Generalized soft rough sets and generated soft ideal rough topological spaces

Generalized soft rough sets and generated soft ideal rough topological spaces

Abstract

Keywords

1 Introduction

2 Preliminaries

4 Deviations of some properties of Pawlak approximation space and generalized soft approximation space

Table 1
Tabular representation of the soft set S

A/U h ₁ h ₂ h ₃ h ₄ h ₅ h ₆

e ₁ 0 0 0 1 0 0

e ₃ 0 0 1 1 1 0

e ₅ 0 1 1 0 0 0

Table 5
Tabular representation of the soft set S

A/U p ₁ p ₂ p ₃ p ₄ p ₅ p ₆

e ₁ 1 0 0 1 1 1

e ₂ 1 1 1 0 0 0

e ₃ 0 1 0 1 1 1

Diabetes 1 1 1 0 0 0

Footnotes

References

Abstract

Keywords

1 Introduction

2 Preliminaries

4 Deviations of some properties of Pawlak approximation space and generalized soft approximation space

Table 1 Tabular representation of the soft set S A/U h 1 h 2 h 3 h 4 h 5 h 6 e 1 0 0 0 1 0 0 e 3 0 0 1 1 1 0 e 5 0 1 1 0 0 0

Table 5 Tabular representation of the soft set S A/U p 1 p 2 p 3 p 4 p 5 p 6 e 1 1 0 0 1 1 1 e 2 1 1 1 0 0 0 e 3 0 1 0 1 1 1 Diabetes 1 1 1 0 0 0

Footnotes

References

Table 1
Tabular representation of the soft set S

A/U h ₁ h ₂ h ₃ h ₄ h ₅ h ₆

e ₁ 0 0 0 1 0 0

e ₃ 0 0 1 1 1 0

e ₅ 0 1 1 0 0 0

Table 5
Tabular representation of the soft set S

A/U p ₁ p ₂ p ₃ p ₄ p ₅ p ₆

e ₁ 1 0 0 1 1 1

e ₂ 1 1 1 0 0 0

e ₃ 0 1 0 1 1 1

Diabetes 1 1 1 0 0 0

Acknowledgments

Acknowledgments

Soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. The main purpose of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. This paper is devoted to the further generalization of the soft rough set theory by using the ideal notion to reduce the soft boundary region. A new soft rough set model, called soft ideal approximation space, is proposed and its properties are derived. The new soft ideal lower and upper approximation operators are presented and their related properties are surveyed.

Rough sets soft rough set Pawlak approximation space soft ideal rough topology

To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertainties presented in these problems. While probability theory, fuzzy set theory [29], rough set theory [19, 20] and other mathematical tools are well known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out in [17, 18]. Rough set theory is a major mathematical method developed by Pawlak in [19, 20].

The rough set philosophy is founded on the assumption that with every object of the universe of discourse we associate some information (data, knowledge). For example, if objects are patients suffering from a certain disease, symptoms of the disease form information about patients. Objects characterized by the same information are indiscernible (similar) in view of the available information about them. The Indiscernible relation generated in this way is the mathematical basis of rough set theory. Each rough set has boundary region cases, i.e., objects which cannot with certainty be classified either as members of the set or of its complement. Obviously crisp sets have no boundary region elements at all. This means that boundary region cases cannot be properly classified by employing available knowledge. The lower approximation consists of all objects which surely belong to the concept and the upper approximation contains all objects which possibly belong to the concept. The difference between the upper and the lower approximation constitutes the boundary region of the vague concept. These approximations are two basic operations in rough set theory. When the available information is insufficient to determine the exact value of a given set, lower and upper approximations can be used by rough set for the representation of the concerned set.

Rough set theory has attracted worldwide attention of many researchers and practitioners, who have contributed essentially to its development and applications. Rough set theory overlaps with many other theories. Despite this, rough set theory may be considered as an independent discipline in its own right. The rough set approach seems to be of fundamental importance in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, image processing, signal analysis, knowledge discovery, decision analysis, and expert systems. Different kinds of generalizations of Pawlak’s rough set model were obtained in [22 , 28]. Rough set theory is expressible in terms of S-approximation spaces [9, 24], which is generalized in [23].

Ideals in topological spaces have been considered by Kuratowski [14] and Jankovic et al. [10]. In 2013, Kandil et al. [12] has generalized the notion of rough sets by using the ideals notion. It’s therefore shown that their results are more generally and decreases the boundary region in comparison with Pawlak’s method [19], Allam’s method [2] and Yao’s method [27].

In 1999, Molodtsov [18] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. This so-called soft set theory is free from the difficulties affecting existing methods. With the establishment of soft set theory, its application has boomed in recent years. In [15], Maji et al. introduced the notion of reduct-soft-set and described the application of soft set theory to a decision-making problem using rough sets. Chen et al. [5] presented a new definition of soft set parameterization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. Kong et al. [13] introduced the notion of normal parameter reduction of soft sets and constructed a reduction algorithm based on the importance degree of parameters. In [4], Bingzhen et al. proposed a new concept of soft fuzzy rough set by combining the fuzzy soft set with the traditional fuzzy rough set. In [11], Jianming et al. introduced a foundation for providing a rough soft tool in considering many problems that contain uncertainties.

