Generalizations of rough set concepts

Let (U, R) be a generalized approximation space. Let τ_xR and τ_Rx be the two topologies generated using the relation R. Then the subset X ⊆ U is called:

1. Semi rough (briefly S₁₂-rough) if $X\subseteq {\overline{R}}_{{\tau }_{\mathit{Rx}}}({\underline{R}}_{{\tau }_{\mathit{xR}}}(X))$ .

2. Pre rough (briefly P₁₂-rough) if $X\subseteq {\overline{R}}_{{\tau }_{\mathit{xR}}}({\underline{R}}_{{\tau }_{\mathit{Rx}}}(A))$ .

3. Semi-pre rough (briefly β₁₂-rough) if $X\subseteq {\overline{R}}_{{\tau }_{\mathit{Rx}}}({\underline{R}}_{{\tau }_{\mathit{xR}}}({\overline{R}}_{{\tau }_{\mathit{Rx}}}(X)))$ .

4. α-Rough (briefly α₁₂-rough) if $X\subseteq {\underline{R}}_{{\tau }_{\mathit{xR}}}({\overline{R}}_{{\tau }_{\mathit{Rx}}}({\underline{R}}_{{\tau }_{\mathit{xR}}}(X)))$ .

5. γ-Rough (briefly γ₁₂-rough) if $X\subseteq {\overline{R}}_{{\tau }_{\mathit{Rx}}}({\underline{R}}_{{\tau }_{\mathit{xR}}}(X))\cup {\underline{R}}_{{\tau }_{\mathit{xR}}}({\overline{R}}_{{\tau }_{\mathit{Rx}}}(X))$ .

The family of all S₁₂-rough (resp. P₁₂-rough, β₁₂-rough, α₁₂-rough and γ₁₂-rough) sets in (U, R) is denoted by FS₁₂(U) (resp. FP₁₂(U), Fβ₁₂(U), Fα₁₂(U) and Fγ₁₂(U)).

The complement of S₁₂-rough (resp. P₁₂-rough, β₁₂-rough, α₁₂-rough and γ₁₂-rough) set is called $S_{12}^c$ -rough (resp. $P_{12}^c$ -rough, ${\beta }_{12}^c$ -rough, ${\alpha }_{12}^c$ -rough and ${\gamma }_{12}^c$ -rough).

The family of all $S_{12}^c$ -rough (resp. $P_{12}^c$ -rough, ${\beta }_{12}^c$ -rough, ${\alpha }_{12}^c$ -rough and ${\gamma }_{12}^c$ -rough) sets of (U, R) is denoted by ${\mathit{FS}}_{12}^c(U)$ (resp. ${\mathit{FP}}_{12}^c(U)\text{,}F{\beta }_{12}^c(U)\text{,}F{\alpha }_{12}^c(U)$ and $F{\gamma }_{12}^c(U))$ .

In the generalized approximation space (U, R), we can prove that:

1. Fα₁₂(U) < FS₁₂(U) < Fγ₁₂(U) < Fβ₁₂(U).

2. Fα₁₂(U) < FP₁₂(U) < Fγ₁₂(U) < Fβ₁₂(U).

Obvious. □

Let R be any binary relation defined on a nonempty set U = {a, b, c, d} defined by R = {(a, a), (a, c), (a, d), (b, b), (b, d), (c, a), (c, b), (c, d), (d, a)}. Hence the subbase of τ_xR is {{a, c, d}, {b, d}, {a, b, d}, {a}} and the subbase of τ_Rx is {{a, c, d}, {b, c}, {a}, {a, b, c}}. Then $$\begin{array}{cc}\hfill & {\tau }_{\mathit{xR}}=\{U\text{,}\varphi \text{,}\{a\text{,}c\text{,}d\}\text{,}\{b\text{,}d\}\text{,}\{a\text{,}b\text{,}d\}\text{,}\{a\}\text{,}\{d\}\text{,}\{a\text{,}d\}\}\text{,}\hfill \\ \hfill & {\tau }_{\mathit{Rx}}=\{U\text{,}\varphi \text{,}\{a\text{,}c\text{,}d\}\text{,}\{a\text{,}c\}\text{,}\{a\text{,}b\text{,}c\}\text{,}\{a\}\text{,}\{c\}\text{,}\{b\text{,}c\}\}\text{.}\hfill \end{array}$$ Consequently, $$F{\alpha }_{12}(U)={\mathit{FS}}_{12}(U)=\{U\text{,}\varphi \text{,}\{a\text{,}c\text{,}d\}\text{,}\{b\text{,}d\}\text{,}\{a\text{,}b\text{,}d\}\text{,}\{a\}\text{,}\{d\}\text{,}\{a\text{,}d\}\}\text{,}$$ $$F{\gamma }_{12}(U)=F{\beta }_{12}(U)=P_{\mathit{rl}}O(U)=\{U\text{,}\varphi \text{,}\{a\}\text{,}\{d\}\text{,}\{a\text{,}b\}\text{,}\{a\text{,}d\}\text{,}\{a\text{,}c\}\text{,}\{a\text{,}b\text{,}c\}\text{,}\{b\text{,}d\}\text{,}\{a\text{,}b\text{,}d\}\text{,}\{a\text{,}c\text{,}d\}\}\text{.}$$

