β-Approach to near set theory

Let ϕ_i ∈ B be general relations, where 1 ⩽ i ⩽ ∣B∣ defined on a nonempty set X, then we can introduce a general neighbourhood of an element x ∈ X as $$(x){\hspace{0.17em}}_{{\varphi }_{i_r}}=\{y\in X:|{\varphi }_i(y)-{\varphi }_i(x)|⩽r\}\text{,}$$ where ∣*∣ is the absolute value of * and r is the length of this neighbourhood.

Let B ⊆ F be a set of functions representing features of x, x′ ∈ X. Objects x and x′ are minimally near each other if ∃ϕ_i ∈ B s.t $x^{\prime }\in (x){\hspace{0.17em}}_{{\varphi }_{i_r}}\text{.}$

Let Y, Y′ ⊆ X and B ⊆ F. Set Y is near to Y′ if ∃x ∈ Y, x′ ∈ Y′ such that x is near to x′

Any subset of X is near to X.

From Definitions 3.2, 3.3, we get the proof obviously. □

Every set X is called near set (near to itself) as every element x ∈ X is near to itself.

Let $(X\text{,}{\tau }_{{\varphi }_i})$ be a topological spaces, where ϕ_i ∈ B, 1 ⩽ i ⩽ ∣B∣. Hence we can define new near lower and upper approximations for any subset A ⊆ X with respect to one probe function ϕ_i as $$\begin{array}{cc}\hfill & {\underset{\_}{N}}_1^{\prime }(A)=\bigcup N_1^{\prime }(B)\text{,}G\subseteq A\}\text{,}\hfill \\ \hfill & {\overline{N}}_1^{\prime }(A)=\bigcap \{F:F\in [N_1^{\prime }(B)]{\hspace{0.17em}}^c\text{,}A\subseteq F\}\text{,}\hfill \end{array}$$ where $N_1^{\prime }(B)=\left\{G:G\in {\bigcup }_{{\varphi }_i\in B}{\tau }_{{\varphi }_i}\right\}$ and ${\tau }_{{\varphi }_i}$ is the topology generated from the family of general neighbourhoods with respect to the probe function ϕ_i ∈ B.

The new near lower and upper approximations with respect to two features of a probe functions are defined as $$\begin{array}{cc}\hfill & {\underset{\_}{N}}_2^{\prime }(A)=\bigcup \{G:G\in N_2^{\prime }(B)\text{,}G\subseteq A\}\text{,}\hfill \\ \hfill & {\overline{N}}_2^{\prime }(A)=\bigcap \{F:F\in [N_2^{\prime }(B)]{\hspace{0.17em}}^c\text{,}A\subseteq F\}\text{,}\hfill \end{array}$$ where $N_2^{\prime }(B)=\{G:G\in {\bigcup }_{{\varphi }_i\text{,}{\varphi }_j\in B}{\tau }_{{\varphi }_i{\varphi }_j}\text{,}i\ne j\}$ and ${\tau }_{{\varphi }_i{\varphi }_j}$ is the topology generated from the family of general neighbourhoods with respect to two features. Consequently, $$\begin{array}{cc}\hfill & {\underset{\_}{N}}_{|B}^{\prime }(A)=\bigcup \{G:G\in N_{|B|}^{\prime }(B)\text{,}G\subseteq A\}\text{,}\hfill \\ \hfill & {\overline{N}}_{|B|}^{\prime }(A)=\bigcap \{F:F\in [N_{|B|}^{\prime }(B)]{\hspace{0.17em}}^c\text{,}A\subseteq F\}\text{,}\hfill \end{array}$$ where $N_{|B|}^{\prime }(B)=\{G:G\in {\bigcup }_{{\varphi }_1\text{,}\dots \text{,}{\varphi }_{|B|}\in B}{\tau }_{{\varphi }_1\dots {\varphi }_{|B|}}\}$ and ${\tau }_{{\varphi }_1\text{,}\dots {\varphi }_{|B|}}$ is the topology generated from the family of general neighbourhoods with respect to all probe functions.