In soft set theory, in most cases the parameters are vague words or sentences involving vague words.

The paper is organized as follows: In Section 2, we will recall some notions and properties of rough sets and soft sets. In Section 3, we will generalize the soft rough set theory by using the ideal notion. Moreover, we will present some properties of the soft ideal rough approximation operators and will introduce a new soft rough set model, which is an improvement of Feng’s et al. model [7, 8]. In Section 4, we will introduce the deviations of some properties of Pawlak’s approximation space and soft (ideal) approximation space, supported by counterexamples. Since the main aim of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. So, we will show that our model reduces the soft boundary in comparison with Feng’s et al. model [7, 8], in Section 5. Especially, we will use the soft ideal rough approximation operators to introduce and study the concept of soft ideal rough topology. In Section 6, we will try to apply the concept of soft rough topology and its base in diabetes mellitus. Overall I think this paper provides a readable introduction to the respective areas with an interesting application.

In this section, we will recall some notions and properties of rough sets and soft sets.

Let U denotes a non-empty finite set. Let R be an equivalence relation on U. The pair (U, R) is called a Pawlak’s approximation space [19, 21]. R will generate a partition U/R = {[x] _R : x ∈ U} on U, where [x] _R is the equivalence class with respect to R containing x. These equivalence classes are referred to as R-elementary sets which are the basic building blocks (concepts) of our knowledge about reality. For each X ⊆ U, the upper approximation $\bar{R} (X)$ and lower approximation $\underline{R} (X)$ of X with respect to (U, R) are defined as [19, 21] $\underline{R} (X) = {x \in U : [x]_{R} \subseteq X},$ (1) $\bar{R} (X) = {x \in U : [x]_{R} \cap X \neq \emptyset}, where$ (2) $[x]_{R} = {y : xRy} .$ (3) X is called definable in (U, R) if $\underline{R} (X) = \bar{R} (X)$ ; otherwise X is called a rough set. The universe can be divided into three disjoint regions using the lower and upper approximations as follows:

POS_R(A) = $\underline{R}$ (X) denotes the positive region of X, NEG_R(X) = U $- \bar{R}$ (X) denotes the negative region of X and BN_R(X) = $\bar{R}$ (X) $- \underline{R}$ (X) denotes the boundary region.

Definition 2.1. [10] A non-empty collection $I$ of subsets of a set X is called an ideal on X, if it satisfies the following conditions

$A \in I$ and $B \in I \Rightarrow A \cup B \in I$ ,

$A \in I$ and $B \subseteq A \Rightarrow B \in I$ .

Definition 2.2. [12] Let R be a binary relation on X, A ⊆ X and $I$ be an ideal on X, The R^*- upper and R_*- lower approximations of A are defined respectively by: $R^{*} (A) = {x \in X : 〈 x 〉_{R} \cap A \notin I},$ (4) $R_{*} (A) = {x \in X : 〈 x 〉_{R} \cap A^{'} \in I} .$ (5)

Definition 2.3. [12] Let R be a binary relation on X, A ⊆ X and $I$ be an ideal on X. The upper approximation of A is defined by $\bar{R} (A) = A \cup R^{*} (A)$ (6) and the lower approximation is defined by: $\underline{R} (A) = {x \in A : 〈 x 〉_{R} \cap A^{'} \in I} .$ (7)

With respect to any subset A ⊆ X. In this case, the upper approximation (6) satisfies Kuratowski’s axioms and induces a topology on X called $τ_{R}^{*}$ given by $τ_{R}^{*} = {A \subseteq X : \bar{R} (A^{c}) = A^{c}} .$ (8)

In such case interior of A, ${int}_{R}^{*} (A)$ , is identical with $\underline{R} (A)$ (7) and closure of A, ${cl}_{R}^{*} (A)$ , is identical with $\bar{R} (A)$ (6).

Definition 2.4. [18] Let X be an initial universe and E be a set of parameters. Let $P (X)$ denote the power set of X and A be a non-empty subset of E. A pair (F, A) is called a soft set over X, where F is a mapping given by $F : A \to P (X)$ .