Let (U, R) be a generalized approximation space and X ⊆ U. Then the general lower (briefly λ₁₂ lower) of X denoted by λ₁₂(X) for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂} is defined by: λ₁₂(X) = ∪ {G ∈ Fλ₁₂(U), G ⊆ X}.

Let (U, R) be a generalized approximation space and X ⊆ U. Then the general upper of X denoted by ${\overline{\lambda }}_{12}(X)$ for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂} is defined by ${\overline{\lambda }}_{12}(X)=\cap \{H\in F{\lambda }_{12}^c(U)\text{,}H\supseteq X\}$ .

Let (U, R) be a generalized approximation space. Then for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂} the topological general lower and topological general upper approximations of any subset X ⊆ U are defined as: ${\underline{R}}_{{\lambda }_{12}}(X)={\underline{\lambda }}_{12}(X)\text{;}{\overline{R}}_{{\lambda }_{12}}(X)={\overline{\lambda }}_{12}(X)$ .

Let (U, R) be a generalized approximation space generated by any binary relation R. Then for any subset X ⊆ U:

1. ${\underline{R}}_{{\tau }_{\mathit{xR}}}(X)\subseteq {\underline{R}}_{{\alpha }_{12}}(X)\subseteq {\underline{R}}_{S_{12}}(X)\subseteq {\underline{R}}_{{\gamma }_{12}}(X)\subseteq {\underline{R}}_{{\beta }_{12}}(X)\subseteq X\subseteq {\overline{R}}_{{\beta }_{12}}(X)\subseteq {\overline{R}}_{{\gamma }_{12}}(X)\subseteq {\overline{R}}_{S_{12}}(X)\subseteq {\overline{R}}_{{\alpha }_{12}}(X)\subseteq {\overline{R}}_{{\tau }_{\mathit{xR}}}(X)$ .

2. ${\underline{R}}_{{\tau }_{\mathit{Rx}}}(X)\subseteq {\underline{R}}_{{\alpha }_{12}}(X)\subseteq {\underline{R}}_{P_{12}}(X)\subseteq {\underline{R}}_{{\gamma }_{12}}(X)\subseteq {\underline{R}}_{{\beta }_{12}}(X)\subseteq X\subseteq {\overline{R}}_{{\beta }_{12}}(X)\subseteq {\overline{R}}_{{\gamma }_{12}}(X)\subseteq {\overline{R}}_{P_{12}}(X)\subseteq {\overline{R}}_{{\alpha }_{12}}(X)\subseteq {\overline{R}}_{{\tau }_{\mathit{Rx}}}(X)$ .

$$(\mathrm{i}){\underline{R}}_{{\tau }_{\mathit{xR}}}(X)=\cup \{G\in {\tau }_{\mathit{xR}}:G\subseteq X\}\subseteq \cup \{G\in F{\alpha }_{12}(U):G\subseteq X\}\subseteq \cup \{G\in {\mathit{FS}}_{12}(U):G\subseteq X\}\subseteq \cup \{G\in F{\gamma }_{12}(U):G\subseteq X\}\subseteq \cup \{G\in F{\beta }_{12}(U):G\subseteq X\}\subseteq X\subseteq \cap \{H\in F{\beta }_{12}^c(U):X\subseteq H\}\subseteq \cap \{H\in F{\gamma }_{12}^c(U):X\subseteq H\}\subseteq \cap \{H\in {\mathit{FS}}_{12}^c(U):X\subseteq H\}\subseteq \cap \{H\in F{\alpha }_{12}^c(U):X\subseteq H\}\subseteq \cap \{H\in {\tau }_{\mathit{xR}}^c:X\subseteq H\}\text{.}$$

