Rough and rough fuzzy sets on two universes via covering approach

Abstract

Rough set theory is an important non-numeric method for dealing with uncertainty and data mining. In this paper, we study rough sets and rough fuzzy sets on two universes via covering models. We introduce a new definition for the lower and upper approximation by taking a covering of the second universe of discourse. Some properties of the new model are revealed. We believe that this model will be more realistic in the sense that rough sets (resp. rough fuzzy sets) are approximated by sets (resp. fuzzy sets) on the same universe. Moreover, some results, examples and counter examples are provided.

Keywords

Rough sets covering approximation space strong covering rough fuzzy sets

1 Introduction

In Pawlak rough set model [20, 21], the approximation space is based on an equivalence relation. A meaningful research direction in rough set theory to generalize Pawlak’s model to more general cases so that rough set theory can be applied to more situations. Thus one of the main directions of research in rough set theory is naturally the generalization of Pawlak rough set approximations. For instance, the notations of approximations are extended to general binary relations [2 , 43], neighborhood systems [12, 37], coverings [5 , 45], completely distributive lattices [4], fuzzy lattices [17] and Boolean algebras [16, 23]. There is another direction for the generalization of rough sets by using fuzzy environment [7 , 26]. The concepts of fuzzy rough and rough fuzzy set based on equivalence relation were first proposed by Dubois and Prade [7] in the Pawlak approximation space. Yao [35] introduced a unified model for both rough fuzzy sets and fuzzy rough sets based on the analysis of level sets of fuzzy sets. Pei [25, 26] and Cock, et al. [6] considered the approximation problems of fuzzy sets in fuzzy information systems result in theory of fuzzy rough sets. Li and Zhang [11] analyzed crisp binary relations and rough fuzzy approximations. Mareay, et al. studied the approximation of fuzzy sets via covering approximation space [10].

On the other hand, rough set can be approximated on two universe of discourse [18 , 40–42]. These models start with a binary relation on two universes of discourse and the approximated sets are located on two different universes of discourse which is inconvenient for knowledge discovery by means of rough set theory. Abd El-Monsef, et al. introduced a comprehensive study of rough and fuzzy sets on two universes using a binary relation [1]. In this paper, we establish the approximation of rough sets and rough fuzzy sets on two universes of discourse by using a covering-based rough set. Since the approximated sets will be on the same universe and the covering is more general, We believe this approximation will be more realistic in data mining.

2 Preliminaries

Let U be a finite and nonempty set called the universe of discourse. The class of all subsets (fuzzy subsets, respectively) of U will be denoted by $\begin{matrix} P \end{matrix} (U)$ ,( $\begin{matrix} F \end{matrix} (U)$ , respectively).

Definition 2.1. [3, 44] Let U be a universe of discourse. Let C be a family of subsets of U. If none subset in C is empty and ⋃C = U, then C is called a covering of U. If P is a covering of U and it is a family of pairwisely disjoint subsets of U, P is called a partition of U.

It’s obvious that the concept of a covering is an extension of the concept of a partition.

Definition 2.2. [33] Let U and V be two finite and nonempty universes and let R be a binary relation from U to V, the triple (U, V, R) is called (two-universe) approximation space. Then the relation R is called:

Serial if for all x ∈ U, there exists y ∈ V such that (x, y) ∈ R,

Inverse serial if for all y ∈ V, there exists x ∈ U such that (x, y) ∈ R,

Compatibility relation, if R is both serial and inverse serial.

Definition 2.3. [34] Let (U, V, R) be a (two-universe) approximation space. Then, a set-valued mapping F from U to $\begin{matrix} P \end{matrix} (V)$ representing the successor neighborhood of x with respect to R, as follows: $F : U \to \begin{matrix} P \end{matrix} (V), F (x) = {y \in V : (x, y) \in R}$

Definition 2.4. [34] Let (U, V, R) be a (two-universe) approximation space. Then, the lower and upper approximations of $Y \in \begin{matrix} P \end{matrix} (V)$ are respectively defined as follows: $\begin{matrix} \underline{R} (Y) & = & {x \in U : F (x) \subseteq Y}, \\ \bar{R} (Y) & = & {x \in U : F (x) \cap Y \neq \emptyset} \end{matrix}$

The ordered set-pair $(\underline{R}, \bar{R})$ is called a generalized rough set. A subset $Y \in \begin{matrix} P \end{matrix} (V)$ is called definable or exact with respect to (U, V, R) if $\underline{R} (Y) = \bar{R} (Y)$ , otherwise it is undefinable or rough.