Definition 2.5. [7, 8] Let S = (F, A) be a soft set over U. Then, the pair P = (U, S) is called a soft approximation space. Based on the soft approximation space P, we define the following two operations: $\begin{matrix} {\underline{apr}}_{P} (X) \\ = {u \in U : \exists e \in A, u \in F (e), F (e) \subseteq X}, \\ {\bar{apr}}_{P} (X) \\ = {u \in U : \exists e \in A, u \in F (e), F (e) \cap X \neq \emptyset}, \end{matrix}$ assigning to every subset X ⊆ U two sets ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ , which are called the soft P-lower approximation and the soft P-upper approximation of X, respectively. In general, we refer to ${\underline{apr}}_{P} (X)$ and ${\bar{apr}}_{P} (X)$ as soft rough approximations of X with respect to P. Moreover, the sets $\begin{matrix} P_{osP} (X) = {\underline{apr}}_{P} (X), \\ N_{egP} (X) = U - {\bar{apr}}_{P} (X), \\ B_{ndP} (X) = {\bar{apr}}_{P} (X) - {\underline{apr}}_{P} (X) . \end{matrix}$ are called the soft P-positive region, the soft P-negative region and the soft P-boundary region of X, respectively. If ${\bar{apr}}_{P} (X) = {\underline{apr}}_{P} (X)$ , then X is said to be soft P-definable. Otherwise, X is called a soft P-rough set.

Theorem 2.6. [8] Let S = (F, A) be a soft set over U. Then, S induces a binary relation ρs ⊆ A × U, which is defined by $(x, y) \in ρ s \Leftrightarrow y \in F (x)$ (9) ∀ x ∈ A and y ∈ U.

Conversely, let ρ be a binary relation from A to U. Define a set-valued mapping F_ρ : A ⟶ P (U) by $F_{ρ} (x) = {y \in U : (x, y) \in ρ}$ (10) for all x ∈ A. Then, S_ρ = (F_ρ, A) is a soft set over U. Moreover, we have that S_ρs = S and ρ_sρ = ρ.

3 Soft ideal rough approximation operators

In this section, we will generalize the soft rough set theory by using the ideal notion. Also, we will present some properties of soft ideal rough approximation operators, and introduce a new soft rough set model, which is an improvement of Feng’s model, by using the ideal notion.

Definition 3.1. Let S = (F, A) be a soft set over U, I be an ideal on U and P = (U, S) be a soft approximation space. Based on the soft approximation space P and the ideal I, we define the following two operations for each X ⊆ U, called as soft ideal rough approximations: $\begin{matrix} {\underline{apr}}_{IP} (X) & = & {u \in U : \exists e \in A such that u \in F (e) \\ and F (e) \cap X^{c} \in I}, \\ {\bar{apr}}_{IP} (X) & = & {u \in U : \exists e \in A such that u \in F (e) \\ and F (e) \cap X \notin I}, \end{matrix}$ which are called the soft IP-lower approximation and the soft IP-upper approximation of X, respectively. If ${\bar{apr}}_{IP} (X) = {\underline{apr}}_{IP} (X)$ , then X is said to be soft IP-definable. Otherwise, X is called a soft IP-rough set. In general, we refer to ${\underline{apr}}_{IP} (X)$ and ${\bar{apr}}_{IP} (X)$ as a generalized soft rough approximations of X with respect to P and I. Moreover, by this definitions of generalized soft rough approximations we obtain preceding definition introduced by Feng et al. [7, 8] and Meng et al. [16]. In other words, soft rough approximations [7 , 16] are special cases of the current definition, as will shown in the following proposition.

Proposition 3.2. If I = {∅} in Definition 3.1, then we get soft rough approximations definition Feng [7, 8] and Meng et al. [16].

Proof. Straightforward.

Proposition 3.3. Let S = (F, E) be a soft set over U, I be an ideal on U and P = (U, S) be a soft approximation space. Then, for each X ⊆ U, we have ${\underline{apr}}_{IP} (X) = ⋃_{e \in E} {F (e) : F (e) \cap X^{c} \in I}$ and ${\bar{apr}}_{IP} (X) = ⋃_{e \in E} {F (e) : F (e) \cap X \notin I}$

Proof. Straightforward.

Theorem 3.4. Let S = (F, E) be a soft set over U, I, J be two ideals on U and P = (U, S) be the corresponding soft approximation space. Then, for each A, B ⊆ U, we have:

${\bar{apr}}_{IP} (U) \subseteq {\underline{apr}}_{IP} (U) = ⋃_{e \in E} F (e) \subseteq U$ ,

${\bar{apr}}_{IP} (\emptyset) = \emptyset \neq {\underline{apr}}_{IP} (\emptyset)$ ,

If A ⊆ B, then ${\bar{apr}}_{IP} (A) \subseteq {\bar{apr}}_{IP} (B)$ ,

If A ⊆ B, then ${\underline{apr}}_{IP} (A) \subseteq {\underline{apr}}_{IP} (B)$ ,

${\bar{apr}}_{IP} (A \cup B) = {\bar{apr}}_{IP} (A) \cup {\bar{apr}}_{IP} (B)$ ,

${\underline{apr}}_{IP} (A \cup B) \supseteq {\underline{apr}}_{IP} (A) \cup {\underline{apr}}_{IP} (B)$ ,

${\underline{apr}}_{IP} (A \cap B) \subseteq {\underline{apr}}_{IP} (A) \cap {\underline{apr}}_{IP} (B)$ ,

