then $x \notin ({^{1} N}_{j R} (X_{1}))^{c}$ and $x \notin {^{1} N}_{j R} (X_{2})$ . Thus $x \in {^{1} N}_{j R} (X_{1})$ and $x \notin {^{1} N}_{j R} (X_{2})$ . Hence, $N_{j} (x) \subseteq X_{1}$ and $N_{j} (x) ⊈ X_{2}$ . So $N_{j} (x) ⊈ X_{1}^{c}$ and $N_{j} (x) \subseteq X_{2}$ . Therefore $N_{j} (x) ⊈ (X_{1}^{c} \cup X_{2})$ , then $x \notin N_{j} (X_{1}^{c} \cup X_{2})$ , i.e., ${^{1} N}_{j}^{R} (X_{1}^{c} \cup X_{2}) \subseteq ({^{1} N}_{j R} (X_{1}))^{c} \cup {^{1} N}_{j R} (X_{2})$ .

The equality of (3) and (6) in Proposition 3.6 does not hold as shown in the following example.

Example 3.7 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {b, d} and X₂ = {a, b, c}, then ${^{1} N}_{j_{2}}^{R} (X_{1} \cap X_{2}) = {b}$ . Hence ${^{1} N}_{j_{2}}^{R} (X_{1}) \cap {^{1} N}_{j_{2}}^{R} (X_{2}) = {a, b}$ . Thus ${^{1} N}_{j_{2}}^{R} (X_{1} \cap X_{1}) \neq {^{1} N}_{j_{2}}^{R} (X_{1}) \cap {^{1} N}_{j_{2}}^{R} (X_{2})$ . Also, if X₁ = {a, b, d} and X₂ = {a, c}, then ${^{1} N}_{j_{2} R} (X_{1}) = {a, d}$ and ${^{1} N}_{j_{2} R} (X_{2}) = {c, d}$ . Hence ${^{1} N}_{j_{2} R} (X_{1}^{c} \cup X_{2}) = {c, d}$ . But $({^{1} N}_{j_{2} R} (X_{1}))^{c} \cup {^{1} N}_{j_{2} R} (X_{2}) = {b, c, d}$ . Therefore ${^{1} N}_{j_{2} R} (X_{1}^{c} \cup X_{2}) \neq ({^{1} N}_{j_{2} R} (X_{1}))^{c} \cup {^{1} N}_{j_{2} R} (X_{2})$ .

Proposition 3.8. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{1} N}_{j R} (Ω) = Ω$ .

(2) If X₁ ⊆ X₂, then ${^{1} N}_{j R} (X_{1}) \subseteq {^{1} N}_{j R} (X_{2})$ .

(3) ${^{1} N}_{j R} (X_{1} \cap X_{2}) = {^{1} N}_{j R} (X_{1}) \cap {^{1} N}_{j R} (X_{2})$ .

(4) ${^{1} N}_{j R} (X_{1}) \cup {^{1} N}_{j R} (X_{2}) \subseteq {^{1} N}_{j R} (X_{1} \cup X_{2})$ .

(5) ${^{1} N}_{j R} (X) = ({^{1} N}_{j}^{R} (X^{c}))^{c}$ .

Proof. Similar to Proposition 3.6.

The equality of (4) in Proposition 3.8 does not hold as shown in the following example.

Example 3.9 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then ${^{1} N}_{j_{2} R} (X_{1}) = {d}$ and ${^{1} N}_{j_{2} R} (X_{2}) = {d}$ . Hence ${^{1} N}_{j_{2} R} (X_{1} \cup X_{2}) = {a, d}$ . Thus ${^{1} N}_{j_{2} R} (X_{1} \cup X_{2}) \neq {^{1} N}_{j_{2} R} (X_{1}) \cup {^{1} N}_{j_{2} R} (X_{2})$ .

Definition 3.10. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. A type 2 of rough set based on j-neighborhood (briefly, type 2 j-neighborhood rough set) of X ⊆ Ω in $(Ω, R, ζ_{j})$ or with respect to ζ_j is a pair $({^{2} N}_{j R} (X), {^{2} N}_{j}^{R} (X))$ , defined by ${^{2} N}_{j R} (X) = ⋃ {N_{j} (x) : N_{j} (x) \subseteq X}$ (called the type 2 lower approximation of X)

and ${^{2} N}_{j}^{R} (X) = [{^{2} N}_{jR} (X^{c})]^{c}$ (called the type 2 upper approximation of X).

Example 3.11 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X = {a, b}, then ${^{2} N}_{j_{2} R} (X) = \emptyset$ and ${^{2} N}_{j_{2}}^{R} (X) = {a, b, d}$ .

Remark 3.12. If we consider j = j₁ in Definition 3.10, then we obtain on Definition 2.3.

Definition 3.13. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space and X ⊆ Ω. A type 2 j-neighborhood boundary region and a type 2 j-neighborhood accuracy measure, denoted by $^{2} B_{j}$ and $^{2} A_{j}$ , respectively, defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $^{2} B_{j} (X) = {^{2} N}_{j}^{R} (X) - {^{2} N}_{j R} (X) .$

(2) $^{2} A_{j} (X) = \frac{| {^{2} N}_{j R} (X) |}{| {^{2} N}_{j}^{R} (X) |} .$

Example 3.14 (Continued from Examples 2.6 and 3.11). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). The type 2 j₂-boundary and the type 2 j₂-accuracy, respectively, are computed as $^{2} B_{j_{2}} (X) = {a, b, d}$ and $^{2} A_{j_{2}} (X) = 0$ .

Proposition 3.15. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{2} N}_{j}^{R} (Ω) = Ω$ .

(2) $X \subseteq {^{2} N}_{j}^{R} (X)$ .

(3) ${^{2} N}_{j R} (X) \subseteq {^{2} N}_{j}^{R} (X)$ .

(4) If X₁ ⊆ X₂, then ${^{2} N}_{j}^{R} (X_{1}) \subseteq {^{2} N}_{j}^{R} (X_{2})$ .

(5) ${^{2} N}_{j}^{R} (X_{1} \cup X_{2}) = {^{2} N}_{j}^{R} (X_{1}) \cup {^{2} N}_{j}^{R} (X_{2})$ .

(6) ${^{2} N}_{j}^{R} (X) = ({^{2} N}_{j R} (X^{c}))^{c}$ .

(7) ${^{2} N}_{j}^{R} ({^{2} N}_{j}^{R} (X)) = {^{2} N}_{j}^{R} (X)$ .

Proof. (1)-(3) Follows from Definition 3.10.

(4) By Definition 3.10 and if X₁ ⊆ X₂, then ${^{2} N}_{j}^{R} (X_{1}) = [{^{2} N}_{jR} ({X_{1}}^{c})]^{c} = ⋂ {{[N}_{j} (x)]^{c} : [N_{j} (x) \subseteq {X_{1}}^{c}]^{c}}$ $\subseteq ⋂ {{[N}_{j} (x)]^{c} : [N_{j} (x) \subseteq {X_{2}}^{c}]^{c}} = {^{2} N}_{j}^{R} (X_{2}) .$

(5) Follows from (4) and by Definition 3.10.

(6) Follows from Definition 3.10.

(7) Follows from (2) and Definition 3.10.

Proposition 3.16. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{2} N}_{j R} (\emptyset) = \emptyset$ .

(2) ${^{2} N}_{j R} (X) \subseteq X$ .

(3) If X₁ ⊆ X₂, then ${^{2} N}_{j R} (X_{1}) \subseteq {^{2} N}_{j R} (X_{2})$ .

(4) ${^{2} N}_{j R} (X_{1}) \cup {^{2} N}_{j R} (X_{2}) \subseteq {^{2} N}_{j R} (X_{1} \cup X_{2})$ .

(5) ${^{2} N}_{j R} (X) = ({^{2} N}_{j}^{R} (X^{c}))^{c}$ .

(6) ${^{2} N}_{j R} ({^{2} N}_{j R} (X)) = {^{2} N}_{j R} (X)$ .

Proof. Similar to Proposition 3.15.

The equality of (4) in Proposition 3.16 does not hold as shown in the following example.

Example 3.17 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then ${^{2} N}_{j_{2} R} (X_{1}) = \emptyset$ and ${^{2} N}_{j_{2} R} (X_{2}) = \emptyset$ . Hence ${^{2} N}_{j_{2} R} (X_{1} \cup X_{2}) = {a}$ . Thus ${^{2} N}_{j_{2} R} (X_{1} \cup X_{2}) \neq {^{2} N}_{j_{2} R} (X_{1}) \cup {^{2} N}_{j_{2} R} (X_{2})$ .

Definition 3.18. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. A type 3 of rough set based on j-neighborhood (briefly, type 3 j-neighborhood rough set) of X ⊆Ω in $(Ω, R, ζ_{j})$ or with respect to ζ_j is a pair $({^{3} N}_{j R} (X), {^{3} N}_{j}^{R} (X))$ , defined by ${^{3} N}_{j R} (X) = [{^{3} N}_{j}^{R} (X^{c})]^{c}$ (called the type 3 lower approximation of X) and

${^{3} N}_{j}^{R} (X) = ⋃ {N_{j} (x) : N_{j} (x) \cap X \neq \emptyset} .$ (called the type 3 upper approximation of X).

Example 3.19 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X = {a, b}, then ${^{3} N}_{j_{2} R} (X) = \emptyset$ and ${^{3} N}_{j_{2}}^{R} (X) = Ω$ .

Remark 3.20. If we consider j = j₁ in Definition 3.18, then we obtain on Definition 2.4.