Proposition 2.1. [34] Let (U, V, R) be a (two-universe) approximation space and let R be a compatibility relation. Then, for all $Y, Y_{1}, Y_{2} \in \begin{matrix} P \end{matrix} (V)$ the following properties are satisfied:

(L₁) $\underline{R} (Y) = (\bar{R} (Y^{c}))^{c}$ , where Y^c denote the complement of the fuzzy subset Y in V,

(L₂) $\underline{R} (V) = U$ , (L₃) $\underline{R} (Y_{1} \cap Y_{2}) = \underline{R} (Y_{1}) \cap \underline{R} (Y_{2})$ , (L₄) $\underline{R} (Y_{1} \cup Y_{2}) \supseteq \underline{R} (Y_{1}) \cup (Y_{2})$ ,

(L₅) $Y_{1} \subseteq Y_{2} \Rightarrow \underline{R} (Y_{1}) \subseteq \underline{R} (Y_{2})$ , (L₆) $\underline{R} (\emptyset) = \emptyset$ .

(U₁) $\bar{R} (Y) = (\underline{R} (Y^{c}))^{c}$ , (U₂) $\bar{R} (\emptyset) = \emptyset$ , (U₃) $\bar{R} (Y_{1} \cup Y_{2}) = \bar{R} (Y_{1}) \cup \bar{R} (Y_{2})$ ,

(U₄) $\bar{R} (Y_{1} \cap Y_{2}) \subseteq \bar{R} (Y_{1}) \cap \bar{R} (Y_{2})$ , (U₅) $Y_{1} \subseteq Y_{2} \Rightarrow \bar{R} (Y_{1}) \subseteq \bar{R} (Y_{2})$ ,

(U₆) $\bar{R} (V) = U$ , (LU) $\underline{R} (Y) \subseteq \bar{R} (Y)$ .

3 Generalized rough sets based on covering

In rough sets on two universe models (U, V, R), subsets of the universe V are approximated by subsets of the other universe U, which seems very unreasonable. Also, for general relation some properties of the approximation space which are true in various generalized rough sets do not hold in two-universe models, because reflexivity, symmetry and transitivity are meaningless for binary relations from U to V. In this section, we introduce a more natural form for rough sets on two universes such that the approximations of subsets of the universe V are subsets of the universe V by using a covering approach.

Definition 3.1. Let $F : U \to \begin{matrix} P \end{matrix} (V)$ , F (x) = {y ∈ V : (x, y) ∈ R}, R is inverse serial relation. Then, C_V = {F (x) , x ∈ U} is a covering of the universe of V and (U, V, C_V) is called two universe approximation space based on covering.

Definition 3.2. Let (U, V, C_V) be two universe covering approximation space. Then we define mapping N from V to $\begin{matrix} P \end{matrix} (V)$ induced by R as follows:

$N : V \to \begin{matrix} P \end{matrix} (V), N (y) = ⋂ {F (x) : y \in F (x), F (x) \in C_{V}}$ .

Definition 3.3. Let (U, V, C_V) be two universe covering approximation space, a covering C of the universe V is called strong covering if for all y₁, y₂ ∈ V, N (y₁)∩ N (y₂) ≠ ∅ implies that N (y₁) = N (y₂).

Lemma 3.1. Let (U, V, C_V) be two universe covering approximation space. If y₁ ∈ N (y₂), then N (y₁) ⊆ N (y₂), ∀y₁, y₂ ∈ V.

Proof. Let y ∈ V such that y ∉ N (y₂). Then there exist at least x₁ ∈ U such that y₂ ∈ F (x₁), y ∉ F (x₁). But y₁ ∈ N (y₂) this leads to y₁ ∈ F (x₁). Hence N (y₁) ⊆ F (x₁). Therefore y ∉ N (y₁) and henceN (y₁) ⊆ N (y₂). □

Lemma 3.2. Let (U, V, C_V) be two universe covering approximation space. Then y ∈ N (y), ∀y ∈ V.

Proof. Since N (y) = ⋂ {F (x) : y ∈ F (x) , F (x) ∈ C_V}. Then y ∈ N (y) , ∀ y ∈ V. □

Definition 3.4. Let (U, V, C_V) be two universe covering approximation space. The covering lower and upper approximation of $Y \in \begin{matrix} P \end{matrix} (V)$ are defined respectively as follows: $\begin{matrix} C_{*} (Y) & = & {y \in V : N (y) \subseteq Y}, \\ C^{*} (Y) & = & {y \in V : N (y) \cap Y \neq \emptyset} . \end{matrix}$

Proposition 3.1. Let (U, V, C_V) be two universe covering approximation space. Then for all $Y, Y_{1}, Y_{2} \in \begin{matrix} P \end{matrix} (V)$ , the approximation operators have the following properties:

$(L_{1}^{*})$ C_* (Y) = (C^* (Y^c)) ^c, $(L_{2}^{*})$ C_* (V) = V.