${\bar{apr}}_{IP} (A \cap B) \supseteq {\bar{apr}}_{IP} (A) \cap {\bar{apr}}_{IP} (B)$ ,

${\underline{apr}}_{IP} (A \cap B) \subseteq {\underline{apr}}_{IP} (A)$ and ${\underline{apr}}_{IP} (A \cap B) \subseteq {\underline{apr}}_{IP} (B)$ ,

${\bar{apr}}_{IP} ({\bar{apr}}_{IP} (A)) \supseteq {\bar{apr}}_{IP} (A)$ ,

${\underline{apr}}_{IP} ({\underline{apr}}_{IP} (A)) = {\underline{apr}}_{IP} (A)$ ,

${\underline{apr}}_{IP} ({\bar{apr}}_{IP} (A)) = {\underline{apr}}_{IP} (A)$ ,

${\bar{apr}}_{IP} ({\bar{apr}}_{IP} (A)) \supseteq {\underline{apr}}_{IP} (A)$ ,

If I ⊆ J, then ${\bar{apr}}_{JP} (A) \subseteq {\bar{apr}}_{IP} (A)$ ,

If I ⊆ J, then ${\underline{apr}}_{IP} (A) \subseteq {\underline{apr}}_{JP} (A)$ ,

${\underline{apr}}_{IP} (A) = ({\bar{apr}}_{IP} (A^{c}))^{c}$ ,

$A ⊈ {\bar{apr}}_{IP} (A)$ , in general.

${\bar{apr}}_{IP} ({\underline{apr}}_{IP} (A)) \supseteq {\underline{apr}}_{IP} (A)$ ,

$[{\underline{apr}}_{IP} (A)]^{c} \subseteq {\bar{apr}}_{IP} (A^{c})$ ,

$[{\bar{apr}}_{IP} (A)]^{c} \subseteq {\underline{apr}}_{IP} (A^{c})$ ,

Proof.

Follows from Proposition 3.3.

Immediate.

Let $u \in {\bar{apr}}_{IP} (A)$ , then ∃ e ∈ E suchthat u ∈ F (e) and F (e) ∩ A ∉ I. Since A ⊆ B, F (e) ∩ A ⊆ F (e) ∩ B. Hence, F (e) ∩ B ∉ I. Therefore, $u \in {\bar{apr}}_{IP} (B)$ .

It is similar to (3).

Let $u \in {\bar{apr}}_{IP} (A \cup B)$ , then ∃ e ∈ Esuch that u ∈ F (e) and F (e) ∩ (A ∪ B) ∉ I. Hence, either F (e) ∩ A ∉ I or F (e) ∩ B ∉ I. Thus, $u \in {\bar{apr}}_{IP} (A)$ or $u \in {\bar{apr}}_{IP} (B)$ . This means, $u \in {\bar{apr}}_{IP} (A) \cup {\bar{apr}}_{IP} (B)$ . It follows, ${\bar{apr}}_{IP} (A \cup B) \subseteq {\bar{apr}}_{IP} (A) \cup {\bar{apr}}_{IP} (B)$ . The other inclusion is clear.

Similar to (5).

Clear.

Let $u \in {\bar{apr}}_{JP} (A)$ . Then, ∃ e ∈ E suchthat u ∈ F (e) and F (e) ∩ A ∉ J. Since I ⊆ J, F (e) ∩ A ∉ I. Hence, $u \in {\bar{apr}}_{IP} (A)$ .

It is similar to (14).

$({\bar{apr}}_{IP} (A^{c}))^{c} = [{u \in U : \exists e \in E suchthat u \in F (e) and F (e) \cap A^{c} \notin I}]^{c} = {u \in U : \exists e \in E such that u \in F (e) and F (e) \cap A^{c} \in I} = {\underline{apr}}_{IP} (A)$ .

It will cleared by a counterexample. See Example 4.2.

Clear.

Let $u \notin {\bar{apr}}_{IP} (A^{c})$ , then ∀ e ∈ E suchthat u ∈ F (e) we have F (e) ∩ A^c ∈ I. Then, $u \in {\underline{apr}}_{IP} (A)$ . Hence, $u \notin [{\underline{apr}}_{IP} (A)]^{c}$ .

It is similar to (19)

In this section, we will introduce the deviations of some properties of Pawlak approximation space and soft (ideal) approximation space, supported by counterexamples.

On accounting of Theorem 3.4 [parts (1), (8) and (17)–(20)] and [Proposition 3.2, [8]] [parts (2), (10) and (12)], there are deviations between some of the properties of lower and upper approximations in soft set theory and its corresponding in Pawlak rough set theory, as will shown in the following examples.