Let $(X\text{,}{\tau }_{{\varphi }_i})$ be topological spaces, where 1 ⩽ i ⩽ ∣B∣. The accuracy measure of any subset A ⊆ X with respect to the probe functions ϕ_i,i = 1, 2, … , ∣B∣ is defined as $${\alpha }_i^{\prime }(A)=\frac{\left|{\underset{\_}{N}}_i^{\prime }(A)\right|}{\left|{\overline{N}}_i^{\prime }(A)\right|}\text{,}A\ne \varphi \text{.}$$

$0⩽{\alpha }_i^{\prime }(A)⩽1$ that measures the degree of exactness of any subset A ⊆ X, if ${\alpha }_i^{\prime }(A)=1$ , then A is exact set with respect to ∣i∣ features.

Let $(X\text{,}{\tau }_{{\varphi }_i})$ be topological spaces, where ϕ_i ∈ B. The generalized lower rough coverage of any subset Yof the family of neighbourhoods with respect to B is defined as $${\nu }_i^{\prime }(Y\text{,}{\underset{\_}{N}}_i^{\prime }(D))=\frac{\left|Y\cap {\underset{\_}{N}}_i^{\prime }(D)\right|}{\left|{\underset{\_}{N}}_i^{\prime }(D)\right|}\text{,}$$ where D is the decision class, means the acceptable objects (Peters, 2007a), ${\underset{\_}{N}}_i^{\prime }(D)\ne \varphi$ . If ${\underset{\_}{N}}_i^{\prime }(D)=\varphi$ , then ${\nu }_i^{\prime }(Y\text{,}{\underset{\_}{N}}_i^{\prime }(D))=1$ .

$0⩽{\nu }_i^{\prime }⩽1$ , it is used to measure the degree that the subset Y covers the sure region ${\underset{\_}{N}}_i^{\prime }(D)$ .

A subset A of a topological space (X, τ) is called β-open (Abd El-Monsef et al., 1983) if $$A\subset \mathit{cl}(\mathit{int}(\mathit{cl}(A)))\text{.}$$ The set of all β-open sets defined by βO(X).

Let $(X\text{,}{\tau }_{{\varphi }_i})$ be topological spaces, where ϕ_i ∈ B, 1 ⩽ i ⩽ ∣B∣. The new β-near lower and upper approximations for any subset A ⊆ X with respect to one feature of the probe functions B are defined as $$\begin{array}{cc}\hfill & {\underset{\_}{N}}_{{\beta }_1}(A)={\bigcup }_{{\beta }_1}(B)\text{,}G\subseteq A\}\text{,}\hfill \\ \hfill & {\overline{N}}_{{\beta }_1}(A)=\bigcap \{F:F\in [N_{{\beta }_1}(B)]{\hspace{0.17em}}^c\text{,}A\subseteq F\}\text{,}\hfill \end{array}$$ where $N_{{\beta }_1}(B)=\{G:G\in {\bigcup }_{i=1\text{,}2\text{,}\dots \text{,}|B|}{\beta }_{i}O(X)\}$ and β_iO(X) is the family of beta open sets with respect to the topology ${\tau }_{{\varphi }_i}\text{.}$ Hence the boundary region of A with respect to one feature is defined as $b_{N_{{\beta }_1}}{\overline{N}}_{{\beta }_1}(A)-{\underset{\_}{N}}_{{\beta }_1}(A)$ .