$(L_{3}^{*})$ C_* (Y₁ ∩ Y₂) = C_* (Y₁) ∩ C_* (Y₂),

$(L_{4}^{*})$ C_* (Y₁ ∪ Y₂) ⊇ C_* (Y₁) ∪ C_* (Y₂),

$(L_{5}^{*})$ Y₁ ⊆ Y₂ ⇒ C_* (Y₁) ⊆ C_* (Y₂),

$(L_{9}^{*})$ C_* (Y) ⊆ C_* (C_* (Y)),

$(U_{1}^{*})$ C^* (Y) = (C_* (Y^c)) ^c,

$(U_{2}^{*})$ C^* (∅) = ∅,

$(U_{3}^{*})$ C^* (Y₁ ∪ Y₂) = C^* (Y₁) ∪ C^* (Y₂),

$(U_{4}^{*})$ C^* (Y₁ ∩ Y₂) ⊆ C^* (Y₁) ∩ C^* (Y₂),

$(U_{5}^{*})$ Y₁ ⊆ Y₂ ⇒ C^* (Y₁) ⊆ C^* (Y₂),

$(U_{9}^{*})$ C^* (C^* (Y)) ⊆ C^* (Y).

Proof. We will prove ( $L_{1}^{*})$ , ( $L_{2}^{*})$ , ( $L_{3}^{*})$ , ( $L_{5}^{*})$ and ( $L_{9}^{*})$ . The proof of other parts is similar.

( $L_{1}^{*})$ (C^* (Y^c)) ^c = {y ∈ V|N (y) ∩ Y^c ≠ ∅} ^c = {y ∈ V|N (y) ∩ Y^c = ∅} = {y ∈ V|N (y) ⊆ Y} = C_* (Y),

( $L_{2}^{*})$ As V is the universe set, hence C_* (V) ⊆ V. Conversely, ∀y ∈ V, N (y) ⊆ V, this implies that y ∈ C_* (V). Thus V ⊆ C_* (V) and soC_* (V) = V.

( $L_{3}^{*})$ Let y ∈ C_* (Y₁ ∩ Y₂) ⇔ N (y) ⊆ (Y₁ ∩ Y₂) ⇔ N (y) ⊆ Y₁ and N (y)⊆ Y₂ ⇔ y ∈ C_* (Y₁) and y∈ C_* (Y₂) ⇔ C_* (Y₁) ∩ C_* (Y₂). Hence C_* (Y₁ ∩ Y₂) = C_* (Y₁) ∩ C_* (Y₂).

( $L_{5}^{*})$ Let y ∈ C_* (Y₁), N (y) ⊆ Y₁ but Y₁ ⊆ Y₂. So N (y) ⊆ Y₂, hence y ∈ C_* (Y₂). Therefore C_* (Y₁) ⊆ C_* (Y₂).

( $L_{9}^{*})$ Let y ∈ C_* (Y), then N (y) ⊆ Y. Let x ∈ N (y), then N (x) ⊆ N (y) ∀ x ∈ N (y), then N (x) ⊆ N (y), and so N (x) ⊆ Y. Consequently, x ∈ C_* (Y). Hence N (y) ⊆ C_* (Y) and hence y ∈ C_* (C_* (Y)). □

Remark 3.1. Let (U, V, R) be two universe approximation space,where R ∈ U × V. Then $\forall Y \in \begin{matrix} P \end{matrix} (V)$ , the following properties do not hold:

$\underline{R} (\emptyset) = \emptyset$ ,

$\underline{R} (Y) \subseteq Y$ ,

$Y \subseteq \underline{R} (\bar{R} (Y))$ ,

$\bar{R} (Y) \subseteq \underline{R} (\bar{R} (Y))$ ,

$\bar{R} (V) = U$ ,

$Y \subseteq \bar{R} (Y)$ ,

$\bar{R} (\underline{R} (Y)) \subseteq Y$ ,

$\bar{R} (\underline{R} (Y)) \subseteq \underline{R} (Y)$ ,

$\underline{R} (Y) \subseteq \bar{R} (Y)$ .

The following example illustrates Remark 3.1.