Example 4.1. Let U = {h₁, h₂, h₃, h₄, h₅, h₆} be the set of six houses under consideration, which Mr. M is going to buy, E = {e₁, e₂, e₃, e₄, e₅} be the set of decision parameters which are stands for “expensive”, “beautiful”, “position”, “wooden” and “green surroundings”, respectively and A = {e₁, e₃, e₅} ⊆ E. Consider the soft approximation P = (U, S), where S = (F, A) is a soft set over U given as a table (Table 1) in the following form.

A/U	h ₂	h ₃	h ₄	h ₅
e ₁	0	0	1	0
e ₃	0	1	1	1
e ₅	1	1	0	0

Let X = {h₁, h₄, h₅}, then ${\underline{apr}}_{P} (X) = {h_{4}}$ , ${\bar{apr}}_{P} (X) = {h_{3}, h_{4}, h_{5}}$ . So, ${\underline{apr}}_{P} (X) \neq {\bar{apr}}_{P} (X)$ . It follows, X is a soft P-rough set. Also, one can calculate the following:

${\underline{apr}}_{P} (U) = {h_{2}, h_{3}, h_{4}, h_{5}} = {\bar{apr}}_{P} (U)$ , ${\bar{apr}}_{P} ({\bar{apr}}_{P} (X)) = {h_{2}, h_{3}, h_{4}, h_{5}}$ , ${\bar{apr}}_{P} ({\underline{apr}}_{P} (X)) = {h_{3}, h_{4}, h_{5}}$ , $[{\underline{apr}}_{P} (X)]^{c} = {h_{1}, h_{2}, h_{3}, h_{5}, h_{6}}$ , ${\bar{apr}}_{P} (X^{c}) = {h_{2}, h_{3}, h_{4}, h_{5}}, [{\bar{apr}}_{P} (X)]^{c} = {h_{1}, h_{2}, h_{6}}, {\underline{apr}}_{P} (X^{c}) = {h_{4}}$ . Thus, we notice that:

${\underline{apr}}_{P} (U) \neq U \neq {\bar{apr}}_{P} (U)$ ,

${\bar{apr}}_{P} ({\bar{apr}}_{P} (X)) ⊈ {\bar{apr}}_{P} (X)$ ,

$X ⊈ {\bar{apr}}_{P} (X)$ ,

${\bar{apr}}_{P} ({\underline{apr}}_{P} (X)) ⊈ {\underline{apr}}_{P} (X)$ ,

$[{\underline{apr}}_{P} (X)]^{c} ⊉ {\bar{apr}}_{P} (X^{c})$ ,

$[{\bar{apr}}_{P} (X)]^{c} ⊉ {\underline{apr}}_{P} (X^{c})$ ,

Example 4.2. Let U = {h₁, h₂, h₃, h₄, h₅, h₆}, E = {e₁, e₂, e₃, e₄, e₅} and A = {e₁, e₃, e₅} ⊆ E. Consider the soft approximation P = (U, S), where S = (F, A) is a soft set over U given as a table (Table 2) in the following form.

Table 2

Tabular representation of the soft set S

A/U	h ₁	h ₂	h ₃	h ₄	h ₅	h ₆
e ₁	0	0	0	1	0	1
e ₃	0	1	0	1	1	0
e ₅	0	1	1	0	0	0

Let X = {h₂, h₅, h₆}, I = {∅ , {h₄} , {h₅} , {h₄, h₅}} and J = {∅ , {h₂} , {h₄} , {h₂, h₄}} be two ideals over U, then ${\underline{apr}}_{IP} (X) = {h_{2}, h_{4}, h_{5}, h_{6}}$ , ${\bar{apr}}_{IP} (X) = {h_{2}, h_{3}, h_{4}, h_{6}}$ . So, ${\underline{apr}}_{IP} (X) \neq {\bar{apr}}_{IP} (X)$ . It follows, X is a soft IP-rough set. Also, one can calculate the following:

${\underline{apr}}_{IP} (U) = {h_{2}, h_{3}, h_{4}, h_{5}, h_{6}} = {\bar{apr}}_{IP} (U), {\bar{apr}}_{IP} ({\bar{apr}}_{IP} (X)) = {h_{2}, h_{3}, h_{4}, h_{5}, h_{6}}$ , ${\bar{apr}}_{IP} ({\underline{apr}}_{IP} (X)) = {h_{2}, h_{3}, h_{4}, h_{5}, h_{6}}$ , $[{\underline{apr}}_{IP} (X)]^{c} = {h_{1}, h_{3}}$ , ${\bar{apr}}_{IP} (X^{c}) = {h_{2}, h_{3}}$ , ${\bar{apr}}_{JP} (X) = {h_{2}, h_{4}, h_{5}, h_{6}}$ , $[{\bar{apr}}_{JP} (X)]^{c} = {h_{1}, h_{3}$ }, ${\underline{apr}}_{JP} (X^{c}) = {h_{2}, h_{3}}$ . Thus, we have