Definition 3.21. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space and X ⊆ Ω. A type 3 j-neighborhood boundary region and a type 3 j-neighborhood accuracy measure, denoted by $^{3} B_{j}$ and $^{3} A_{j}$ , respectively, defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $^{3} B_{j} (X) = {^{3} N}_{j}^{R} (X) - {^{3} N}_{j R} (X) .$

(2) $^{3} A_{j} (X) = \frac{| {^{3} N}_{j R} (X) |}{| {^{3} N}_{j}^{R} (X) |} .$

Example 3.22 (Continued from Examples 2.6 and 3.19). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). The type 3 j₂-boundary and the type 3 j₂-accuracy are computed as $^{3} B_{j_{2}} (X) = Ω$ and $^{3} A_{j_{2}} (X) = 0$ .

Proposition 3.23. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{3} N}_{j}^{R} (\emptyset) = \emptyset$ .

(2) ${^{3} N}_{j}^{R} ({^{3} N}_{j R} (X)) \subseteq X$ .

(3) If X₁ ⊆ X₂, then ${^{3} N}_{j}^{R} (X_{1}) \subseteq {^{3} N}_{j}^{R} (X_{2})$ .

(4) ${^{3} N}_{j}^{R} (X_{1} \cap X_{2}) \subseteq {^{3} N}_{j}^{R} (X_{1}) \cap {^{3} N}_{j}^{R} (X_{2})$ .

(5) ${^{3} N}_{j}^{R} (X_{1} \cup X_{2}) = {^{3} N}_{j}^{R} (X_{1}) \cup {^{3} N}_{j}^{R} (X_{2})$ .

(6) ${^{3} N}_{j}^{R} (X) = ({^{3} N}_{j R} (X^{c}))^{c}$ .

(7) ${^{3} N}_{j}^{R} ({X_{1}}^{c} \cup X_{2}) \subseteq ({^{3} N}_{j R} (X_{1}))^{c} \cup {^{3} N}_{j R} (X_{2})$ .

Proof. We only prove (7).

(7) By contradiction, let $x \notin ({^{3} N}_{j R} (X_{1}))^{c} \cup {^{3} N}_{j R} (X_{2})$ . Then $x \notin ({^{3} N}_{j R} (X_{1}))^{c}$ and $x \notin {^{3} N}_{j R} (X_{2})$ . Thus $x \in {^{3} N}_{j R} (X_{1})$ and $x \notin {^{3} N}_{j R} (X_{2})$ . Hence, $N_{j} (x) \subseteq X_{1}$ and $N_{j} (x) ⊈ X_{2}$ . So $N_{j} (x) ⊈ {X_{1}}^{c}$ and $N_{j} (x) \subseteq X_{2}$ . Therefore $N_{j} (x) ⊈ ({X_{1}}^{c} \cup X_{2})$ , then $x \notin N_{j} ({X_{1}}^{c} \cup X_{2})$ , i.e., ${^{3} N}_{j}^{R} ({X_{1}}^{c} \cup X_{2}) \subseteq ({^{3} N}_{j R} (X_{1}))^{c} \cup {^{3} N}_{j R} (X_{2})$ .

The equality of (4) in Proposition 3.23 does not hold as shown in the following example.

Example 3.24 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then ${^{3} N}_{j_{2}}^{R} (X_{1}) = {a, d}$ and ${^{3} N}_{j_{2}}^{R} (X_{2}) = {a, d}$ . Hence ${^{3} N}_{j_{2}}^{R} (X_{1} \cap X_{2}) = \emptyset$ . Thus ${^{3} N}_{j_{2}}^{R} (X_{1} \cap X_{2}) \neq {^{3} N}_{j_{2}}^{R} (X_{1}) \cap {^{3} N}_{j_{2}}^{R} (X_{2})$ .

Proposition 3.25. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{3} N}_{j R} (Ω) = Ω$ .

(2) $X \subseteq {^{3} N}_{j R} ({^{3} N}_{j}^{R} (X))$ .

(3) If X₁ ⊆ X₂, then ${^{3} N}_{j R} (X_{1}) \subseteq {^{3} N}_{j R} (X_{2})$ .

(4) ${^{3} N}_{j R} (X_{1} \cap X_{2}) = {^{3} N}_{j R} (X_{1}) \cap {^{3} N}_{j R} (X_{2})$ .

(5) ${^{3} N}_{j R} (X_{1}) \cup {^{3} N}_{j R} (X_{2}) \subseteq {^{3} N}_{j R} (X_{1} \cup X_{2})$ .

(6) ${^{3} N}_{j R} (X) = ({^{3} N}_{j}^{R} (X^{c}))^{c}$ .

Proof. Similar to Proposition 3.23.

The equality of (5) in Proposition 3.25 does not hold as shown in the following example.

Example 3.26 (Continued from Example 2.6). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then ${^{3} N}_{j_{2} R} (X_{1}) = \emptyset$ and ${^{3} N}_{j_{2} R} (X_{2}) = \emptyset$ . Hence ${^{3} N}_{j_{2} R} (X_{1} \cup X_{2}) = {a, d}$ . Thus ${^{3} N}_{j_{2} R} (X_{1} \cup X_{2}) \neq {^{3} N}_{j_{2} R} (X_{1}) \cup {^{3} N}_{j_{2} R} (X_{2})$ .

4 Generalized three types of rough set models based on j-adhesion neighborhood space

In this section, we will extent Definition 2.5 to j-adhesion neighborhood (which there are some insufficiency in Definition 2.5, see Example 3.2) and also give another three types of rough set models based on j-adhesion neighborhood as indicated below.

Definition 4.1. Let $R$ be an arbitrary binary relation on Ω. The j-adhesion neighborhood of element x ∈ Ω are defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $P_{j_{1}} (x) = {y \in X : N_{j_{1}} (x) = N_{j_{1}} (y)}$ .

(2) $P_{j_{2}} (x) = {y \in X : N_{j_{2}} (x) = N_{j_{2}} (y)}$ .

(3) $P_{j_{3}} (x) = {y \in X : ⋂_{x \in N_{j_{1}} (y)} N_{j_{1}} (y) = ⋂_{y \in N_{j_{1}} (x)} N_{j_{1}} (x)}$ .

(4) $P_{j_{4}} (x) = {y \in X : ⋂_{x \in N_{j_{2}} (y)} N_{j_{2}} (y) = ⋂_{y \in N_{j_{2}} (x)} N_{j_{2}} (x)}$ .

(5) $P_{j_{5}} (x) = P_{j_{1}} (x) \cup P_{j_{2}} (x)$ .

(6) $P_{j_{6}} (x) = P_{j_{1}} (x) \cap P_{j_{2}} (x)$ .

(7) $P_{j_{7}} (x) = P_{j_{3}} (x) \cup P_{j_{4}} (x)$ .

(8) $P_{j_{8}} (x) = P_{j_{3}} (x) \cap P_{j_{4}} (x)$ .

According to above Definition 4.1, we call $(Ω, R, ζ_{j})$ a j-adhesion neighborhood space.

In the following example, we explain the above definition as:

Example 4.2 (Continued from Example 2.6). The j-adhesion neighborhood are computed by

(1) As j = j₁, we have $P_{j_{1}} (a) = {a, d}, P_{j_{1}} (b) = {b},$ $P_{j_{1}} (c) = {c}, P_{j_{1}} (d) = {a, d} .$

(2) As j = j₂, we have $P_{j_{2}} (a) = {a}, P_{j_{2}} (b) = {b},$ $P_{j_{2}} (c) = {c}, P_{j_{2}} (d) = {d} .$

(3) As j = j₃, we have $P_{j_{3}} (a) = {a, d}, P_{j_{3}} (b) = {b},$ $P_{j_{3}} (c) = {c}, P_{j_{3}} (d) = {a, d} .$

(4) As j = j₄, we have $P_{j_{4}} (a) = {a}, P_{j_{4}} (b) = {b},$ $P_{j_{4}} (c) = {c}, P_{j_{4}} (d) = {d} .$

(5) As j = j₅, we have $P_{j_{5}} (a) = {a, d}, P_{j_{5}} (b) = {b},$ $P_{j_{5}} (c) = {c}, P_{j_{5}} (d) = {a, d} .$

(6) As j = j₆, we have $P_{j_{6}} (a) = {a}, P_{j_{6}} (b) = {b},$ $P_{j_{6}} (c) = {c}, P_{j_{6}} (d) = {d} .$

(7) As j = j₇, we have $P_{j_{7}} (a) = {a, d}, P_{j_{7}} (b) = {b},$ $P_{j_{7}} (c) = {c}, P_{j_{7}} (d) = {a, d} .$

(8) As j = j₈, we have $P_{j_{8}} (a) = {a}, P_{j_{8}} (b) = {b},$ $P_{j_{8}} (c) = {c}, P_{j_{8}} (d) = {d} .$

Definition 4.3. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. A type 4 of rough set based on j-adhesion neighborhood (briefly, type 4 j-adhesion neighborhood rough set) of X ⊆ Ω in $(Ω, R, ζ_{j})$ or with respect to ζ_j is a pair $({^{4} N}_{j R} (X), {^{4} N}_{j}^{R} (X))$ , defined by ${^{4} N}_{j R} (X) = {x \in Ω : P_{j} (x) \subseteq X}$ (called the type 4 lower approximation of X)

and ${^{4} N}_{j}^{R} (X) = {x \in Ω : P_{j} (x) \cap X \neq \emptyset}$ (called the type 4 upper approximation of X).

As noticed from the above Definition 4.3, it is more accurate than the Definition 3.1 and has handled some of its problems as referred in Propositions 4.7 and 4.9.