The new β-near lower and upper approximations with respect to two features take the form $$\begin{array}{cc}\hfill & {\underset{\_}{N}}_{{\beta }_2}(A)=\bigcup \{G:G\in N_{{\beta }_2}(B)\text{,}G\subseteq A\}\text{,}\hfill \\ \hfill & {\overline{N}}_{{\beta }_2}(A)=\bigcap \{F:F\in [N_{{\beta }_2}(B)]{\hspace{0.17em}}^c\text{,}A\subseteq F\}\text{,}\hfill \end{array}$$ where $N_{{\beta }_2}(B)=\{G:G\in {\bigcup }_{i\text{,}j=1\text{,}2\text{,}\dots \text{,}|B|}{\beta }_{i\text{,}j}O(X)\text{,}i\ne j\}$ and β_i,jO(X) is the family of beta open sets with respect to the topology ${\tau }_{{\varphi }_i{\varphi }_j}\text{.}$ Consequently, $$\begin{array}{cc}\hfill & {\underset{\_}{N}}_{{\beta }_{|B|}}(A)=\bigcup \{G:G\in N_{{\beta }_{|B|}}(B)\text{,}G\subseteq A\}\text{,}\hfill \\ \hfill & {\overline{N}}_{{\beta }_{|B|}}(A)=\bigcap \{F:F\in [N_{{\beta }_{|B|}}(B)]{\hspace{0.17em}}^c\text{,}A\subseteq F\}\text{,}\hfill \end{array}$$ where $N_{{\beta }_{|B|}}(B)=\{G:G\in {\beta }_{1\text{,}2\text{,}\dots \text{,}|B|}O(X)\}$ and β_{1,2,…,∣B∣}O(X) is the family of beta open sets with respect to the topology ${\tau }_{{\varphi }_1{\varphi }_2\dots {\varphi }_{|B|}}\text{.}$

Let $(X\text{,}{\tau }_{{\varphi }_i})$ be topological spaces, where ϕ_i ∈ B, 1 ⩽ i ⩽ ∣B∣, hence we can define the β-near accuracy measure of any subset A ⊆ X with respect to i features of the probe functions B as $${\alpha }_{N_{{\beta }_i}}(A)=\frac{|{\underset{\_}{N}}_{{\beta }_i}(A)|}{|{\overline{N}}_{{\beta }_i}(A)|}\text{,}A\ne \varphi \text{.}$$

$0⩽{\alpha }_{N_{{\beta }_i}}(A)⩽1$ , it means the degree of exactness of any subset A ⊆ X. If ${\alpha }_{N_{{\beta }_i}}(A)=1$ , then A is $N_{{\beta }_i}$ -exact set with respect to ifeatures.

For any subset A ⊆ X, ${\underset{\_}{N}}_{{\beta }_i}(A)$ is near to ${\overline{N}}_{{\beta }_i}(A)$ , where 1 ⩽ i ⩽ ∣B∣.

From Definition 4.2 and Remark 4.1, we can deduce that ${\underset{\_}{N}}_{{\beta }_i}(A)\subseteq {\overline{N}}_{{\beta }_i}(A)\text{.}$ Hence from Theorem 3.1, we get the proof. □

For any subset A ⊆ X, $b_{N_{{\beta }_i}}(A)$ is near to ${\overline{N}}_{{\beta }_i}(A)$ , where 1 ⩽ i ⩽ ∣B∣.

A set A with a boundary $|b_{N_{{\beta }_i}}(A)|⩾0$ , is a near set.

Every rough set is a near set but not every near set is a rough set.

There are two cases to consider

1. $|b_{N_{{\beta }_i}}(A)|>0\text{.}$ Given a set A ⊆ X that has been approximated with a nonempty boundary, this means A is a rough set as well as a near set.

2. $|b_{N_{{\beta }_i}}(A)|=0\text{.}$ Given a set A ⊆ X that has been approximated with an empty boundary, this means A is a near set but not a rough set. □

Let $(X\text{,}{\tau }_{{\varphi }_i})$ be topological spaces, where ϕ_i  ∈ B,1 ⩽ i ⩽ ∣B∣. The new generalized lower rough coverage of any subset Y ⊆ Xwith respect to the sure region of the decision class D is defined as $${\nu }_{{\beta }_i}(Y\text{,}{\underset{\_}{N}}_{{\beta }_i}(D))=\frac{|Y\cap {\underset{\_}{N}}_{{\beta }_i}(D)t|}{|{\underset{\_}{N}}_{{\beta }_i}(D)|}\text{,}{\underset{\_}{N}}_{{\beta }_i}(D)\ne \varphi \text{.}$$

$0⩽{\nu }_{N_{{\beta }_i}}(Y\text{,}{\underset{\_}{N}}_{{\beta }_i}(D))⩽1$ , it is used to measure the degree that the subset Y covers the acceptable objects. If ${\underset{\_}{N}}_{{\beta }_i}(D)=\varphi$ , then ${\nu }_{N_{{\beta }_i}}(Y\text{,}{\underset{\_}{N}}_{{\beta }_i}(D))=1$

Let s, a, r be three features defined on a nonempty set X = {x₁, x₂, x₃, x₄} as in Table 1.