${\underline{apr}}_{IP} (U) \neq U \neq {\bar{apr}}_{IP} (U)$ ,

${\bar{apr}}_{IP} ({\bar{apr}}_{IP} (X)) ⊈ {\bar{apr}}_{IP} (X)$ ,

$X ⊈ {\bar{apr}}_{IP} (X)$ ,

${\bar{apr}}_{IP} ({\underline{apr}}_{IP} (X)) ⊈ {\underline{apr}}_{IP} (X)$ ,

$[{\underline{apr}}_{IP} (X)]^{c} ⊉ {\bar{apr}}_{IP} (X^{c})$ ,

$[{\bar{apr}}_{JP} (X)]^{c} ⊉ {\underline{apr}}_{JP} (X^{c})$ ,

5 Soft ideal rough topology

In this section, we will use the soft ideal rough approximation operators to introduce the concept of soft ideal rough topology.

Definition 5.1. Let U be the universe, A ⊆ U and I be an ideal on U. The soft upper approximation of A is defined by ${\bar{R}}_{I} (A) = A \cup {\bar{apr}}_{IP} (A)$ (11) and the soft lower approximation is defined by: ${\underline{R}}_{I} (A) = ⋃_{e \in A} {F (e) : F (e) \cap A^{c} \in I}$ (12)

Moreover, the universe set can be divided into three disjoint regions using the soft lower and soft upper approximations as follows: $\begin{matrix} P_{osIP} (A) = {\underline{R}}_{I} (A), \\ N_{egIP} (A) = U - {\bar{R}}_{I} (A), \\ B_{ndIP} (A) = {\bar{R}}_{I} (A) - {\underline{R}}_{I} (A) . \end{matrix}$

Proposition 5.2. Let S = (F, E) be a soft set over U, I be an ideal on U and P = (U, S) be the corresponding soft approximation space. For all A, B ⊆ U, the soft lower and soft upper approximations defined by (11) and (12), respectively satisfy the following properties:

${\bar{R}}_{I} (\emptyset) = \emptyset$ ,

$A \subseteq {\bar{R}}_{I} (A)$ ,

If A ⊆ B, then ${\bar{R}}_{I} (A) \subseteq {\bar{R}}_{I} (B)$ ,

${\bar{R}}_{I} (A \cup B) = {\bar{R}}_{I} (A) \cup {\bar{R}}_{I} (B)$ ,

${\bar{R}}_{I} ({\bar{R}}_{I} (A)) = {\bar{R}}_{I} (A)$ ,

${\bar{R}}_{I} (A \cap B) \subseteq {\bar{R}}_{I} (A) \cap {\bar{R}}_{I} (B)$ ,

${\underline{R}}_{I} (A) \subseteq {\bar{R}}_{I} (A)$ ,

If A ⊆ B, then ${\underline{R}}_{I} (A) \subseteq {\underline{R}}_{I} (B)$ ,

${\underline{R}}_{I} (A) \subseteq A$ ,

${\underline{R}}_{I} ({\underline{R}}_{I} (A)) = {\underline{R}}_{I} (A)$ ,

${\underline{R}}_{I} (A) \cup {\underline{R}}_{I} (B) \subseteq {\underline{R}}_{I} (A \cup B)$ .

Proof. Follows immediately from Theorem 3.4.

Definition 5.3. [3] Let U be the universe, X ⊆ U and P = (U, S) be a soft approximation space. Then, the collection $τ_{SR} (X) = {U, \emptyset, {\bar{apr}}_{P} (X), {\underline{apr}}_{P} (X), B_{ndP} (X)}$ , forms a topology on U called as the soft rough topology on U w.r.t X.

Definition 5.4. [3] Let (U, τ_SR (X) , E) be a soft rough topological space w.r.t. X ⊆ U. The soft rough interior of each A ⊆ U is defined as the union of all soft rough open subsets of A and it is denoted by Sint_R (A). Also, the soft rough closure of A is defined as the intersection of all soft rough closed subsets containing A and it is denoted by Scl_R (A). i.e ${Scl}_{R} (A) = A \cup {\bar{apr}}_{P} (A) .$

Moreover, the collection $β = {U, {\underline{apr}}_{P} (X), B_{ndP} (X)}$ is a soft base of τ_SR.

Corollary 5.5. On accounting of Proposition 5.2, the soft upper approximation (11) satisfies Kuratowski’s axioms and induces a topology on U, denoted by τ_{R
_I}, called the soft ideal rough topology, given as follows: $τ_{R_{I}} = {A \subseteq X : \bar{R_{I}} (A^{c}) = A^{c}} .$ (13)

In such case the soft rough interior of A w. r. t. I, denoted by Sint_{R
_I} (A), is identical with ${\underline{R}}_{I} (A)$ (12) and the soft rough closure of A w. r. t. I, denoted by Scl_{R
_I} (A), is identical with $\bar{R_{I}} (A)$ (11).

Theorem 5.6. Let (X, τ_{R
_I}) be a soft ideal rough topology. Then, Scl_{R
_I} (A) ⊆ Scl_R (A).