Example 4.4 (Continued from Example 2.6). If X = {a, b}. Then we have

(1) As j = j₁, we have ${^{4} N}_{j_{1} R} (X) = {b}, {^{4} N}_{j_{1}}^{R} (X) = {a, b, d} .$

(2) As j = j₂, we have ${^{4} N}_{j_{2} R} (X) = {a, b}, {^{4} N}_{j_{2}}^{R} (X) = {a, b} .$

(3) As j = j₃, we have ${^{4} N}_{j_{3} R} (X) = {b}, {^{4} N}_{j_{3}}^{R} (X) = {a, b, d} .$

(4) As j = j₄, we have ${^{4} N}_{j_{4} R} (X) = {a, b}, {^{4} N}_{j_{4}}^{R} (X) = {a, b} .$

(5) As j = j₅, we have ${^{4} N}_{j_{8} R} (X) = {b}, {^{4} N}_{j_{5}}^{R} (X) = {a, b, d} .$

(6) As j = j₆, we have ${^{4} N}_{j_{6} R} (X) = {a, b}, {^{4} N}_{j_{6}}^{R} (X) = {a, b} .$

(7) As j = j₇, we have ${^{4} N}_{j_{7} R} (X) = {b}, {^{4} N}_{j_{7}}^{R} (X) = {a, b, d} .$

(8) As j = j₈, we have ${^{4} N}_{j_{8} R} (X) = {a, b}, {^{4} N}_{j_{8}}^{R} (X) = {a, b} .$

Definition 4.5. Let $(Ω, R, ζ_{j})$ be a j-neighborhood space and X ⊆ Ω. A type 4 j-adhesion neighborhood boundary region and a type 4 j-adhesion neighborhood accuracy measure, denoted by $^{4} B_{j}$ and $^{4} A_{j}$ , respectively, defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $^{4} B_{j} (X) = {^{4} N}_{j}^{R} (X) - {^{4} N}_{j R} (X) .$

(2) $^{4} A_{j} (X) = \frac{| {^{4} N}_{j R} (X) |}{| {^{4} N}_{j}^{R} (X) |} .$

Example 4.6. (Continued from Examples 2.6, 4.2, and 4.4). Take j = j₂ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈): are similarly). The type 4 j₂-boundary and the type 4 j₂-accuracy, respectively, are computed as $^{4} B_{j_{2}} (X) = \emptyset$ and $^{4} A_{j_{2}} (X) = 1$ .

Proposition 4.7. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{4} N}_{j R} (X) = ({^{4} N}_{j}^{R} (X^{c}))^{c}$ .

(2) ${^{4} N}_{j R} (\emptyset) = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{4} N}_{j R} (X_{1}) \subseteq {^{4} N}_{j R} (X_{2})$ .

(4) ${^{4} N}_{j R} (X_{1} \cap X_{2}) = {^{4} N}_{j R} (X_{1}) \cap {^{4} N}_{j R} (X_{2})$ .

(5) ${^{4} N}_{j R} (X_{1}) \cup {^{4} N}_{j R} (X_{2}) \subseteq {^{4} N}_{j R} (X_{1} \cup X_{2})$ .

(6) ${^{4} N}_{j R} (Ω) = Ω$ .

(7) ${^{4} N}_{j R} (X) \subseteq X$ .

(8) $X \subseteq {^{4} N}_{j R} ({^{4} N}_{j}^{R} (X))$ .

(9) ${^{4} N}_{j R} ({^{4} N}_{j R} (X)) = {^{4} N}_{j R} (X)$ .

(10) ${^{4} N}_{j}^{R} (X) = {^{4} N}_{j R} ({^{4} N}_{j}^{R} (X))$ .

Proof. (1) If $x \in {^{4} N}_{j R} (X)$ for every x ∈ X, there exists $P_{j} (x) \subseteq X$ . Then for every x ∈ Ω - [Ω - X], there exists $P_{j} (x)$ such that $P_{j} (x) \cap [Ω - X] = \emptyset$ . So $x \notin {^{4} N}_{j}^{R} [Ω - X]$ , $x \in Ω - [{^{4} N}_{j}^{R} (Ω - X)]$ . Therefore, ${^{4} N}_{j R} (X) = Ω - [{^{4} N}_{j}^{R} (Ω - X)]$ $= ({^{4} N}_{j}^{R} (X^{c}))^{c}$ .

(2) ${^{4} N}_{j R} (\emptyset) = {x \in Ω : P_{j} (x) \subseteq \emptyset} = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{4} N}_{j R} (X_{1}) = {x \in Ω : P_{j} (x) \subseteq X_{1}}$ $\subseteq {x \in Ω : P_{j} (x) \subseteq X_{2}}$ $= {^{4} N}_{j R} (X_{2})$ .

(4) It is sufficient to show ${^{4} N}_{j R} (X_{1} \cap X_{2})$ $= {^{4} N}_{j R} (X_{1}) \cap {^{4} N}_{j R} (X_{2})$ . By Definition 4.3, we have ${^{4} N}_{j R} (X_{1} \cap X_{2})$ $= {x \in Ω : P_{j} (x) \subseteq (X_{1} \cap X_{2})}$ . Since (X₁ ∩ X₂) ⊆ X₁ and (X₁ ∩ X₂) ⊆ X₂, then $P_{j} (x) \subseteq X_{1}$ and $P_{j} (x) \subseteq X_{2}$ . Thus by (3), we have ${^{4} N}_{j R} (X_{1} \cap X_{2}) \subseteq {^{4} N}_{j R} (X_{1})$ and ${^{4} N}_{j R} (X_{1} \cap X_{2}) \subseteq {^{4} N}_{j R} (X_{2})$ . Therefore, ${^{4} N}_{j R} (X_{1}) \cap {^{4} N}_{j R} (X_{2})$ $= {x \in Ω : P_{j} (x) \subseteq X_{1}}$ and $= {x \in Ω : P_{j} (x) \subseteq X_{2}}$ $= {x \in Ω : P_{j} (x) \subseteq (X_{1} \cap X_{2})}$ $= {^{4} N}_{j R} (X_{1} \cap X_{2})$ $= {^{4} N}_{j R} (X_{1}) \cap {^{4} N}_{j R} (X_{2})$ .

(5) The proof is similar to (4).

(6) Follows directly by (2) and Definition 4.3.

(7) and (8) Follows from Definition 4.3.

(9) Follows from (7) and Definition 4.3.

(10) Follows from (8) and Definition 4.3.

The equality of (5) in Proposition 4.7 does not hold as shown in the following example.

Example 4.8 (Continued from Example 4.2). Take j = j₁ (and also (j = j₂, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then $^{4} N_{j_{1} R} (X_{1}) = \emptyset$ and ${^{4} N}_{j_{1} R} (X_{2}) = \emptyset$ . Hence ${^{4} N}_{j_{1} R} (X_{1}) \cup {^{4} N}_{j_{1} R} (X_{2}) = \emptyset$ . But ${^{4} N}_{j_{1} R} (X_{1} \cup X_{2}) = {a, d}$

${^{1} N}_{j R} (X) ⊈ {^{1} N}_{j}^{R} (X)) .$ . Therefore ${^{4} N}_{j_{1} R} (X_{1}) \cup {^{4} N}_{j_{1} R} (X_{2}) \neq {^{4} N}_{j_{1} R} (X_{1} \cup X_{2})$ .

Proposition 4.9. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{4} N}_{j}^{R} (X) = ({^{4} N}_{j R} (X^{c}))^{c}$ .

(2) ${^{4} N}_{j}^{R} (\emptyset) = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{4} N}_{j}^{R} (X_{1}) \subseteq {^{4} N}_{j}^{R} (X_{2})$ .

(4) ${^{4} N}_{j}^{R} (X_{1} \cap X_{2}) \subseteq {^{4} N}_{j}^{R} (X_{1}) \cap {^{4} N}_{j}^{R} (X_{2})$ .

(5) ${^{4} N}_{j}^{R} (X_{1} \cup X_{2}) = {^{4} N}_{j}^{R} (X_{1}) \cup {^{4} N}_{j}^{R} (X_{2})$ .

(6) ${^{4} N}_{j}^{R} (Ω) = Ω$ .

(7) $X \subseteq {^{4} N}_{j}^{R} (X)$ .

(8) ${^{4} N}_{j}^{R} ({^{4} N}_{j R} (X)) \subseteq X$ .

(9) ${^{4} N}_{j}^{R} ({^{4} N}_{j}^{R} (X)) = {^{4} N}_{j}^{R} (X)$ .

(10) ${^{4} N}_{j R} (X) = {^{4} N}_{j}^{R} ({^{4} N}_{j R} (X))$ .

(11) ${^{4} N}_{j R} ({X_{1}}^{c} \cup X_{2}) \subseteq ({^{4} N}_{j R} (X_{1}))^{c} \cup {^{4} N}_{j R} (X_{2})$ .

(12) ${^{4} N}_{j R} (X) \subseteq {^{4} N}_{j}^{R} (X)$ .

Proof. The proof analogues to Proposition 4.7.

The equality of (4) and (11) in Proposition 4.9 does not hold as shown in the following example.