Example 3.1. Let U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇}, V = {y₁, y₂, y₃, y₄, y₅, y₆} and R ∈ U × V be a binary relation defined as Table 1:

Hence we have $\underline{R} (\emptyset) = {x_{2}} \neq \emptyset$ and $\bar{R} (V) = {x_{1}, x_{3}, x_{4}, x_{5}, x_{6}} \neq U$ , i.e. L₆ and U₆ do not hold. If Y₁ = {y₁, y₄, y₆},then $\underline{R} (Y_{1}) = {x_{1}, x_{2}, x_{4}}$ , $\bar{R} (Y_{1}) = {x_{1}, x_{3}, x_{4}, x_{5}, x_{6}} = \bar{R} (\underline{R} (Y_{1}))$ , hence $\underline{R} (Y_{1}) ⊈ Y_{1}, \underline{R} (Y_{1}) ⊈ \bar{R} (Y_{1}), \bar{R} (\underline{R} Y_{1}) ⊈ (Y_{1})$ and $\bar{R} (\underline{R} (Y_{1}) ⊈ \underline{R} (Y_{1})$ , i.e.,L₇, LU, U₈and U₁₀ do not hold. If Y₂ = {y₁, y₂, y₅},then $\bar{R} (Y_{2}) = {x_{1}, x_{5}, x_{6}}$ and $\underline{R} (\bar{R} (Y_{2})) = {x_{2}}$ . Thus $Y_{2} ⊈ \bar{R} (Y_{2}), Y_{2} ⊈ \underline{R} (\bar{R} (Y_{2}))$ and $\bar{R} (Y_{2}) ⊈ \underline{R} (\bar{R} (Y_{2}))$ . Therefore U₇, L₈ and L₁₀ do not hold.

Proposition 3.2. Let (U, V, C_V) be two universe covering approximation space. Then for all $Y \in \begin{matrix} P \end{matrix} (V)$ , the approximation operators have the following properties:

$(L_{6}^{*})$ C_* (∅) = ∅, $(L_{7}^{*})$ C_* (Y) ⊆ Y,

$(U_{6}^{*})$ C^* (V) = V, $(U_{7}^{*})$ Y ⊆ C^* (Y),

(LU^*) C_* (Y) ⊆ C^* (Y),

Proof. By the duality of approximation operators, we only need to prove the properties $L_{6}^{*}$ , $L_{7}^{*}$ and LU*.

( $L_{6}^{*}$ ) Since C is a covering of the universe V, then by Lemma 3.2. y ∈ N (y) for all y ∈ V, thus there does not exist y ∈ V such that N (y)⊆ ∅,hence C_* (∅) = ∅.

( $L_{7}^{*}$ ) Let y ∈ C_* (Y), then N (y) ⊆ Y but y ∈ N (y), thus y ∈ Y. Therefore C_* (Y) ⊆ Y.

(LU*) The proof of LU comes from L₇ and U₇. □

Remark 3.2. Let (U, V, R) be two universe approximation space, where R ∈ U × V. Then $\forall Y \in \begin{matrix} P \end{matrix} (V)$ , the following properties do not hold:

(L₈) $Y \subseteq \underline{R} (\bar{R} (Y))$ , (L₁₀) $\bar{R} (Y) \subseteq \underline{R} (\bar{R} (Y))$ , (U₈) $\bar{R} (\underline{R} (Y)) \subseteq Y$ , (U₁₀) $\bar{R} (\underline{R} (Y)) \subseteq \underline{R} (Y)$ .

Example 3.2. Let U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇}, V = {y₁, y₂, y₃, y₄, y₅, y₆} and R ∈ U × V be any binary relation defined as Table 2: If Y₁ = {y₃, y₄, y₅, y₆},then $\underline{R} (Y_{1}) = {x_{3}, x_{4}, x_{5}}$ and $\bar{R} (\underline{R} (Y_{1})) = {x_{2}, x_{3}, x_{4}, x_{5}}$ , hence $\bar{R} (\underline{R} (Y_{1})) ⊈ (Y_{1})$ and $\bar{R} (\underline{R} (Y_{1})) ⊈ \underline{R} (Y_{1})$ ,i.e. U₈ and U₁₀ do not hold. If Y₂ = {y₂, y₃, y₆}, then $\bar{R} (Y_{2}) = {x_{1}, x_{2}, x_{3}, x_{6}}$ and $\bar{R} (\underline{R} (Y_{2})) = {x_{1}, x_{6}}$ . Thus $Y_{2} ⊈ \underline{R} (\bar{R} (Y_{2}))$ and $\bar{R} (Y_{2}) ⊈ \underline{R} (\bar{R} (Y_{2}))$ . Therefore L₈ and L₁₀ do not hold.

Proposition 3.3. Let (U, V, C_V) be two universe covering approximation space, C_V is strong covering of V. Then for all $Y \in \begin{matrix} P \end{matrix} (V)$ , the approximation operators have the following properties:

$(L_{8}^{*})$ Y ⊆ C_* (C^* (Y)), $(L_{10}^{*})$ C^* (Y) ⊆ C_* (C^* (Y)), $(U_{8}^{*})$ C^* (C_* (Y)) ⊆ Y,

$(U_{10}^{*})$ C^* (C_* (Y)) ⊆ C_* (Y).