Proof. Let x ∈ Scl_{R
_I} (A). Hence x ∈ A or $x \in {\bar{apr}}_{IP} (A)$ . This means, ∃ e ∈ A such that u ∈ F (e) and F (e) ∩ A ∉ I. It follows, F (e)∩ A ≠ ∅. Hence, x ∈ Scl_R (A).

In the following corollary, we show that our results which have obtained here are generalization to such in [3].

Corollary 5.7. Let U be the universe, then τ_{R
_I} is finer than τ_SR.

Proof. Follows from Theorem 5.6.

The following examples shows that our generated topology is a generalization to that’s has obtained in [3].

Example 5.8. Let U = {h₁, h₂, h₃}, E = {e₁, e₂, e₃, e₄, e₅} and A = {e₁, e₂, e₃} ⊆ E. Consider the soft approximation P = (U, S), where S = (F, A) is a soft set over U given as a table (Table 3) in the following form.

Table 3

Tabular representation of the soft set S

A/U	h ₁	h ₂	h ₃
e ₁	0	0	1
e ₂	1	1	0
e ₃	0	1	1

It follows, ${\underline{apr}}_{P} (U) = U$ , ${\bar{apr}}_{P} (U) = U$ and B_ndP (U) =∅. Hence, τ_SR = {U, ∅}. Let I = {∅ , {h₁}} be an ideal on U, we have: ${\bar{R}}_{I} (\emptyset) = \emptyset$ , so U ∈ τ_{R
_I}, ${\bar{R}}_{I} (U) = U$ , so ∅ ∈ τ_{R
_I}, ${\bar{R}}_{I} ({h_{2}, h_{3}}) = {h_{2}, h_{3}}$ , so {h₁} ∈ τ_{R
_I}, ${\bar{R}}_{I} ({h_{1}, h_{2}}) = {h_{1}, h_{2}}$ , so {h₃} ∈ τ_{R
_I}, ${\bar{R}}_{I} ({h_{2}}) = {h_{2}}$ , so {h₁, h₃} ∈ τ_{R
_I}, ${\bar{R}}_{I} ({h_{1}}) = {h_{1}, h_{2}} \neq {h_{1}}$ , so {h₂, h₃} ∉ τ_{R
_I}, ${\bar{R}}_{I} ({h_{3}}) = {h_{3}}$ , so {h₁, h₂} ∈ τ_{R
_I}, ${\bar{R}}_{I} ({h_{1}, h_{3}}) = U \neq {h_{1}, h_{3}}$ , so {h₂} ∉ τ_{R
_I}.

Therefore, τ_{R
_I} = {U, ∅ , {h₁} , {h₃} , {h₁, h₂} , {h₁, h₃}} is the soft ideal rough topology on U. Thus, τ_SR ⊆ τ_{R
_I}.

Theorem 5.9. Let U be the universe, I and J be two ideals on U. If I ⊆ J, then B_ndJP (A) ⊆ B_ndIP (A) for each A ⊆ U.

Proof. Let x ∈ B_ndJP (A). Then, $x \in {\bar{R}}_{J} (A)$ and $x \in [{\underline{R}}_{J} (A)]^{c}$ . By Theorem 3.4 (14), $x \in {\bar{R}}_{I} (A)$ and $x \in [{\underline{R}}_{I} (A)]^{c}$ . Hence, x ∈ B_ndIP (A).

In the following example, we see that the current method in Definition 5.1 reduce the soft boundary in comparison of with [7, 8] method.

Example 5.10. Consider the soft approximation and the soft set in Example 5.8. Let I = {∅ , {h₁}} be an ideal on U and X ⊆ U. The following table (Table 4) shows that our method reduced the soft boundary comparing with [7, 8] method.

Table 4

Tabular representation of the comparison between [7, 8] method and the present method

X	[7, 8] model	Our model	[7, 8] model	Our model	[7, 8] model	Our model
	${\underline{apr}}_{P} (X)$	${\underline{R}}_{I} (X)$	${\bar{apr}}_{P} (X)$	${\bar{R}}_{I} (X)$	B_ndP (X)	B_ndIP (X)
∅	∅	∅	∅	∅	∅	∅
X	X	X	X	X	∅	∅
{h₁}	∅	{h₁, h₂}	{h₁, h₂}	{h₁, h₂}	{h₁, h₂}	∅
{h₂}	∅	∅	U	{h₂}	U	{h₂}
{h₃}	{h₃}	{h₂, h₃}	{h₂, h₃}	{h₃}	{h₂}	∅
{h₁, h₂}	{h₁, h₂}	{h₁, h₂}	U	{h₁, h₂}	{h₃}	∅
{h₁, h₃}	{h₃}	U	U	U	{h₁, h₂}	∅
{h₂, h₃}	{h₂, h₃}	{h₂, h₃}	U	{h₂, h₃}	{h₁}	∅

6 Application

In this section, we will apply the concept of soft rough topology in Diabetes mellitus (DM) [3], commonly referred to as diabetes, is a group of metabolic diseases in which there are high blood sugar levels over a prolonged period. Symptoms of high blood sugar include frequent urination, increased thirst and increased hunger. If left untreated, diabetes can cause many complications. Acute complications can include diabetic ketoacidosis, nonketotic hyperosmolar coma or death. Serious long-term complications include heart disease, stroke, chronic kidney failure, foot ulcers, and damage to the eyes.