Example 4.10 (Continued from Example 4.2). Take j = j₁ (and also (j = j₂, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a, b} and X₂ = {b, d}, then ${^{4} N}_{j_{1}}^{R} (X_{1} \cap X_{2}) = {b}$ . Hence ${^{4} N}_{j_{1}}^{R} (X_{1}) \cap {^{4} N}_{j_{1}}^{R} (X_{2}) = {a, b, d}$ . Thus ${^{4} N}_{j_{1}}^{R} (X_{1}) \cap {^{4} N}_{j_{1}}^{R} (X_{2}) \neq {^{4} N}_{j_{1}}^{R} (X_{1} \cap X_{2})$ . Also, if X₁ = {a, b} and X₂ = {b, d}, then ${^{4} N}_{j_{1} R} (X_{1}) = {b}$ and ${^{4} N}_{j_{1} R} (X_{2}) = {b}$ . Then ${^{4} N}_{j_{1} R} ({X_{1}}^{c} \cup X_{2}) = {b, c}$ . But $({^{4} N}_{j_{1} R} (X_{1}))^{c} \cup {^{4} N}_{j_{1} R} (X_{2}) = Ω$ . Therefore ${^{4} N}_{j_{1} R} ({X_{1}}^{c} \cup X_{2}) \neq ({^{4} N}_{j_{1} R} (X_{1}))^{c} \cup {^{4} N}_{j_{1} R} (X_{2})$ .

Definition 4.11. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. A type 5 of rough set based on j-adhesion neighborhood (briefly, type 5 j-adhesion neighborhood rough set) of X ⊆ Ω in $(Ω, R, ζ_{j})$ or with respect to ζ_j is a pair $({^{5} N}_{j R} (X), {^{5} N}_{j}^{R} (X))$ , defined by ${^{5} N}_{j R} (X) = ⋃ {P_{j} (x) : P_{j} (x) \subseteq X}$ (called the type 5 lower approximation of X) and ${^{5} N}_{j}^{R} (X) = [{^{5} N}_{j R} (X^{c})]^{c}$ (called the type 5 upper approximation of X).

As noticed from the above Definition 4.11, it is more accurate than the Definition 3.10 and has handled some of its problems as referred in Propositions 4.15 and 4.17.

Example 4.12 (Continued from Example 4.2). Take j = j₁ (and also (j = j₂, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X = {b, c}, then ${^{5} N}_{j_{1} R} (X) = {b, c}$ and ${^{5} N}_{j_{1}}^{R} (X) = {b, c}$ .

Definition 4.13. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space and X ⊆ Ω. A type 5 j-adhesion neighborhood boundary region and a type 5 j-adhesion neighborhood accuracy measure based on j-adhesion neighborhood, denoted by $^{5} B_{j}$ and $^{5} A_{j}$ , respectively, defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $^{5} B_{j} (X) = {^{5} N}_{j}^{R} (X) - {^{5} N}_{j R} (X) .$

(2) $^{5} A_{j} (X) = \frac{| {^{5} N}_{j R} (X) |}{| {^{5} N}_{j}^{R} (X) |} .$

Example 4.14. (Continued from Examples 2.6, 4.2, and 4.12). Take j = j₁ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). The type 5 j₁-boundary and the type 5 j₁-accuracy, respectively, are computed as $^{5} B_{j_{1}} (X) = \emptyset$ and $^{5} A_{j_{1}} (X) = 1$ .

Proposition 4.15. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{5} N}_{j R} (X) = ({^{5} N}_{j}^{R} (X^{c}))^{c}$ .

(2) ${^{5} N}_{j R} (\emptyset) = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{5} N}_{j R} (X_{1}) \subseteq {^{5} N}_{j R} (X_{2})$ .

(4) ${^{5} N}_{j R} (X_{1} \cap X_{2}) = {^{5} N}_{j R} (X_{1}) \cap {^{5} N}_{j R} (X_{2})$ .

(5) ${^{5} N}_{j R} (X_{1}) \cup {^{5} N}_{j R} (X_{2}) \subseteq {^{5} N}_{j R} (X_{1} \cup X_{2})$ .

(6) ${^{5} N}_{j R} (Ω) = Ω$ .

(7) ${^{5} N}_{j R} (X) \subseteq X$ .

(8) $X \subseteq {^{5} N}_{j R} ({^{5} N}_{j}^{R} (X))$ .

(9) ${^{5} N}_{j R} ({^{5} N}_{j R} (X)) = {^{5} N}_{j R} (X)$ .

(10) ${^{5} N}_{j}^{R} (X) = {^{5} N}_{j R} ({^{5} N}_{j}^{R} (X))$ .

Proof. (1) If $x \in {^{5} N}_{j R} (X)$ for every x ∈ X, there exists $P_{j} (x) \subseteq X$ . Then for every x ∈ Ω - [Ω - X], there exists $P_{j} (x)$ such that $P_{j} (x) \cap [Ω - X] = \emptyset$ . So $x \notin {^{5} N}_{j}^{R} [Ω - X]$ , $x \in Ω - [{^{5} N}_{j}^{R} (Ω - X)]$ . Therefore, ${^{5} N}_{j R} (X) = Ω - [{^{5} N}_{j}^{R} (Ω - X)]$ $= ({^{5} N}_{j}^{R} (X^{c}))^{c}$ .

(2) ${^{5} N}_{j R} (\emptyset) = ⋃ {P_{j} (x) : P_{j} (x) \subseteq \emptyset} = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{5} N}_{j R} (X_{1}) = ⋃ {P_{j} (x) : P_{j} (x) \subseteq X_{1}}$ $\subseteq ⋃ {P_{j} (x) : P_{j} (x) \subseteq X_{2}}$ $= {^{5} N}_{j R} (X_{2})$ .

(4) It is sufficient to show ${^{5} N}_{j R} (X_{1} \cap X_{2})$ $= {^{5} N}_{j R} (X_{1}) \cap {^{5} N}_{j R} (X_{2})$ . By Definition 4.11, we have ${^{5} N}_{j R} (X_{1} \cap X_{2})$ $= ⋃ {P_{j} (x) : P_{j} (x) \subseteq (X_{1} \cap X_{2})}$ . Since (X₁ ∩ X₂) ⊆ X₁ and (X₁ ∩ X₂) ⊆ X₂, then $P_{j} (x) \subseteq X_{1}$ and $P_{j} (x) \subseteq X_{2}$ . Thus by (3), we have ${^{5} N}_{j R} (X_{1} \cap X_{2}) \subseteq {^{5} N}_{j R} (X_{1})$ and ${^{5} N}_{j R} (X_{1} \cap X_{2}) \subseteq {^{5} N}_{j R} (X_{2})$ . Therefore, ${^{5} N}_{j R} (X_{1}) \cap {^{5} N}_{j R} (X_{2})$ $= ⋃ {P_{j} (x) : P_{j} (x) \subseteq X_{1}}$ and $= ⋃ {P_{j} (x) : P_{j} (x) \subseteq X_{2}}$ $= ⋃ {P_{j} (x) : P_{j} (x) \subseteq (X_{1} \cap X_{2})}$ $= {^{5} N}_{j R} (X_{1} \cap X_{2})$ $= {^{5} N}_{j R} (X_{1}) \cap {^{5} N}_{j R} (X_{2})$ .

(5) The proof is similar to (4).

(6) Follows directly by (2) and Definition 4.11.

(7) and (8) Follow from Definition 4.11.

(9) Follows from (7) and Definition 4.11.

(10) Follows from (8) and Definition 4.11.

The equality of (5) in Proposition 4.15 does not hold as shown in the following example.

Example 4.16 (Continued from Example 4.2). Take j = j₁ (and also (j = j₂, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then ${^{5} N}_{j_{1} R} (X_{1}) = \emptyset$ and ${^{5} N}_{j_{1} R} (X_{2}) = \emptyset$ . Then ${^{5} N}_{j_{1} R} (X_{1}) \cup {^{5} N}_{j_{1} R} (X_{2}) = \emptyset$ . But ${^{5} N}_{j_{1} R} (X_{1} \cup X_{2}) = {a, d}$ . Therefore ${^{5} N}_{j_{1} R} (X_{1})$ $\cup {^{5} N}_{j_{1} R} (X_{2}) \neq {^{5} N}_{j_{1} R} (X_{1} \cup X_{2})$ .

Proposition 4.17. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{5} N}_{j}^{R} (X) = ({^{5} N}_{j R} (X^{c}))^{c}$ .

(2) ${^{5} N}_{j}^{R} (\emptyset) = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{5} N}_{j}^{R} (X_{1}) \subseteq {^{5} N}_{j}^{R} (X_{2})$ .

(4) ${^{5} N}_{j}^{R} (X_{1} \cap X_{2}) \subseteq {^{5} N}_{j}^{R} (X_{1}) \cap {^{5} N}_{j}^{R} (X_{2})$ .

(5) ${^{5} N}_{j}^{R} (X_{1} \cup X_{2}) = {^{5} N}_{j}^{R} (X_{1}) \cup {^{5} N}_{j}^{R} (X_{2})$ .

(6) ${^{5} N}_{j}^{R} (Ω) = Ω$ .

(7) $X \subseteq {^{5} N}_{j}^{R} (X)$ .

(8) ${^{5} N}_{j}^{R} ({^{5} N}_{j R} (X)) \subseteq X$ .

(9) ${^{5} N}_{j}^{R} ({^{5} N}_{j}^{R} (X)) = {^{5} N}_{j}^{R} (X)$ .

(10) ${^{5} N}_{j R} (X) = {^{5} N}_{j}^{R} ({^{5} N}_{j R} (X))$ .

(11) ${^{5} N}_{j R} ({X_{1}}^{c} \cup X_{2}) \subseteq ({^{5} N}_{j R} (X_{1}))^{c} \cup {^{5} N}_{j R} (X_{2})$ .

(12) ${^{5} N}_{j R} (X) \subseteq {^{5} N}_{j}^{R} (X)$ .