Proof. ( $L_{10}^{*})$ Let y ∉ C_* (C^* (Y)), then N (y) ⊈ C^* (Y), i.e., there exists z ∈ V such that z ∈ N (y) and z ∉ C^* (Y), this leads to N (z)∩ Y = ∅. Since C is strong covering of V, then N (z) = N (y). Consequently, N (y)∩ Y = ∅, i.e., y ∉ C^* (Y). Therefore C^* (Y) ⊆ C_* (C^* (Y)).

The proof of $L_{8}^{*}$ comes from $L_{7}^{*}$ and $L_{10}^{*}$ . The proofs of $U_{8}^{*}$ and $U_{10}^{*}$ are the same. □

4 Generalized rough fuzzy sets on two universes

A rough fuzzy set is a generalization of rough set, derived from the approximation of fuzzy set in a crisp approximation space. W.-Z. Wu, et. al. introduced rough fuzzy sets on two universes using relation.

Definition 4.1. Let U and V be two finite non-empty universes of discourse and R ∈ (U × V) a binary relation from U to V. The ordered triple (U, V, R) is called a (two-universe) approximation space. For any fuzzy set $Y \in \begin{matrix} F (V) \end{matrix}$ ,the lower and upper approximation of $Y, \underline{R} (Y)$ and $\bar{R} (Y)$ , with respect to the approximation space are fuzzy sets of Uwhose membership functions, for each x ∈ U, are defined,respectively, by $\begin{matrix} \underline{R} (Y) (x) & = & \min {Y (y) | y \in F (x)}, \\ \bar{R} (Y) (x) & = & \max {Y (y) | y \in F (x)} . \end{matrix}$

Where F (x)is the right neighborhood of x defined in Definition 2.3.

The ordered set pair $(\underline{R} (Y), \bar{R} (Y))$ is referred to as generalized rough fuzzy set, and $\underline{R} (Y)$ and $\bar{R} : F (V) \to F (U)$ are referred to as lower and upper generalized rough fuzzy approximation operators, respectively.

Proposition 4.1. [41] In a (two-universe) model (U, V, R) with compatibility relation R, the approximation operators satisfy the following properties for all $Y, Y_{1}, Y_{2} \in F (V)$ :

(L₁) $\underline{R} (Y) = (\bar{R} (Y^{c}))^{c}$ , where Y^c denote the complement of the fuzzy subset Y in V.

(L₂) $\underline{R} (V) = U$ . (L₃) $\underline{R} (Y_{1} \cap Y_{2}) = \underline{R} (Y_{1}) \cap \underline{R} (Y_{2})$ , (L₄) $\underline{R} (Y_{1} \cup Y_{2}) \supseteq \underline{R} (Y_{1}) \cup (Y_{2})$ ,

(L₅) $Y_{1} \subseteq Y_{2} \Rightarrow \underline{R} (Y_{1}) \subseteq \underline{R} (Y_{2})$ , (L₆) $\underline{R} (\emptyset) = \emptyset$ ,

(U₁) $\bar{R} (Y) = (\underline{R} (Y^{c}))^{c}$ , (U₂) $\bar{R} (\emptyset) = \emptyset$ ,

(U₃) $\bar{R} (Y_{1} \cup Y_{2}) = \bar{R} (Y_{1}) \cup \bar{R} (Y_{2})$ ,

(U₄) $\bar{R} (Y_{1} \cap Y_{2}) \subseteq \bar{R} (Y_{1}) \cap \bar{R} (Y_{2})$ , (U₅) $Y_{1} \subseteq Y_{2} \Rightarrow \bar{R} (Y_{1}) \subseteq \bar{R} (Y_{2})$ , (U₆) $\bar{R} (V) = U$ ,

(LU) $\underline{R} (Y) \subseteq \bar{R} (Y)$ .

Proposition 4.2. [41] Let R ∈ (U × U) be an arbitrary binary relation on U. Then $\forall X \in F (U)$ ,

R is reflexive ⇔ $(L_{7}) \underline{R} (X) \subseteq X$

⇔ $(U_{7}) X \subseteq \bar{R} (X)$ .

R is symmetric ⇔ $(L_{8}) \bar{R} (\underline{R} (X)) \subseteq X$

⇔ $(U_{8}) X \subseteq \underline{R} (\bar{R} (X))$ .

R is transitive ⇔ $(L_{9}) \underline{R} (X) \subseteq \underline{R} (\underline{R} (X))$

⇔ $(U_{9}) \bar{R} (\bar{R} (X)) \subseteq \bar{R} (X)$ .

R is Euclidean ⇔ $(L_{10}) \bar{R} (\underline{R} (X)) \subseteq \underline{R} (X)$

⇔ $(U_{10}) \bar{R} (X) \subseteq \underline{R} (\bar{R} (X))$ .