Consider the following information table giving data about 6 patients as a random representative. The rows of the table represent the attributes (the symptoms for Diabetes) and the columns represent the objects (the patients). Let U = {p₁, p₂, p₃, p₄, p₅, p₆} and A = {e₁ (Increased Hunger) , e₂ (Frequent Urination) , e₃ (Increased Thirst)}. Consider the soft approximation P = (U, S), where S = (F, A) is a soft set over U given as a table (Table 5) in the following form.

Let X = {p₁, p₂, p₃} be the set of patients having diabetes. Then, we have ${\underline{apr}}_{P} (X) = {p_{1}, p_{2}, p_{3}}$ , ${\bar{apr}}_{P} (X) = U$ and B_ndP (X) = {p₄, p₅, p₆}. Hence, τ_SR = {U, ∅ , {p₁, p₂, p₃} , {p₄, p₅, p₆}} is the soft rough topology on U and its soft basis β = {U, {p₁, p₂, p₃} , {p₄, p₅, p₆}}.

If the attribute ‘Increased Hunger’ is removed, we have ${\underline{apr}}_{P - e_{1}} (X) = {p_{1}, p_{2}, p_{3}}$ , ${\bar{apr}}_{P - e_{1}} (X) = U$ and B_ndP-e₁ (X) = {p₄, p₅, p₆}. Therefore, τ_SR-e₁ = {U, ∅ , {p₁, p₂, p₃} , {p₄, p₅, p₆}} is a soft rough topology on U and its soft basis β - e₁ = {U, {p₁, p₂, p₃} , {p₄, p₅, p₆}} = β.

Again, if the attribute ‘Frequent Urination’ is removed, we have ${\underline{apr}}_{P - e_{2}} (X) = \emptyset$ , ${\bar{apr}}_{P - e_{2}} (X) = {p_{1}, p_{2}, p_{4}, p_{5}, p_{6}}$ and B_ndP-e₂ (X) = {p₁, p₂, p₄, p₅, p₆}. Therefore, τ_SR-e₂ = {U, ∅ , {p₁, p₂, p₄, p₅, p₆}} is a soft rough topology on U and its soft basis β - e₂ = {U, ∅ , {p₁, p₂, p₄, p₅, p₆}} ≠ β.

Finally, if the attribute ‘Increased Thirst’ is removed, we have ${\underline{apr}}_{P - e_{3}} (X) = {p_{1}, p_{2}, p_{3}}$ , ${\bar{apr}}_{P - e_{3}} (X) = U$ and B_ndP-e₃ (X) = {p₄, p₅, p₆}. Therefore, τ_SR-e₃ = {U, ∅ , {p₁, p₂, p₃} , {p₄, p₅, p₆}} is a soft rough topology on U and its soft basis β - e₃ = {U, {p₁, p₂, p₃} , {p₄, p₅, p₆}} = β.

Therefore, CORE (SR) = {e₂}, i.e ‘Frequent Urination’ is the key attribute that has close connection to the disease diabetes.

Algorithm:

Step 1: Given a finite universe U, a finite set A of attributes represent the data as an information table, rows of which are labeled by attributes (C),columns by objects and entries of the table are attribute values.

Step 2: Find the soft P-lower approximation, soft P-upper approximation and the soft P-boundary region of X ⊆ U.

Step 3: Generate the soft rough topology τ_SR on U and its soft basis β.

Step 4: Remove an attribute x from conditions of attributes (C) and find the soft P-lower and soft P-upper approximations and the soft P-boundary region of X on C - (x).

Step 5: Generate the soft rough topology τ_SR-x on U and its soft basisβ - x.

Step 6: Repeat steps 4 and 5 for all attributes in C.

Step 7: Those attributes in C for which β - x ≠ β forms the core (SR).

7 Conclusion

In this paper, we generalized soft rough set theory by introducing concepts of soft IP-lower and soft IP-upper approximation. Some of their basic properties with the help of counterexamples were investigated and the interrelation between them and Pawlak’s approximation were obtained. Furthermore, we used the soft ideal rough approximation operators to introduce and study the concept of soft ideal rough topology and application. In future, we will introduces the notion of fuzzy soft ideal rough approximation and the future research will be undertaken in this direction.

The author express his sincere thanks to the reviewers for their valuable suggestions. The author is also thankful to the editors-in-chief and managing editors for their important comments which helped to improve the presentation of the paper.

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Diabetes