If the length of the neighbourhood of the feature s (resp a and r) equals to 0.2 (resp 0.9 and 0.5), then $$\begin{array}{cc}\hfill & N_1(B)=\{\xi (s_{0.2})\text{,}\xi (a_{0.9})\text{,}\xi (r_{0.5})\}\text{,}\text{where}\hfill \\ \hfill & \xi (s_{0.2})=\{\{x_1\text{,}{x}_2\}\text{,}\{x_1\text{,}{x}_2\text{,}{x}_3\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}\{x_3\text{,}{x}_4\}\}\text{,}\hfill \\ \hfill & \xi (a_{0.9})=\{\{x_1\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\}\text{,}\hfill \\ \hfill & \xi (r_{0.5})=\{\{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}\{x_2\}\}.\text{So}\hfill \\ \hfill & {\tau }_{s_{0.2}}=\{\{x_2\}\text{,}\{x_3\}\text{,}\{x_1\text{,}{x}_2\}\text{,}\{x_2\text{,}{x}_3\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\hfill \\ \hfill & \{x_1\text{,}{x}_2\text{,}{x}_3\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}X\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_{a_{0.9}}=\{\{x_3\}\text{,}\{x_4\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\}\text{,}\{x_1\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}\hfill \\ \hfill & \{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}X\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_{r_{0.5}}=\{\{x_2\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}X\text{,}\varphi \}\text{.}\hfill \end{array}$$ Hence $$N_1^{\prime }(B)=\{\{x_2\}\text{,}\{x_3\}\text{,}\{x_4\}\text{,}\{x_1\text{,}{x}_2\}\text{,}\{x_1\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\{x_1\text{,}{x}_2\text{,}{x}_3\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}X\text{,}\varphi \}\text{.}$$ Also, we can get $$\begin{array}{cc}\hfill & N_2(B)=\{\xi (s_{0.2}\text{,}{a}_{0.9})\text{,}\xi (S_{0.2}\text{,}{r}_{0.5})\text{,}\xi (a_{0.9}\text{,}{r}_{0.5})\}\text{,}\text{where}\hfill \\ \hfill & \xi (s_{0.2}\text{,}{a}_{0.9})=\{\{x_1\}\text{,}\{x_2\text{,}{x}_3\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}\{x_3\text{,}{x}_4\}\}\text{,}\hfill \\ \hfill & \xi (s_{0.2}\text{,}{r}_{0.5})=\{\{x_1\}\text{,}\{x_2\}\text{,}\{x_3\text{,}{x}_4\}\}\text{,}\hfill \\ \hfill & \xi (a_{0.9}\text{,}{r}_{0.5})=\{\{x_1\text{,}{x}_4\}\text{,}\{x_2\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\}.\text{So}\hfill \\ \hfill & {\tau }_{s_{0.2}{a}_{0.9}}=\{\{x_1\}\text{,}\{x_3\}\text{,}\{x_2\text{,}{x}_3\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\{x_1\text{,}{x}_3\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}\hfill \\ \hfill & \{x_1\text{,}{x}_2\text{,}{x}_3\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}X\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_{s_{0.2}{r}_{0.5}}=\{\{x_1\}\text{,}\{x_2\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\{x_1\text{,}{x}_2\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}\hfill \\ \hfill & X\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_{a_{0.9}{r}_{0.5}}=\{\{x_2\}\text{,}\{x_4\}\text{,}\{x_1\text{,}{x}_4\}\text{,}\{x_3\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_4\}\text{,}\{x_1\text{,}{x}_3\text{,}{x}_4\}\text{,}\hfill \\ \hfill & \{x_1\text{,}{x}_2\text{,}{x}_4\}\text{,}\{x_2\text{,}{x}_3\text{,}{x}_4\}\text{,}X\text{,}\varphi \}\text{.}\hfill \end{array}$$ So $N_2^{\prime }(B)=N_3^{\prime }(B)=\{\{x_1\}\text{,}\{x_2\}\text{,}\{x_3\}\text{,}\{x_4\}\text{,}\dots \text{,}X\text{,}\varphi \}$ . Consequently ${\tau }_{S_{0.2}{r}_{0.5}}\equiv {\tau }_{S_{0.2}{r}_{0.5}{a}_{0.9}}$ . That means the reduct of these features is {s, r}, so the feature {a} can be cancelled.