5 Revised rough fuzzy sets on two universes based on covering

In the above model for generalized rough fuzzy sets, fuzzy subsets of the universe V are approximated by fuzzy subsets of the other universe U. This seems very unreasonable. Furthermore there exists no relation between the set and its lower and upper approximations. Since the operators $\underline{R} \underline{R}$ , $\bar{R} \underline{R}$ , $\underline{R} \bar{R}$ and $\bar{R} \bar{R}$ are not defined, so the properties (L₇) – (L₁₀) and (U₇) – (U₁₀) which are true in various generalized rough set models do not hold in (two-universe) models. In this section, we introduce a natural form for rough sets on two universes based on covering such that the approximations of subsets of the universe V are subsets of the universe V.

Definition 5.1. Let (U, V, C_V) be two universe covering approximation space. The covering lower and upper approximation of $Y \in \begin{matrix} F \end{matrix} (V)$ are defined respectively as follows: $\begin{matrix} C_{*} (Y) (y) & = & \min {Y (z) : z \in N (y)}, \\ C^{*} (Y) (y) & = & \max {Y (z) : z \in N (y)} . \end{matrix}$

The pair (C_* (Y) , C^* (Y)) are referred as covering rough fuzzy set, and $C_{*}, C^{*} : F (V) \to F (V)$ are referred to as lower and upper covering rough fuzzy approximation operators, respectively.

Proposition 5.1. Let (U, V, C_V) be two universe covering approximation space. Then for all $Y, Y_{1}, Y_{2} \in \begin{matrix} F \end{matrix} (V)$ , the approximation operators have the following properties:

$(L_{1}^{*})$ C_* (Y) = (C^* (Y^c)) ^c, $(L_{2}^{*})$ C_* (V) = V,

$(L_{3}^{*})$ C_* (Y₁ ∩ Y₂) = C_* (Y₁) ∩ C_* (Y₂),

$(L_{4}^{*})$ C_* (Y₁ ∪ Y₂) ⊇ C_* (Y₁) ∪ C_* (Y₂),

$(L_{5}^{*})$ Y₁ ⊆ Y₂ ⇒ C_* (Y₁) ⊆ C_* (Y₂),

$(L_{9}^{*})$ C_* (Y) ⊆ C_* (C_* (Y)),

$(U_{1}^{*})$ C^* (Y) = (C_* (Y^c)) ^c, $(U_{2}^{*})$ C^* (∅) = ∅,

$(U_{3}^{*})$ C^* (Y₁ ∪ Y₂) = C^* (Y₁) ∪ C^* (Y₂),

$(U_{4}^{*})$ C^* (Y₁ ∩ Y₂) ⊆ C^* (Y₁) ∩ C^* (Y₂),

$(U_{5}^{*})$ Y₁ ⊆ Y₂ ⇒ C^* (Y₁) ⊆ C^* (Y₂),

$(U_{9}^{*})$ C^* (C^* (Y)) ⊆ C^* (Y).

Proof. We will prove the properties $L_{1}^{*}, L_{2}^{*}, L_{3}^{*}, L_{4}^{*}, L_{5}^{*}$ and $L_{9}^{*}$ . By the duality of approximation operators, the other parts are proved.

( $L_{1}^{*})$ Since ∀y ∈ V,(C^* (Y^c)) ^c (y) =1 - {max {Y^c (z) : z ∈ N (y)}} =1 - {max {1 - Y (z) : z ∈ N (y)}} =1 - {1 - min {Y (z) : z ∈ N (y)}} =1 - {max {Y (z) : z ∈ N (y)}} = min {Y (z) : z ∈ N (y)} = C_* (Y) (y). Hence C_* (Y) = (C^* (Y^c)) ^c

( $L_{2}^{*})$ Since ∀y ∈ V, V (y) =1 and N (y) ⊆ V, then min {V (z) : z ∈ N (y)} =1. Thus C_* (V) (y) = min {V (z) : z ∈ N (y)} =1. Therefore C_* (V) = V.

( $L_{3}^{*})$ Since ∀y ∈ V,C_* (Y₁ ∩ Y₂) (y) = min {(Y₁ ∩ Y₂) (z) : z ∈ N (y)} = min {Y₁ (z) ∧ Y₂ (z) : z ∈ N (y)} = min {Y₁ (z) : z ∈ N (y)} ∧ min {Y₂ (z) : z ∈ N (y)} = C_* (Y₁) (y) ∧ C_* (Y₂) (y) = C_* (Y₁) ∩ C_* (Y₂). Hence C_* (Y₁ ∩ Y₂) = C_* (Y₁) ∩ C_* (Y₂).