Proof. The proof analogues to Proposition 4.15.

The equality of (4) and (11) in Proposition 4.17 does not hold as shown in the following example.

Example 4.18 (Continued from Example 4.2). Take j = j₁ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a, b} and X₂ = {b, d}, then ${^{5} N}_{j_{1}}^{R} (X_{1} \cap X_{2}) = {b}$ . Hence ${^{5} N}_{j_{1}}^{R} (X_{1}) \cap {^{5} N}_{j_{1}}^{R} (X_{2}) = {a, b, d}$ . Thus ${^{5} N}_{j_{1}}^{R} (X_{1}) \cap {^{5} N}_{j_{1}}^{R} (X_{2}) \neq {^{5} N}_{j_{1}}^{R} (X_{1} \cap X_{2})$ . Also, if X₁ = {a, b} and X₂ = {b, d}, then ${^{5} N}_{j_{1} R} (X_{1}) = {b}$ and ${^{5} N}_{j_{1} R} (X_{2}) = {b}$ . Hence ${^{5} N}_{j_{1} R} ({X_{1}}^{c} \cup X_{2}) = {b, c}$ . But $({^{5} N}_{j_{1} R} (X_{1}))^{c} \cup {^{5} N}_{j_{1} R} (X_{2}) = Ω$ . Therefore ${^{5} N}_{j_{1} R} ({X_{1}}^{c} \cup X_{2}) \neq ({^{5} N}_{j_{1} R} (X_{1}))^{c} \cup {^{5} N}_{j_{1} R} (X_{2})$ .

Definition 4.19. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. A type 6 of rough set based on j-adhesion neighborhood (briefly, type 6 j-adhesion neighborhood rough set) of X ⊆ Ω in $(Ω, R, ζ_{j})$ or with respect to ζ_j is a pair $({^{6} N}_{j R} (X), {^{6} N}_{j}^{R} (X))$ , defined by ${^{6} N}_{j R} (X) = [{^{6} N}_{j}^{R} (X^{c})]^{c}$ (called the type 6 lower approximation of X) and ${^{6} N}_{j}^{R} (X) = ⋃ {P_{j} (x) : P_{j} (x) \cap X \neq \emptyset}$ (called the type 6 upper approximation of X).

As noticed from the above Definition 4.19, it is more accurate than the Definition 3.18 and has handled some of its problems as referred in Propositions 4.23 and 4.25.

Example 4.20 (Continued from Example 4.2). Take j = j₁ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X = {b, c}, then ${^{6} N}_{j_{1} R} (X) = {b, c}$ and ${^{6} N}_{j_{1}}^{R} (X) = {b, c}$ .

Definition 4.21. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space and X ⊆ Ω. A type 6 j-adhesion neighborhood boundary region and a type 6 j-adhesion neighborhood accuracy measure, denoted by $^{6} B_{j}$ and $^{6} A_{j}$ , respectively, defined as follows (j = j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈):

(1) $^{6} B_{j} (X) = {^{6} N}_{j}^{R} (X) - {^{6} N}_{j R} (X) .$

(2) $^{6} A_{j} (X) = \frac{| {^{6} N}_{j R} (X) |}{| {^{6} N}_{j}^{R} (X) |} .$

Example 4.22. (Continued from Examples 2.6, 4.2, and 4.20). Take j = j₁ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). The type 6 j₁-boundary and the type 6 j₁-accuracy, respectively, are computed as $^{6} B_{j_{1}} (X) = \emptyset$ and $^{6} A_{j_{1}} (X) = 1$ .

Proposition 4.23. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{6} N}_{j R} (X) = ({^{6} N}_{j}^{R} (X^{c}))^{c}$ .

(2) ${^{6} N}_{j R} (\emptyset) = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{6} N}_{j R} (X_{1}) \subseteq {^{6} N}_{j R} (X_{2})$ .

(4) ${^{6} N}_{j R} (X_{1} \cap X_{2}) = {^{6} N}_{j R} (X_{1}) \cap {^{6} N}_{j R} (X_{2})$ .

(5) ${^{6} N}_{j R} (X_{1}) \cup {^{6} N}_{j R} (X_{2}) \subseteq {^{6} N}_{j R} (X_{1} \cup X_{2})$ .

(6) ${^{6} N}_{j R} (Ω) = Ω$ .

(7) ${^{6} N}_{j R} (X) \subseteq X$ .

(8) $X \subseteq {^{6} N}_{j R} ({^{6} N}_{j}^{R} (X))$ .

(9) ${^{6} N}_{j R} ({^{6} N}_{j R} (X)) = {^{6} N}_{j R} (X)$ .

(10) ${^{6} N}_{j}^{R} (X) = {^{6} N}_{j R} ({^{6} N}_{j}^{R} (X))$ .

Proof. (1) If $x \in {^{6} N}_{j R} (X)$ for every x ∈ X, there exists $P_{j} (x) \subseteq X$ . Then for every x ∈ Ω - [Ω - X], there exists $P_{j} (x)$ such that $P_{j} (x) \cap [Ω - X] = \emptyset$ . So $x \notin {^{6} N}_{j}^{R} [Ω - X]$ , $x \in Ω - [{^{6} N}_{j}^{R} (Ω - X)]$ . Therefore, ${^{6} N}_{j R} (X) = Ω - [{^{6} N}_{j}^{R} (Ω - X)]$ $= ({^{6} N}_{j}^{R} (X^{c}))^{c}$ .

(2) ${^{6} N}_{j R} (\emptyset) = ⋂ {P_{j} (x) : P_{j} (x) \cap Ω \neq \emptyset}^{c} = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{6} N}_{j R} (X_{1}) = [⋃ {P_{j} (x) : P_{j} (x) \cap (X_{1})^{c} \neq \emptyset}]^{c}$ $\subseteq [⋃ {P_{j} (x) : P_{j} (x) \cap (X_{2})^{c} \neq \emptyset}]^{c}$ $= {^{6} N}_{j R} (X_{2})$ .

(4) It is sufficient to show ${^{6} N}_{j R} (X_{1} \cap X_{2})$ $= {^{6} N}_{j R} (X_{1}) \cap {^{6} N}_{j R} (X_{2})$ . By Definition 4.19, we have ${^{6} N}_{j R} (X_{1} \cap X_{2})$ $= ⋂ [{P_{j} (x) : P_{j} (x) \cap (X_{1} \cap X_{2})^{c} \neq \emptyset}]^{c}$ . Since (X₁ ∩ X₂) ⊆ X₁ and (X₁ ∩ X₂) ⊆ X₂, then $P_{j} (x) \subseteq X_{1}$ and $P_{j} (x) \subseteq X_{2}$ . Thus by (3), we have ${^{6} N}_{j R} (X_{1} \cap X_{2}) \subseteq {^{6} N}_{j R} (X_{1})$ and ${^{6} N}_{j R} (X_{1} \cap X_{2}) \subseteq {^{6} N}_{j R} (X_{2})$ . Therefore, ${^{6} N}_{j R} (X_{1}) \cap {^{6} N}_{j R} (X_{2})$ $= ⋂ [{P_{j} (x) : P_{j} (x) \cap X_{1}^{c} \neq \emptyset}]^{c}$ and $= ⋂ [{P_{j} (x) : P_{j} (x) \cap X_{2}^{c} \neq \emptyset}]^{c}$ $= ⋂ [{P_{j} (x) : P_{j} (x) \cap (X_{1} \cap X_{2})^{c} \neq \emptyset}]^{c}$ $= {^{6} N}_{j R} (X_{1} \cap X_{2}) = {^{6} N}_{j R} (X_{1}) \cap {^{6} N}_{j R} (X_{2})$ .

(5) The proof is similar to (4).

(6) Follows directly by (2) and Definition 4.19.

(7) and (8) Follows from Definition 4.19.

(9) Follows from (7) and Definition 4.19.

(10) Follows From (8) and Definition 4.19.

The equality of (5) in Proposition 4.23 does not hold as shown in the following example.

Example 4.24 (Continued from Example 4.2). Take j = j₁ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a} and X₂ = {d}, then ${^{6} N}_{j_{1} R} (X_{1}) = \emptyset$ and ${^{6} N}_{j_{1} R} (X_{2}) = \emptyset$ . Then ${^{6} N}_{j_{1} R} (X_{1}) \cup {^{6} N}_{j_{1} R} (X_{2}) = \emptyset$ . But ${^{6} N}_{j_{1} R} (X_{1} \cup X_{2}) = {a, d}$ . Therefore ${^{6} N}_{j_{1} R} (X_{1}) \cup {^{6} N}_{j_{1} R} (X_{2}) \neq {^{6} N}_{j_{1} R} (X_{1} \cup X_{2})$ .

Proposition 4.25. Let $(Ω, R, ζ_{j})$ be a j-adhesion neighborhood space. The following conditions hold for every X, X₁, X₂ ⊆ Ω:

(1) ${^{6} N}_{j}^{R} (X) = ({^{6} N}_{j R} (X^{c}))^{c}$ .

(2) ${^{6} N}_{j}^{R} (\emptyset) = \emptyset$ .

(3) If X₁ ⊆ X₂, then ${^{6} N}_{j}^{R} (X_{1}) \subseteq {^{6} N}_{j}^{R} (X_{2})$ .

(4) ${^{6} N}_{j}^{R} (X_{1} \cap X_{2}) \subseteq {^{6} N}_{j}^{R} (X_{1}) \cap {^{6} N}_{j}^{R} (X_{2})$ .