For β-approach to near sets, we get $$N_{{\beta }_1}(B)=N_{{\beta }_2}(B)=N_{{\beta }_3}(B)=\{\{x_1\}\text{,}\{x_2\}\text{,}\{x_3\}\text{,}\{x_4\}\text{,}\dots \text{,}X\text{,}\varphi \}\text{.}$$ The following example deduces a comparison between the classical and two new approaches by using the accuracy measure.

From Example 4.1, we can introduce Table 2, where Q(X) is a subset of X.

From Table 2, we can note that the classical approximations of near sets is more strong than the classical approximations of rough sets, but when we use the first generalized approach to near sets, we find that many sets will be completely exact.

Also, we find that all sets become $N_{{\beta }_1}$ -exact. It means all sets of this example become exact with respect to only one feature when we use β-near approach to near sets.

So our β-near approach is the best of our study. Hence we can consider that, our approximations is the start point to apply of our life applications in many fields of science.

In Example 4.1, if the decision class is D = {x₁, x₃} and we consider the following groups of the patients: {x₁, x₃}, {x₂, x₃}, {x₃, x₄}, {x₁, x₂, x₃}, and {x₂, x₃, x₄}.

Then, we get the following results:

$N_1(B){\hspace{0.17em}}_{*}D=\varphi \text{,}{N}_2(B){\hspace{0.17em}}_{*}D=N_3(B){\hspace{0.17em}}_{*}D=\{x_1\}\text{,}{\underset{\_}{N}}_1^{\prime }(D)=\{x_3\}\text{,}{\underset{\_}{N}}_2^{\prime }(D)={\underset{\_}{N}}_3^{\prime }(D)={\underset{\_}{N}}_{{\beta }_1}(D)={\underset{\_}{N}}_{{\beta }_2}(D)={\underset{\_}{N}}_{{\beta }_3}(D)=\{x_1\text{,}{x}_3\}\text{.}$ So these sets cover the acceptable objects by the degrees introduced in Table 3, where $I={\nu }_{N_2}^{\prime }={\nu }_{N_3}^{\prime }={\nu }_{N_{{\beta }_1}}={\nu }_{N_{{\beta }_2}}={\nu }_{N_{{\beta }_3}}$ and Q(X) is a family of subsets of X.

If we want to determine the degree that, the lower covers these sets then we use the following formulas: $$\begin{array}{cc}\hfill & {\nu }_i^{*}(B_i(x)\text{,}{N}_r(B){\hspace{0.17em}}_{*}D)=\frac{\left|B_i(x)\cap N_r(B){\hspace{0.17em}}_{*}D\right|}{|B_i(x)|}\text{,}{B}_i(x)\ne \varphi \text{,}\hfill \\ \hfill & {\nu }_{N_i}^{*\prime }(Y\text{,}{\underset{\_}{N}}_i(D))=\frac{|Y\cap {\underset{\_}{N}}_i(D)|}{|Y|}\text{,}Y\ne \varphi \text{,}\hfill \\ \hfill & {\nu }_{{\beta }_{N_i}}^{*\prime }(Y\text{,}{\underset{\_}{N}}_{{\beta }_i}(D))=\frac{|Y\cap {\underset{\_}{N}}_{{\beta }_i}(D)|}{|Y|}\text{,}Y\ne \varphi \text{.}\hfill \end{array}$$

In Example 5.1, if we interest in the degree that the acceptable objects (sure region) cover these groups, then we get Table 4, where $\mathit{II}={\nu }_{N_2}^{*\prime }={\nu }_{{\beta }_{N_1}}^{*\prime }={\nu }_{{\beta }_{N_2}}^{*\prime }={\nu }_{{\beta }_{N_3}}^{*\prime }$ and Q(X) is a family of subsets of X.