Hence, ${\underline{R}}_{{\tau }_{\mathit{xR}}}(X)\subseteq {\underline{R}}_{{\alpha }_{12}}(X)\subseteq {\underline{R}}_{S_{12}}(X)\subseteq {\underline{R}}_{{\gamma }_{12}}(X)\subseteq {\underline{R}}_{{\beta }_{12}}(X)\subseteq X\subseteq {\overline{R}}_{{\beta }_{12}}(X)\subseteq {\overline{R}}_{{\gamma }_{12}}(X)\subseteq {\overline{R}}_{S_{12}}(X)\subseteq {\overline{R}}_{{\alpha }_{12}}(X)\subseteq {\overline{R}}_{{\tau }_{\mathit{xR}}}(X)$ .

(ii) By the same manner as (i). □

According to Example 3.1, if X = {a, b} and Y = {d}. Then ${\underline{R}}_{{\alpha }_{12}}(X)=\{a\}\text{,}{\underline{R}}_{P_{12}}(X)=\{a\text{,}b\}\text{,}{\overline{R}}_{{\alpha }_{12}}(Y)=\{b\text{,}c\text{,}d\}$ and ${\overline{R}}_{P_{12}}(Y)=\{d\}$ . So ${\underline{R}}_{{\alpha }_{12}}(X)\subset {\underline{R}}_{P_{12}}(X)$ and ${\overline{R}}_{P_{12}}(Y)\subset {\overline{R}}_{{\alpha }_{12}}(Y)$ .

Let (U, R) be a generalized approximation space generated by any binary relation R. Then for any two subsets X, Y ⊆ U we have for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}:

1. ${\underline{R}}_{{\lambda }_{12}}(\varphi )={\overline{R}}_{{\lambda }_{12}}(\varphi )=\varphi$ and ${\underline{R}}_{{\lambda }_{12}}(U)={\overline{R}}_{{\lambda }_{12}}(U)=U$ .

2. If X ⊆ Y, then ${\underline{R}}_{{\lambda }_{12}}(X)\subseteq {\underline{R}}_{{\lambda }_{12}}(Y)$ .

3. If X ⊆ Y, then ${\overline{R}}_{{\lambda }_{12}}(X)\subseteq {\overline{R}}_{{\lambda }_{12}}(Y)$ .

4. ${\underline{R}}_{{\lambda }_{12}}(X\cup Y)\supseteq {\underline{R}}_{{\lambda }_{12}}(X)\cup {\underline{R}}_{{\lambda }_{12}}(Y)$ .

5. ${\overline{R}}_{{\lambda }_{12}}(X\cup Y)\supseteq {\overline{R}}_{{\lambda }_{12}}(X)\cup {\overline{R}}_{{\lambda }_{12}}(Y)$ .

6. ${\underline{R}}_{{\lambda }_{12}}(X\cap Y)\subseteq {\underline{R}}_{{\lambda }_{12}}(X)\cap {\underline{R}}_{{\lambda }_{12}}(Y)$ .

7. ${\overline{R}}_{{\lambda }_{12}}(X\cap Y)\subseteq {\overline{R}}_{{\lambda }_{12}}(X)\cap {\overline{R}}_{{\lambda }_{12}}(Y)$ .

8. ${\underline{R}}_{{\lambda }_{12}}(X^c)=({\overline{R}}_{{\lambda }_{12}}(X)){\hspace{0.17em}}^c$ .

9. ${\overline{R}}_{{\lambda }_{12}}(X^c)=({\underline{R}}_{{\lambda }_{12}}(X)){\hspace{0.17em}}^c$ .

By using the properties of λ₁₂ and ${\overline{\lambda }}_{12}$ for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂} the proof is complete. □

According to Example 3.1 if X = {a} andY = {c, d}, then ${\underline{R}}_{{\alpha }_{12}}(X)=\{a\}\text{,}{\underline{R}}_{{\alpha }_{12}}(Y)=\{d\}$ , and ${\underline{R}}_{{\alpha }_{12}}(X\cup Y)\ne {\underline{R}}_{{\alpha }_{12}}(X)\cup {\underline{R}}_{{\alpha }_{12}}(Y)$ .

The following example shows that the inverse of the Properties (v) and (vi) in Proposition 3.2, in general are not true for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}, we consider λ₁₂ = β₁₂.