( $L_{4}^{*})$ Since ∀y ∈ V, C_* (Y₁ ∪ Y₂) (y) = min {(Y₁ ∪ Y₂) (z) : z ∈ N (y)} = min {Y₁ (z) ∨ Y₂ (z) : z ∈ N (y)} ≥ min {Y₁ (z) : z ∈ N (y)} = C_* (Y₁) (y). Also C_* (Y₁ ∪ Y₂) (y) = min {(Y₁ ∪ Y₂) (z) : z ∈ N (y)} = min {Y₂ (z) ∨ Y₂ (z) : z ∈ N (y)} ≥ min {Y₂ (z) : z ∈ N (y)} = C_* (Y₂) (y). Then C_* (Y₁ ∪ Y₂) (y) ≥ max {C_* (Y₁) (y) , C_* (Y₂) (y)} = (C_* (Y₁) ∪ C_* (Y₂)) (y). Therefore C_* (Y₁ ∪ Y₂) ⊇ C_* (Y₁) ∪ C_* (Y₂).

( $L_{5}^{*})$ Since Y₁ ⊆ Y₂, then ∀y ∈ V, Y₁ (y) ≤ Y₂ (y). Thus C_* (Y₁) (y) = min {(Y₁) (z) : z ∈ N (y)} ≤ min {(Y₂) (z) : z ∈ N (y)} = C_* (Y₂) (y). Therefore C_* (Y₁) ⊆ C_* (Y₂).

( $L_{9}^{*})$ According to Lemma 3.1, if z ∈ N (y), then N (z) ⊆ N (y). Thus C_* (C_* (Y)) (y) = min {C_* (Y) (z) : z ∈ N (y)} = min {min {Y (w) : w ∈ N (z)} : z ∈ N (y)} ≥ min {Y (w) : w ∈ N (z)} = C_* (Y) (y). Hence C_* (Y) ⊆ C_* (C_* (Y)) □.

Remark 5.1. Let (U, V, R) be two universe approximation space, where R ∈ U × V any binary relation. Then $\forall Y \in \begin{matrix} F \end{matrix} (V)$ , the following properties do not hold:

(L₆) $\underline{R} (\emptyset) = \emptyset$ , (L₇) $\underline{R} (Y) \subseteq Y$ ,

(L₈) $Y \subseteq \underline{R} (\bar{R} (Y))$ , (L₁₀) $\bar{R} (Y) \subseteq \underline{R} (\bar{R} (Y))$ ,

(U₆) $\bar{R} (V) = U$ , (U₇) $Y \subseteq \bar{R} (Y)$ ,

(U₈) $\bar{R} (\underline{R} (Y)) \subseteq Y$ , (U₁₀) $\bar{R} (\underline{R} (Y)) \subseteq \underline{R} (Y)$ ,

(LU) $\underline{R} (Y) \subseteq \bar{R} (Y)$ .

The following example shows Remark 5.1.

Example 5.1. Let U = {x₁, x₂, x₃, x₄, x₅, x₆}, V = {y₁, y₂, y₃, y₄, y₅, y₆} and R ∈ U × V be a binary relation defined as Table 3: Let Y be a fuzzy subset of V defined as: Y (y₁) =0.2, Y (y₂) =0.5, Y (y₃) =0.1, Y (y₄) =0.7, Y (y₅) =0.3, Y (y₆) =0.8, then we have

Hence, from Table 4 we have $\underline{R} (\emptyset) \neq \emptyset$ , $\bar{R} (V) \neq U$ , $\underline{R} Y) ⊈ Y$ , $Y ⊈ \bar{R} (Y)$ , $Y ⊈ \underline{R} (\bar{R} (Y))$ , $\bar{R} (\underline{R} (Y)) ⊈ Y$ , $\bar{R} (Y) ⊈ \underline{R} (\bar{R} (Y))$ , $\bar{R} (\underline{R} (Y)) ⊈ \underline{R} (Y)$ and $\underline{R} (Y) ⊈ \bar{R} (Y)$ . Therefore, L₆, U₆, L₇, U₇, L₈, U₈, L₁₀, U₁₀ and LU do not hold.

Proposition 5.2. Let (U, V, C_V) be two universe covering approximation space. Then for all $Y \in \begin{matrix} F \end{matrix} (V)$ , the approximation operators have the following properties:

$(L_{6}^{*})$ C_* (∅) = ∅, $(L_{7}^{*})$ C_* (Y) ⊆ Y, $(U_{6}^{*})$ C^* (V) = V, $(U_{7}^{*})$ Y ⊆ C^* (Y),

(LU^*) C_* (Y) ⊆ C^* (Y).