(5) ${^{6} N}_{j}^{R} (X_{1} \cup X_{2}) = {^{6} N}_{j}^{R} (X_{1}) \cup {^{6} N}_{j}^{R} (X_{2})$ .

(6) ${^{6} N}_{j}^{R} (Ω) = Ω$ .

(7) $X \subseteq {^{6} N}_{j}^{R} (X)$ .

(8) ${^{6} N}_{j}^{R} ({^{6} N}_{j R} (X)) \subseteq X$ .

(9) ${^{6} N}_{j}^{R} ({^{6} N}_{j}^{R} (X)) = {^{6} N}_{j}^{R} (X)$ .

(10) ${^{6} N}_{j R} (X) = {^{6} N}_{j}^{R} ({^{6} N}_{j R} (X))$ .

(11) ${^{6} N}_{j R} ({X_{1}}^{c} \cup X_{2}) \subseteq ({^{6} N}_{j R} (X_{1}))^{c} \cup {^{6} N}_{j R} (X_{2})$ .

(12) ${^{6} N}_{j R} (X) \subseteq {^{6} N}_{j}^{R} (X)$ .

Proof. The proof analogues to Proposition 4.23.

The equality of (4) and (11) in Proposition 4.25 does not hold as shown in the following example.

Example 4.26 (Continued from Example 4.2). Take j = j₁ (and also (j = j₁, j₃, j₄, j₅, j₆, j₇, j₈) are similarly). If X₁ = {a, b} and X₂ = {b, d}, then ${^{6} N}_{j_{1}}^{R} (X_{1} \cap X_{2}) = {b}$ . Hence ${^{6} N}_{j_{1}}^{R} (X_{1}) \cap {^{6} N}_{j_{1}}^{R} (X_{2}) = {a, b, d}$ . Thus ${^{6} N}_{j_{1}}^{R} (X_{1}) \cap {^{6} N}_{j_{1}}^{R} (X_{2}) \neq {^{6} N}_{j_{1}}^{R} (X_{1} \cap X_{2})$ . Also, if X₁ = {a, b} and X₂ = {b, d}, then ${^{6} N}_{j_{1} R} (X_{1}) = {b}$ and ${^{6} N}_{j_{1} R} (X_{2}) = {b}$ . Hence ${^{6} N}_{j_{1} R} ({X_{1}}^{c} \cup X_{2}) = {b, c}$ . But $({^{6} N}_{j_{1} R} (X_{1}))^{c} \cup {^{6} N}_{j_{1} R} (X_{2}) = Ω$ . Therefore ${^{6} N}_{j_{1} R} ({X_{1}}^{c} \cup X_{2}) \neq ({^{6} N}_{j_{1} R} (X_{1}))^{c} \cup {^{6} N}_{j_{1} R} (X_{2})$ .

The relationship between our models and Definitions 3.1, 3.10, and 3.18 as shown in Figures 1-3.

Fig. 1

The relationships between our models which are given in Definitions 3.1 and 4.3.

Fig. 2

The relationships between our models which are given in Definitions 3.10 and 4.11.

Fig. 3

The relationships between our models which are given in Definitions 3.18 and 4.19.

In the following, we will give the comparison between the properties of rough sets based on Definitions 3.1, 3.10, and 3.18 by using j-neighborhood and Definitions 4.3, 4.11, and 4.19 by using j-adhesion neighborhood as shown in Table 1.

Table 1

Comparison between the properties of rough sets based on j-neighborhood and j-adhesion neighborhood

Property	Definition 3.1	Definition 3.10	Definition 3.18	Definition 4.3	Definition 4.11	Definition 4.19
$N_{j R} (X) = (N_{j}^{R} (X^{c}))^{c}$	×	×	×	×	×	×
$N_{j R} (\emptyset) = \emptyset$		×		×	×	×
$N_{j R} (X_{1}) \subseteq N_{j R} (X_{2})$	×	×	×	×	×	×
$N_{j R} (X_{1} \cap X_{2}) = N_{j R} (X_{1}) \cap N_{j R} (X_{2})$	×		×	×	×	×
$N_{j R} (X_{1}) \cup N_{j R} (X_{2}) \subseteq N_{j R} (X_{1} \cup X_{2})$	×	×	×	×	×	×
$N_{j R} (Ω) = Ω$	×		×	×	×	×
$N_{j R} (X) \subseteq X$		×		×	×	×
$X \subseteq N_{j R} (N_{j}^{R} (X))$			×	×	×	×
$N_{j R} (N_{j R} (X)) = N_{j R} (X)$		×		×	×	×
$N_{j}^{R} (X) = N_{j R} (N_{j}^{R} (X))$				×	×	×
$N_{j}^{R} (X) = (N_{j R} (X^{c}))^{c}$	×	×	×	×	×	×
$N_{j}^{R} (\emptyset) = \emptyset$	×		×	×	×	×
$N_{j}^{R} (X_{1}) \subseteq N_{j}^{R} (X_{2})$	×	×	×	×	×	×
$N_{j}^{R} (X_{1} \cap X_{2}) \subseteq N_{j}^{R} (X_{1}) \cap N_{j}^{R} (X_{2})$	×		×	×	×	×
$N_{j}^{R} (X_{1} \cup X_{2}) = N_{j}^{R} (X_{1}) \cup N_{j}^{R} (X_{2})$	×	×	×	×	×	×
$N_{j}^{R} (Ω) = Ω$		×		×	×	×
$X \subseteq N_{j}^{R} (X)$		×		×	×	×
$N_{j}^{R} (N_{j R} (X)) \subseteq X$			×	×	×	×
$N_{j}^{R} (N_{j}^{R} (X)) = N_{j}^{R} (X)$		×		×	×	×
$N_{j R} (X) = N_{j}^{R} (N_{j R} (X))$				×	×	×
$N_{j R} ({X_{1}}^{c} \cup X_{2}) \subseteq (N_{j R} (X_{1}))^{c} \cup N_{j R} (X_{2})$	×		×	×	×	×
$N_{j R} (X) \subseteq N_{j}^{R} (X)$		×		×	×	×

A cross (×) indicates that the property is satisfied.

5 Comparison between the Yao’s method [16] and the present method

The main aim of comparison between the Yao’s method [16] and the present method is to increase the accuracy measure and reduce the boundary region of sets by increasing the lower approximations and decreasing the upper approximations. So, in this section, we apply an Example 5.1 to compare between the Yao’s method [16] (see Definitions 2.2–2.4) (i.e., Definitions 3.1, 3.10, and 3.18 when j = j₁) and the current method (i.e., Definitions 4.3, 4.11, and 4.19 when j = j₁), respectively.

Example 5.1. Let Ω = {x₁, x₂, x₃, x₄, x₅} and $R = {(x_{1}, x_{1}), (x_{1}, x_{4}), (x_{2}, x_{1}), (x_{2}, x_{2}), (x_{2}, x_{3}), (x_{2}, x_{4}),$ (x₂, x₅) , (x₃, x₁) , (x₃, x₂) , (x₃, x₃) , (x₃, x₄) , (x₃, x₅) , (x₄, x₄) , (x₅, x₁) , (x₅, x₄) , (x₅, x₅)}.

From the above example, we will compute the type 1 lower approximation, the type 1 upper approximation, the type 1 j-neighborhood boundary region, and the type 1 j-neighborhood accuracy measure for subsets of X ⊆ Ω in Yao’s method [16] (as in Definition 2.2) (i.e., Definition 3.1 when j = j₁) and the type 4 lower approximation, the type 4 upper approximation, the type 4 j-adhesion neighborhood boundary region, and the type 4 j-adhesion neighborhood accuracy measure for subsets of X ⊆ Ω in present method in Definition 4.3, respectively, as shown in Table 2. Further, we will compute the type 2 lower approximation, the type 2 upper approximation, the type 2 j-neighborhood boundary region, and the type 2 j-neighborhood accuracy measure for subsets of X ⊆ Ω in Yao’s method [16] (as in Definition 2.3) (i.e., Definition 3.10 when j = j₁) and the type 5 lower approximation, the type 5 upper approximation, the type 5 j-adhesion neighborhood boundary region, and the type 5 j-adhesion neighborhood accuracy measure for subsets of X ⊆ Ω in present method in Definition 4.11, respectively, as shown in Table 3. Finally, we will compute the type 3 lower approximation, the type 3 upper approximation, the type 3 j-neighborhood boundary region, and the type 3 j-neighborhood accuracy measure for subsets of X ⊆ Ω in Yao’s method [16] (as in Definition 2.4) (i.e., Definition 3.18 when j = j₁) and the type 6 lower approximation, the type 6 upper approximation, the type 6 j-adhesion neighborhood boundary region, and the type 6 j-adhesion neighborhood accuracy measure for subsets of X ⊆ Ω in present method in Definition 4.19, respectively, as shownin Table 4.

The comparison between the Yao’s method [16] and the present method is shown in Tables 2-4.