According to Example 3.1, if X₁ = {a}, X₂ = {d}, Y₁ = {a, b} and Y₂ = {b, d}, then ${\overline{R}}_{{\beta }_{12}}(X_1)=\{a\text{,}c\}\text{,}{\overline{R}}_{{\beta }_{12}}(X_2)=\{d\}\text{,}{\overline{R}}_{{\beta }_{12}}(X_1\cup X_2)=U\text{,}{\underline{R}}_{{\beta }_{12}}(Y_1)=\{a\text{,}b\}\text{,}{\underline{R}}_{{\beta }_{12}}(Y_2)=\{b\text{,}d\}$ , and ${\underline{R}}_{{\beta }_{12}}(Y_1\cap Y_2)=\varphi$ , hence ${\overline{R}}_{{\beta }_{12}}(X_1\cup X_2)\ne {\overline{R}}_{{\beta }_{12}}(X_1)\cup {\overline{R}}_{{\beta }_{12}}(X_2)$ , and ${\underline{R}}_{{\beta }_{12}}(Y_1\cap Y_2)\ne {\underline{R}}_{{\beta }_{12}}(Y_1)\cap {\underline{R}}_{{\beta }_{12}}(Y_2)$ .

The following example shows that the Property (vii) in Proposition 3.2, in general are not true for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}, we consider here λ₁₂ = P₁₂.

According to Example 3.1, if X = {a, c, d} and Y = {b, c, d}. Then ${\overline{R}}_{P_{12}}(X)=U\text{,}{\overline{R}}_{P_{12}}(Y)=\{b\text{,}c\text{,}d\}$ and ${\overline{R}}_{P_{12}}(X\cap Y)=\{c\text{,}d\}$ , hence ${\overline{R}}_{P_{12}}(X\cap Y)\ne {\overline{R}}_{P_{12}}(X)\cap {\overline{R}}_{P_{12}}(Y)$ .

Let (X, R) be a generalized approximation space defined on any binary relation R. Then for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}, and for any X ⊆ U, the following properties do not hold:

1. ${\underline{R}}_{{\lambda }_{12}}({\underline{R}}_{{\lambda }_{12}}(X))={\overline{R}}_{{\lambda }_{12}}({\underline{R}}_{{\lambda }_{12}}(X))={\underline{R}}_{{\lambda }_{12}}(X)$ .

2. ${\overline{R}}_{{\lambda }_{12}}({\overline{R}}_{{\lambda }_{12}}(X))={\underline{R}}_{{\lambda }_{12}}({\overline{R}}_{{\lambda }_{12}}(X))={\overline{R}}_{{\lambda }_{12}}(X)$ .

The following example illustrates the above proposition, using λ₁₂ = β₁₂.

According to Example 3.1, if X = {a} and Y = {c, d}. Then ${\underline{R}}_{{\beta }_{12}}(X)=\{a\}\text{,}{\underline{R}}_{{\beta }_{\mathit{rl}}}{\underline{R}}_{{\beta }_{\mathit{rl}}}(A)=\{a\}\text{,}{\overline{R}}_{{\beta }_{\mathit{rl}}}{\underline{R}}_{{\beta }_{\mathit{rl}}}(A)=\{a\text{,}c\}\text{,}{\overline{R}}_{{\beta }_{\mathit{rl}}}(B)=\{c\text{,}d\}\text{,}{\overline{R}}_{{\beta }_{\mathit{rl}}}{\overline{R}}_{{\beta }_{\mathit{rl}}}(B)=\{c\text{,}d\}$ , and ${\underline{R}}_{{\beta }_{12}}({\overline{R}}_{{\beta }_{12}}(Y))=\{d\}$ , hence ${\underline{R}}_{{\beta }_{12}}({\underline{R}}_{{\beta }_{12}}(X))\ne {\overline{R}}_{{\beta }_{12}}({\underline{R}}_{{\beta }_{12}}(X))$ , and ${\overline{R}}_{{\beta }_{12}}({\overline{R}}_{{\beta }_{12}}(Y))\ne {\underline{R}}_{{\beta }_{12}}({\overline{R}}_{{\beta }_{12}}(Y))$ .

Let (U, R) be a generalized approximation space, and for any X ⊆ U, then $({\mathit{Cl}}_{{\lambda }_{12}}(X)){\hspace{0.17em}}^c={\mathit{int}}_{{\lambda }_{12}}(X^c)$ for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}.