Proof. The proof comes from Definition 5.1 and Lemma 3.2. □

Example 5.2. Let U = {x₁, x₂, x₃, x₄, x₅, x₆}, V = {y₁, y₂, y₃, y₄, y₅, y₆} and R ∈ U × V be a binary relation defined as Table 5: Let Y be a fuzzy subset of V defined as: Y (y₁) =0.2, Y (y₂) =0.5, Y (y₃) =0.1, Y (y₄) =0.7, Y (y₅) =0.3, Y (y₆) =0.8. Therefor, from Table 6 we have $Y ⊈ \underline{R} (\bar{R} (Y)), \bar{R} (\underline{R} (Y)) ⊈ Y, \bar{R} (Y) ⊈ \underline{R} (\bar{R} (Y))$ , and $\bar{R} (\underline{R} (Y)) ⊈ \bar{R} (Y)$ . Hence L₈, U₈, L₁₀, and U₁₀ do not hold.

Proposition 5.3. Let (U, V, C_V) be two universe covering approximation space, C is strong covering. Then for all $Y \in \begin{matrix} F \end{matrix} (V)$ , the approximation operators have the following properties:

$(L_{8}^{*})$ Y ⊆ C_* (C^* (Y)), $(L_{10}^{*})$ C^* (Y) ⊆ C_* (C^* (Y)),

$(U_{8}^{*})$ C^* (C_* (Y)) ⊆ Y, $(U_{10}^{*})$ C^* (C_* (Y)) ⊆ C_* (Y).

Proof. ( $L_{10}^{*}$ ) Since C is strong covering, then if z ∈ N (y) implies that N (z) = N (y) for all y, z ∈ V. Thus C_* (C^* (Y)) (y) = min {C^* (Y) (z) : z ∈ N (y)} = min {max {Y (w) : w ∈ N (z)} : z ∈ N (y)} = max {Y (w) : w ∈ N (y)} = C^* (Y) (y). Hence,C^* (Y) ⊆ C_* (C^* (Y)).

The proofs of $L_{8}^{*}, U_{8}^{*}$ , and $U_{10}^{*}$ are similar. □

Example 5.3. (a real-life application). Let U = {x₁, x₂, x₃, x₄} be a set of four computers, V = {y₁, y₂, y₃, y₄} be a set of four network connectivity devices where y₁ is a hub, y₂ switch, y₃ bridge, y₄ and A = {A₁, A₂, A₃} be the attributes of network connectivity devices, where A₁ =Connection = {a₁, b₁, c₁, d₁, e₁, f₁, g₁}, where a₁ =connect all computers on each side of the network, b₁ =connect two segments of the same LAN, c₁ =connect two local area networks (LANs), d₁ =connect two similar network, e₁ =connect two dissimilar network, f₁ = divide a busy network into two segments and g₁ = select the best path to route a message based on the destination address and origin. A₂ =read or cannot read the addresses= {a₂, b₂, c₂, d₂} = {a₁, b₁, c₁, d₁, e₁, f₁, g₁} where a₂ =read the addressees of all computers on each side of the network, b₂ =cannot read the addressees of any computer in the network, c₂ =read the addressees of bridges on the network and e₂ =read the addressees of other products on the network. A₃ =cost = {a₃, b₃, c₃}, where a₃ = 15dollar,b₃ = 45dollar and c₃ = 75dollar. By taking the first attribute, we can get F (x₁) = {y₁, y₂}, F (x₂) = {y₃, y₄}, F (x₃) = {y₁, y₂, y₃} and F (x₄) = {y₁, y₂, y₄}. Therefore, C_V = {{y₁, y₂} , {y₃, y₄} , {y₁, y₂, y₃} , {y₁, y₂, y₄}}. Hence, N (y₁) = {y₁, y₂}, N (y₂) = {y₁, y₂}, N (y₃) = {y₁, y₂, y₃} and N (y₄) = {y₄}. Suppose $Y \in \begin{matrix} P \end{matrix} (V)$ such that Y = {y₁, y₂, y₃}, then C_* (Y) = {y₁, y₂, y₃} and C^* (Y) = {y₁, y₂, y₃}. So our approach can reduce the boundary region. Also, let $Y \in \begin{matrix} F \end{matrix} (V)$ such that Y (y₁) =0.1, Y (y₂) =0.6, Y (y₃) =1, Y (y₄) =0.8, then C_* (Y) (y₁) =0.2, C_* (Y) (y₂) =0.2, C_* (Y) (y₃) =0.2, C_* (Y) (y₄) =0.8 and C^* (Y) (y₁) =0.6, C^* (Y) (y₂) =0.6, C^* (Y) (y₃) =1, C^* (Y) (y₄) =0.8.

6 Conclusion

We studied the two universes approximation spaces via covering-based rough sets. The lower and upper approximation on two universes based on covering are introduced. All the properties of rough sets have been simulated by employing covering approach on two universes. Finally, we expanded our approach for rough fuzzy sets on two universes and the properties of this type are examined.

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