Table 2
Comparison between the boundary region and accuracy measure by using Yao’s approach [16] and our approach in Definitions 3.1 and 4.3, respectively

X Yao’s method [16] (as in Definition 2.2) The current method in Definition 4.3

(i.e., Definition 3.1 when j = j₁)

${^{1} N}_{j_{1} R} (X)$ ${^{1} N}_{j_{1}}^{R} (X)$ $^{1} B_{j_{1}} (X)$ $^{1} A_{j} (X)$ ${^{4} N}_{j_{1} R} (X)$ ${^{4} N}_{j_{1}}^{R} (X)$ $^{4} B_{j_{1}} (X)$ $^{4} A_{j} (X)$

{x₁} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁} {x₁} ∅ 1

{x₂} ∅ {x₂, x₃} {x₂, x₃} 0 ∅ {x₂, x₃} {x₂, x₃} 0

{x₃} ∅ {x₂, x₃} {x₂, x₃} 0 ∅ {x₂, x₃} {x₂, x₃} 0

{x₄} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₄} {x₄} ∅ 1

{x₅} ∅ {x₂, x₃, x₅} {x₂, x₃, x₅} 0 {x₅} {x₅} ∅ 1

{x₁, x₂} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁} {x₁, x₂, x₃} {x₂, x₃} $\frac{1}{3}$

{x₁, x₃} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁} {x₁, x₂, x₃} {x₂, x₃} $\frac{1}{3}$

{x₁, x₄} {x₁, x₄} Ω {x₂, x₃, x₅} $\frac{2}{3}$ {x₁, x₄} {x₁, x₄} ∅ 1

{x₁, x₅} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁, x₅} {x₁, x₅} ∅ 1

{x₂, x₃} ∅ {x₂, x₃} {x₂, x₃} 0 {x₂, x₃} {x₂, x₃} ∅ 1

{x₂, x₄} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₄} {x₂, x₃, x₄} {x₂, x₃} $\frac{1}{3}$

{x₂, x₅} ∅ {x₂, x₃, x₅} {x₂, x₃, x₅} 0 {x₅} {x₂, x₃, x₅} {x₂, x₃} $\frac{1}{3}$

{x₃, x₄} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₄} {x₂, x₃, x₄} {x₂, x₃} $\frac{1}{3}$

{x₃, x₅} ∅ {x₂, x₃, x₅} {x₂, x₃, x₅} 0 {x₅} {x₂, x₃, x₅} {x₂, x₃} $\frac{1}{3}$

{x₄, x₅} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₄, x₅} {x₄, x₅} ∅ 1

{x₁, x₂, x₃} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁, x₂, x₃} {x₁, x₂, x₃} ∅ 1

{x₁, x₂, x₄} {x₁, x₄} Ω {x₂, x₃, x₅} $\frac{2}{5}$ {x₁, x₄} {x₁, x₂, x₃, x₄} {x₂, x₃} $\frac{1}{2}$

{x₁, x₂, x₅} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁, x₅} {x₁, x₂, x₃, x₅} {x₂, x₃} $\frac{1}{2}$

{x₁, x₃, x₄} {x₁, x₄} Ω {x₂, x₃, x₅} $\frac{2}{5}$ {x₁, x₄} {x₁, x₂, x₃, x₄} {x₂, x₃} $\frac{1}{2}$

{x₁, x₃, x₅} ∅ {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} 0 {x₁, x₅} {x₁, x₂, x₃, x₅} {x₂, x₃} $\frac{1}{2}$

{x₁, x₄, x₅} {x₁, x₄, x₅} Ω {x₂, x₃} $\frac{3}{5}$ {x₁, x₄, x₅} {x₁, x₄, x₅} ∅ 1

{x₂, x₃, x₄} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₂, x₃, x₄} {x₂, x₃, x₄} ∅ 1

{x₂, x₃, x₅} ∅ {x₂, x₃, x₅} {x₂, x₃, x₅} 0 {x₂, x₃, x₅} {x₂, x₃, x₅} ∅ 1

{x₂, x₄, x₅} {x₂, x₄, x₅} Ω {x₁, x₃} $\frac{3}{5}$ {x₂, x₄, x₅} {x₂, x₄, x₅} ∅ 1

{x₃, x₄, x₅} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₄, x₅} {x₂, x₃, x₄, x₅} {x₂, x₃} $\frac{1}{2}$

{x₁, x₂, x₃, x₄} {x₁, x₄} Ω {x₂, x₃, x₅} $\frac{2}{5}$ {x₁, x₂, x₃, x₄} {x₁, x₂, x₃, x₄} ∅ 1

{x₁, x₂, x₃, x₅} ∅ Ω Ω 0 {x₁, x₂, x₃, x₅} {x₁, x₂, x₃, x₅} ∅ 1

{x₁, x₂, x₄, x₅} {x₁, x₄, x₅} Ω {x₂, x₃} $\frac{3}{5}$ {x₁, x₄, x₅} Ω {x₂, x₃} $\frac{3}{5}$

{x₁, x₃, x₄, x₅} {x₁, x₄, x₅} Ω {x₂, x₃} $\frac{3}{5}$ {x₁, x₄, x₅} Ω {x₂, x₃} $\frac{3}{5}$

{x₂, x₃, x₄, x₅} {x₄} Ω {x₁, x₂, x₃, x₅} $\frac{1}{5}$ {x₂, x₃, x₄, x₅} {x₂, x₃, x₄, x₅} ∅ 1

Ω Ω Ω ∅ 1 Ω Ω ∅ 1

X		Yao’s method [16] (as in Definition 2.2)			The current method in Definition 4.3
{x₁}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁}	∅	1
{x₂}	∅	{x₂, x₃}	{x₂, x₃}	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₃}	∅	{x₂, x₃}	{x₂, x₃}	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₄}	∅	1
{x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₅}	∅	1
{x₁, x₂}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₃}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{3}$	{x₁, x₄}	{x₁, x₄}	∅	1
{x₁, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₅}	∅	1
{x₂, x₃}	∅	{x₂, x₃}	{x₂, x₃}	0	{x₂, x₃}	{x₂, x₃}	∅	1
{x₂, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₂, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄, x₅}	{x₄, x₅}	∅	1
{x₁, x₂, x₃}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₂, x₃}	{x₁, x₂, x₃}	∅	1
{x₁, x₂, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	{x₁, x₄, x₅}	∅	1
{x₂, x₃, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₂, x₃, x₄}	{x₂, x₃, x₄}	∅	1
{x₂, x₃, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₂, x₃, x₅}	{x₂, x₃, x₅}	∅	1
{x₂, x₄, x₅}	{x₂, x₄, x₅}	Ω	{x₁, x₃}	$\frac{3}{5}$	{x₂, x₄, x₅}	{x₂, x₄, x₅}	∅	1
{x₃, x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄, x₅}	{x₂, x₃, x₄, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₃, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₂, x₃, x₄}	{x₁, x₂, x₃, x₄}	∅	1
{x₁, x₂, x₃, x₅}	∅	Ω	Ω	0	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	∅	1
{x₁, x₂, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$
{x₁, x₃, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$
{x₂, x₃, x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₂, x₃, x₄, x₅}	{x₂, x₃, x₄, x₅}	∅	1
Ω	Ω	Ω	∅	1	Ω	Ω	∅	1

Table 3

Comparison between the boundary region and accuracy measure by using Yao’s approach [16] and our approach in Definitions 3.10 and 4.11, respectively

X		Yao’s method [16] (as in Definition 2.3)			The current method in Definition 4.11
		(i.e., Definition 3.10 when j = j₁)
	${^{2} N}_{j_{1} R} (X)$	${^{2} N}_{j_{1}}^{R} (X)$	$^{2} B_{j_{1}} (X)$	$^{2} A_{j} (X)$	${^{5} N}_{j_{1} R} (X)$	${^{5} N}_{j_{1}}^{R} (X)$	$^{5} B_{j_{1}} (X)$	$^{5} A_{j} (X)$
{x₁}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁}	∅	1
{x₂}	∅	{x₂, x₃}	{x₂, x₃}	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₃}	∅	{x₂, x₃}	{x₂, x₃}	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₄}	∅	1
{x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₅}	∅	1
{x₁, x₂}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₃}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{3}$	{x₁, x₄}	{x₁, x₄}	∅	1
{x₁, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₅}	∅	1
{x₂, x₃}	∅	{x₂, x₃}	{x₂, x₃}	0	{x₂, x₃}	{x₂, x₃}	∅	1
{x₂, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₂, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄, x₅}	{x₄, x₅}	∅	1
{x₁, x₂, x₃}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₂, x₃}	{x₁, x₂, x₃}	∅	1
{x₁, x₂, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	{x₁, x₄, x₅}	∅	1
{x₂, x₃, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₂, x₃, x₄}	{x₂, x₃, x₄}	∅	1
{x₂, x₃, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₂, x₃, x₅}	{x₂, x₃, x₅}	∅	1
{x₂, x₄, x₅}	{x₂, x₄, x₅}	Ω	{x₁, x₃}	$\frac{3}{5}$	{x₂, x₄, x₅}	{x₂, x₄, x₅}	∅	1
{x₃, x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄, x₅}	{x₂, x₃, x₄, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₃, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₂, x₃, x₄}	{x₁, x₂, x₃, x₄}	∅	1
{x₁, x₂, x₃, x₅}	∅	Ω	Ω	0	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	∅	1
{x₁, x₂, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$
{x₁, x₃, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$
{x₂, x₃, x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₂, x₃, x₄, x₅}	{x₂, x₃, x₄, x₅}	∅	1
Ω	Ω	Ω	∅	1	Ω	Ω	∅	1

Table 4

Comparison between the boundary region and accuracy measure by using Yao’s approach [16] and our approach in Definitions 3.18 and 4.19, respectively