Let X ⊆ U, then for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}, we get: $$({\mathit{Cl}}_{{\lambda }_{12}}(X)){\hspace{0.17em}}^c=U-{\mathit{Cl}}_{{\lambda }_{12}}(X)=U-\cap \{F\subseteq U:F\text{is}{\lambda }_{12}\text{upper}\text{set and}X\subseteq F\}=\cup \{(U-F)\subseteq U:(U-F)\mathrm{is}{\lambda }_{12}\text{lower set,}(U-F)\subseteq (U-X)={\mathit{int}}_{{\lambda }_{12}}(U-X)\text{.}$$

Thus $({\mathit{Cl}}_{{\lambda }_{12}}(X)){\hspace{0.17em}}^c={\mathit{int}}_{{\lambda }_{12}}(X^c)$ . □

Let (U, R) be a generalized approximation space defined on any binary relation R for any two subsets X, Y ⊆ U we have: ${\underline{R}}_{{\lambda }_{12}}(X-Y)\subseteq {\underline{R}}_{{\lambda }_{12}}(X)-{\underline{R}}_{{\lambda }_{12}}(Y)$ , for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}.

As X − Y = X ∩ Y^c, then: $${\underline{R}}_{{\lambda }_{12}}(X-Y)={\mathit{int}}_{{\lambda }_{12}}(X-Y)={\mathit{int}}_{{\lambda }_{12}}(X\cap Y^c)\subseteq {\mathit{int}}_{{\lambda }_{12}}(X)\cap {\mathit{int}}_{{\lambda }_{12}}(Y^c)\text{.}$$

By Lemma 3.1, we have: $${\underline{R}}_{{\lambda }_{12}}(X-Y)\subseteq {\mathit{int}}_{{\lambda }_{12}}(X)\cap ({\mathit{Cl}}_{{\lambda }_{12}}(Y)){\hspace{0.17em}}^c={\mathit{int}}_{{\lambda }_{12}}(X)-{\mathit{Cl}}_{{\lambda }_{12}}(Y)\subseteq {\mathit{int}}_{{\lambda }_{12}}(X)-{\mathit{int}}_{{\lambda }_{12}}(Y)$$

Thus ${\underline{R}}_{{\lambda }_{12}}(X-Y)\subseteq {\underline{R}}_{{\lambda }_{12}}(X)-{\underline{R}}_{{\lambda }_{12}}(Y)$ .

The next example illustrates that the inverse of Proposition 3.5, in general does not hold with respect to λ₁₂ = β₁₂. □

According to Example 3.1, if X = {a, b} and Y = {a}, then ${\underline{R}}_{{\beta }_{12}}(Y)=\{a\}\text{,}{\underline{R}}_{{\beta }_{12}}(X)=\{a\text{,}b\}\text{,}$ and ${\underline{R}}_{{\beta }_{12}}(X-Y)=\varphi$ , thus ${\underline{R}}_{{\beta }_{12}}(X-Y)\ne {\underline{R}}_{{\beta }_{12}}(X)-{\underline{R}}_{{\beta }_{12}}(Y)$ .

Let (U, R) be a generalized approximation space defined on any binary relation R. Then for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂} and for any X ⊆ U we define:

1. X is totally topological λ₁₂-definable (λ₁₂-exact) set if ${\underline{R}}_{{\lambda }_{12}}(X)={\overline{R}}_{{\lambda }_{12}}(X)=X$ .

2. X is internally topological λ₁₂-definable set if ${\underline{R}}_{{\lambda }_{12}}(X)=X\text{,}\mathrm{and}{\overline{R}}_{{\lambda }_{12}}(X)\ne X$ .

3. X is externally topological λ₁₂-definable set if ${\underline{R}}_{{\lambda }_{12}}(X)\ne X\text{,}\mathrm{and}{\overline{R}}_{{\lambda }_{12}}(X)=X$ .

4. X is topologically λ₁₂-indefinable (λ₁₂-rough) set if ${\underline{R}}_{{\lambda }_{12}}(X)\ne A\text{,}\mathrm{and}{\overline{R}}_{{\lambda }_{12}}(X)\ne X$ .

According to Example 3.1, for subsets X = {d} and Y = {c, d}, X is topologicallyβ₁₂-exact set, X is topologically internally α₁₂-definable set, Y is topologically S₁₂-rough set, and Y is topologically externally P₁₂-definable set.

Let (U, R) be a generalized approximation space defined on any binary relation R. Then we can introduce the generalized accuracy measure for any set X ⊆ U as the following: $${\mathit{Acc}}_{{\lambda }_{12}}(X)=\frac{\left|{\underline{R}}_{{\lambda }_{12}}(X)\right|}{\left|{\overline{R}}_{{\lambda }_{12}}(X)\right|}\text{,}{\overline{R}}_{{\lambda }_{12}}(X)\ne \varphi \text{,}$$ where λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂},and ∣X∣ denoted the cardinality of the set X.

The number ${\mathit{Acc}}_{{\lambda }_{12}}$ of the above definition is a measure of the degree of exactness of any subset X ⊆ U. So by this measure we will determine, what is the best of our definitions for the λ₁₂ lower and λ₁₂ upper approximations. We can notice that:

1. $0⩽{\mathit{Acc}}_{{\tau }_{\mathit{xR}}}(X)⩽{\mathit{Acc}}_{{\alpha }_{12}}(X)⩽{\mathit{Acc}}_{S_{12}}(X)⩽{\mathit{Acc}}_{{\gamma }_{12}}(X)⩽{\mathit{Acc}}_{{\beta }_{12}}(X)⩽1$ .

2. $0⩽{\mathit{Acc}}_{{\tau }_{\mathit{xR}}}(X)⩽{\mathit{Acc}}_{{\alpha }_{12}}(X)⩽{\mathit{Acc}}_{P_{12}}(X)⩽{\mathit{Acc}}_{{\gamma }_{12}}(X)⩽{\mathit{Acc}}_{{\beta }_{12}}(X)⩽1$ .

The next example studies the comparison between β₁₂ and S₁₂.

According to Example 3.1, we have the following table:

By using the definitions of rough concepts at λ₁₂ = β₁₂ we can tends to exactness of many sets. This will lead to accurate results in many data reduction applications using new topological approaches. Next works shall deal with more types of applications in data reductions, data processing, image processing and rule extraction.

Let (U, R) be a generalized approximation space defined on any binary relation R. Then for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂} and for any X, Y ⊆ U we define:

1. $X\underset{\sim {\lambda }_{12}}{\subset }Y$ if ${\underline{R}}_{{\lambda }_{12}}(X)\subseteq {\underline{R}}_{{\lambda }_{12}}(Y)$ .

2. $X{\tilde{\subset }}_{{\lambda }_{12}}Y$ if ${\overline{R}}_{{\lambda }_{12}}(X)\subseteq {\overline{R}}_{{\lambda }_{12}}(Y)$ .

   The illustration of the facts of the above definition are given as below example.

According to Example 3.1, let X₁ = {a, c, d}, X₂ = {b, c, d}, X₃ = {b, d} and X₄ = {c, d}, then we have: $X_4\underset{\approx p_{12}}{\subset }{X}_3\text{,}{X}_2{\tilde{\subset }}_{s_{12}}{X}_1$ and $X_2{\tilde{\subset }}_{{\beta }_{12}}{X}_1$ .

Let (U, R) be a generalized approximation space defined on any binary relation R. For any subset X ⊆ U and any element x ∈ U, for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}, we define:

1. $x\underset{\approx {\lambda }_{12}}{\in }X$ if $x\in {\underline{R}}_{{\lambda }_{12}}(X)$ .

2. $x{\tilde{\in }}_{{\lambda }_{12}}X$ if $x\in {\overline{R}}_{{\lambda }_{12}}(X)$ .

Let (U, R) be a generalized approximation space defined on any binary relation R. For any subset X ⊆ U and any element x ∈ U, for all λ₁₂ ∈ {S₁₂, P₁₂, β₁₂, α₁₂, γ₁₂}, we have:

1. if $x\underset{\approx {\lambda }_{12}}{\in }X$ then x ∈ X.

2. if $x{\tilde{\notin }}_{{\lambda }_{12}}X$ then x ∉ X.

The proof is direct from definitions. □

According to Example 3.1, let X = {a, b, c} and Y = {a, d}, then we have b ∈ X, but $b\underset{\approx s_{12}}{\notin }X$ and $b\underset{\approx {\alpha }_{12}}{\notin }X$ . Also, we have b ∉ Y, but $b{\tilde{\notin }}_{s_{12}}Y\text{,}b{\tilde{\notin }}_{{\alpha }_{12}}Y\text{,}b{\tilde{\notin }}_{p_{\mathit{rL}}}D\text{,}b{\tilde{\notin }}_{{\gamma }_{12}}Y$ and $b{\tilde{\notin }}_{{\beta }_{12}}Y$ .