X		Yao’s method [16] (as in Definition 2.4)			The current method in Definition 4.19
		(i.e., Definition 3.18 when j = j₁)
	${^{3} N}_{j_{1} R} (X)$	${^{3} N}_{j_{1}}^{R} (X)$	$^{3} B_{j_{1}} (X)$	$^{3} A_{j} (X)$	${^{6} N}_{j_{1} R} (X)$	${^{6} N}_{j_{1}}^{R} (X)$	$^{6} B_{j_{1}} (X)$	$^{6} A_{j} (X)$
{x₁}	∅	Ω	Ω	0	{x₁}	{x₁}	∅	1
{x₂}	∅	Ω	Ω	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₃}	∅	Ω	Ω	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₄}	∅	Ω	Ω	0	{x₄}	{x₄}	∅	1
{x₅}	∅	Ω	Ω	0	{x₅}	{x₅}	∅	1
{x₁, x₂}	∅	Ω	Ω	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₃}	∅	Ω	Ω	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₄}	∅	Ω	Ω	0	{x₁, x₄}	{x₁, x₄}	∅	1
{x₁, x₅}	∅	Ω	Ω	0	{x₁, x₅}	{x₁, x₅}	∅	1
{x₂, x₃}	∅	Ω	Ω	0	{x₂, x₃}	{x₂, x₃}	∅	1
{x₂, x₄}	∅	Ω	Ω	0	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₂, x₅}	∅	Ω	Ω	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₄}	∅	Ω	Ω	0	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₅}	∅	Ω	Ω	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₄, x₅}	∅	Ω	Ω	0	{x₄, x₅}	{x₄, x₅}	∅	1
{x₁, x₂, x₃}	∅	Ω	Ω	0	{x₁, x₂, x₃}	{x₁, x₂, x₃}	∅	1
{x₁, x₂, x₄}	∅	Ω	Ω	0	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₅}	∅	Ω	Ω	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₄}	∅	Ω	Ω	0	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₅}	∅	Ω	Ω	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₄, x₅}	∅	Ω	Ω	0	{x₁, x₄, x₅}	{x₁, x₄, x₅}	∅	1
{x₂, x₃, x₄}	∅	Ω	Ω	0	{x₂, x₃, x₄}	{x₂, x₃, x₄}	∅	1
{x₂, x₃, x₅}	∅	Ω	Ω	0	{x₂, x₃, x₅}	{x₂, x₃, x₅}	∅	1
{x₂, x₄, x₅}	∅	Ω	Ω	0	{x₂, x₄, x₅}	{x₂, x₄, x₅}	∅	1
{x₃, x₄, x₅}	∅	Ω	Ω	0	{x₄, x₅}	{x₂, x₃, x₄, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₃, x₄}	∅	Ω	Ω	0	{x₁, x₂, x₃, x₄}	{x₁, x₂, x₃, x₄}	∅	1
{x₁, x₂, x₃, x₅}	∅	Ω	Ω	0	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	∅	1
{x₁, x₂, x₄, x₅}	∅	Ω	Ω	0	{x₁, x₄, x₅}	X	{x₂, x₃}	$\frac{3}{5}$
{x₁, x₃, x₄, x₅}	∅	Ω	Ω	0	{x₁, x₄, x₅}	X	{x₂, x₃}	$\frac{3}{5}$
{x₂, x₃, x₄, x₅}	∅	Ω	Ω	0	{x₂, x₃, x₄, x₅}	{x₂, x₃, x₄, x₅}	∅	1
Ω	Ω	Ω	∅	1	Ω	Ω	∅	1

From Table 2, for example, if we take {x₁, x₂, x₃} , then the type 4 j-adhesion neighborhood boundary and the type 4 j-adhesion neighborhood accuracy by Definition 4.3 are ∅ and 1, whereas the type 1 j-neighborhood boundary and the type 1 j-neighborhood accuracy by using Yao’s method [16] (as in Definition 2.2) (i.e., Definition 3.1 when j = j₁) are {x₁, x₂, x₃, x₅} and 0, respectively. Also, from Table 3, for example, if we take {x₂, x₃} , then the type 5 j-adhesion neighborhood boundary and the type 5 j-adhesion neighborhood accuracy by Definition 4.11 are ∅ and 1, whereas the type 2 j-neighborhood boundary and the type 2 j-neighborhood accuracy by using Yao’s method [16] (as in Definition 2.3) (i.e., Definition 3.10 when j = j₁) are {x₂, x₃} and 0, respectively. Additionally, from Table 4, for example, if we take {x₄} , then the type 6 j-adhesion neighborhood boundary and the type 6 j-adhesion neighborhood accuracy by Definition 4.19 are ∅ and 1, whereas the type 3 j-neighborhood boundary and the type 3 j-neighborhood accuracy by using Yao’s method [16] (as in Definition 2.4) (i.e., Definition 3.18 when j = j₁) are Ω and $\frac{1}{5}$ , respectively.

From Tables 2-4, we can see the types 4-6 of lower and upper approximations based on j-adhesion neighborhood space are achieved to the increase the accuracy measure and reduce the boundary region of set X. Therefore, in this paper, the results by Definitions 4.3, 4.11, and 4.19 when j = j₁ are better than the Yao’s method [16] (as Definitions 2.2–2.4) (i.e., Definitions 3.1, 3.10, and 3.18 when j = j₁), respectively.

6 Conclusions

In recent decades, many applications of the rough set theory have emerged in various fields. Therefore, in this paper, we aim to reduce the boundary region and increase the accuracy measure of decision-making. Besides, we introduce a generalization for three types of rough set models based on j-neighborhood space. A new three types of rough set models based on j-adhesion neighborhood space as, type 4 j-adhesion neighborhood rough set, type 5 j-adhesion neighborhood rough set, and type 6 j-adhesion neighborhood rough set are defined. Also, some of the fundamental properties of these types of rough j-adhesion neighborhood space are discussed. The relationship between the properties of different types are achieved. Finally, we made a comparison of Yao’s methods and our methods of finding boundary region and accuracy. In the future, the present work can be extended to present some new rough set approximations under j-adhesion neighborhood space [42, 43] and provide a comparison between our approach and the approach of Abo-Tabl [24] as referred in [44]. We will also consider control strategies (related to Traditional Chinese Medicine) and mathematical models (related to this work and fuzzy quantifiers and their integral semantics [45]) of incipient stage of COVID-19 (i.e., Corona Virus Disease) in the group of old people. We believe that the six types of rough approximations developed here are expected to attract the researchers working in these related areas and also in [46 –50].

Footnotes

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions. The authors wish to thank the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia, for their support for their research. Further, This work was supported by the National Natural Science Foundations of China (11771263, 61967001) and the Key Research and Development Project of Shaanxi Province of China (2018KW-050).

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X		Yao’s method [16] (as in Definition 2.2)			The current method in Definition 4.3
		(i.e., Definition 3.1 when j = j₁)
	${^{1} N}_{j_{1} R} (X)$	${^{1} N}_{j_{1}}^{R} (X)$	$^{1} B_{j_{1}} (X)$	$^{1} A_{j} (X)$	${^{4} N}_{j_{1} R} (X)$	${^{4} N}_{j_{1}}^{R} (X)$	$^{4} B_{j_{1}} (X)$	$^{4} A_{j} (X)$
{x₁}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁}	∅	1
{x₂}	∅	{x₂, x₃}	{x₂, x₃}	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₃}	∅	{x₂, x₃}	{x₂, x₃}	0	∅	{x₂, x₃}	{x₂, x₃}	0
{x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₄}	∅	1
{x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₅}	∅	1
{x₁, x₂}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₃}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁}	{x₁, x₂, x₃}	{x₂, x₃}	$\frac{1}{3}$
{x₁, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{3}$	{x₁, x₄}	{x₁, x₄}	∅	1
{x₁, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₅}	∅	1
{x₂, x₃}	∅	{x₂, x₃}	{x₂, x₃}	0	{x₂, x₃}	{x₂, x₃}	∅	1
{x₂, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₂, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄}	{x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{3}$
{x₃, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₅}	{x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{3}$
{x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄, x₅}	{x₄, x₅}	∅	1
{x₁, x₂, x₃}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₂, x₃}	{x₁, x₂, x₃}	∅	1
{x₁, x₂, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₄}	{x₁, x₂, x₃, x₄}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₃, x₅}	∅	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	0	{x₁, x₅}	{x₁, x₂, x₃, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	{x₁, x₄, x₅}	∅	1
{x₂, x₃, x₄}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₂, x₃, x₄}	{x₂, x₃, x₄}	∅	1
{x₂, x₃, x₅}	∅	{x₂, x₃, x₅}	{x₂, x₃, x₅}	0	{x₂, x₃, x₅}	{x₂, x₃, x₅}	∅	1
{x₂, x₄, x₅}	{x₂, x₄, x₅}	Ω	{x₁, x₃}	$\frac{3}{5}$	{x₂, x₄, x₅}	{x₂, x₄, x₅}	∅	1
{x₃, x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₄, x₅}	{x₂, x₃, x₄, x₅}	{x₂, x₃}	$\frac{1}{2}$
{x₁, x₂, x₃, x₄}	{x₁, x₄}	Ω	{x₂, x₃, x₅}	$\frac{2}{5}$	{x₁, x₂, x₃, x₄}	{x₁, x₂, x₃, x₄}	∅	1
{x₁, x₂, x₃, x₅}	∅	Ω	Ω	0	{x₁, x₂, x₃, x₅}	{x₁, x₂, x₃, x₅}	∅	1
{x₁, x₂, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$
{x₁, x₃, x₄, x₅}	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$	{x₁, x₄, x₅}	Ω	{x₂, x₃}	$\frac{3}{5}$
{x₂, x₃, x₄, x₅}	{x₄}	Ω	{x₁, x₂, x₃, x₅}	$\frac{1}{5}$	{x₂, x₃, x₄, x₅}	{x₂, x₃, x₄, x₅}	∅	1
Ω	Ω	Ω	∅	1	Ω	Ω	∅	1