Rough approximations of bipolar soft sets by soft relations and their application in decision making

Abstract

The rough set theory is an effective method for analyzing data vagueness, while bipolar soft sets can handle data ambiguity and bipolarity in many cases. In this article, we apply Pawlak’s concept of rough sets to the bipolar soft sets and introduce the rough bipolar soft sets by defining a rough approximation of a bipolar soft set in a generalized soft approximation space. We study their structural properties and discuss how the soft binary relation affects the rough approximations of a bipolar soft set. Two sorts of bipolar soft topologies induced by soft binary relation are examined. We additionally discuss some similarity relations between the bipolar soft sets, depending on their roughness. Such bipolar soft sets are very useful in the problems related to decision-making such as supplier selection problem, purchase problem, portfolio selection, site selection problem etc. A methodology has been introduced for this purpose and two algorithms are presented based upon the ongoing notions of foresets and aftersets respectively. These algorithms determine the best/worst choices by considering rough approximations over two universes i.e. the universe of objects and universe of parameters under a single framework of rough bipolar soft sets.

Keywords

Rough sets bipolar soft sets rough bipolar soft sets bipolar soft topology decision making

1 Introduction

In modern society, the facts gathered and studied for several purposes have vagueness and uncertainty in lots of ideas in engineering, economics, social science, environmental science, medical science and many other fields. This ambiguity together with volume and complexity grows quickly in such information. All the mathematical notions in classical mathematics must be precise and accurate. It is therefore not always a successful tool for dealing with uncertain problems. Over the years, many researchers and scientists have been attempting to find out a few appropriate tools to deal with these uncertainties. They created much equipment for this purpose. The rough set theory [26] and the soft set theory [10, 24] are among the most popular of all these. These theories brought down the gap between traditional mathematical designs and ambiguous real-world data.

Pawlak [26] initially developed the rough set theory in 1982. The rough set theory successfully provided a systematic scheme to handle the imprecision and uncertainty in the information. Pawlak utilized the lower and upper approximations of a collection of objects to investigate how close the objects are to the information attached to them. There are many precious applications to that theory. Therefore, this theory attracted a lot of researchers and scientists and initiated research in many ways.

The innovative idea of soft sets, a revolutionary mathematical method to deal with ambiguity and uncertainty, was pioneered by Molodtsov [24]. The knowledge parameters play a vital role in the inspection and interpretation of a data or in making a decision. The soft sets theory has been demonstrated as an effective method for parameterization. This theory therefore magically solved many problems, which occurred while using the old theories. This theory has gained much attention from researchers and scientists because of its diverse applications. It is possible to see a rapid growth in soft sets research in the last few years. Maji et al. [22] have described some basic operations on soft sets. Maji et al. [21] have applied the soft set theory to a problem of decision-making. Ali et al. [1] described certain new operations on soft sets. Ali et al. [2] examined the algebraic structures of soft sets relating to new operations. In 2010, Shabir and Naz [27] proposed the concept of soft topological space which was later investigated in diverse areas with further theoretical enhancements e.g. [8 , 14]. A hybrid approach of rough approximation using soft sets is introduced and investigated in details by Shabir et al. in [17 , 31].

The bipolarity of information is a core feature to be recognized in many forms of data analysis when creating mathematical models for many problems. Bipolarity talks about the information’s positive and negative characteristics. Positive information is what is expected to be true, whereas negative information is what is unlikely, prohibited or certainly wrong. The theory behind the presence of bipolar knowledge is that the foundation of bipolar judgemental thought is a wide range of human decision-making. Shabir and Naz introduced the concept of a Bipolar Soft set in [28]. Naz and Shabir [25] have also contributed to the algebraic structure of bipolar soft sets in fuzzy environment. The bipolar soft sets can control the bipolarity of certain objects using two mappings. One mapping deals with the information’s positivity while the other mapping examines the negativity. Shabir and Bakhtawar [29] studied the underlying topological structural properties within bipolar soft sets. Karaaslan [19] and Karaaslan and Çağman [20] initiated the notion of bipolar soft rough sets which is a fusion of rough set with bipolarity and discussed its applications in decision-making. Shabir and Gul [31] pointed out some basic problems in the definition of bipolar soft rough set in [20] and redefined as a new version of bipolar soft rough sets known as modified rough bipolar soft sets.

Motivated by the above ideas and recent works in bipolar soft sets, we will investigate a more general case of roughness with approximations through bipolar soft binary relations. We introduce the concept of approximation of a bipolar soft set with a given bipolar soft relation and this gives rise to the approximations in two-universes i.e. V and W. In the end, we get a very neat way of three-way decision-making which sorts out the set of objects and set of parameters in a separate manner. This paper is intended to establish the idea of roughness in the bipolar soft sets in the light of soft binary relations. We direct an evaluation of rough approximations by obtaining two bipolar soft sets. Several properties are viewed by taking soft reflexive and soft symmetric binary relations. This paper has a layout as follows: Section 2 recalls some fundamentals concepts and definitions. Section 3 is dedicated to the study of rough bipolar soft sets by describing the lower and upper approximations of bipolar soft sets in a generalized soft approximation space. Section 4 is dedicated to the study of bipolar soft topologies induced by soft reflexive relations. Section 5 describes some similarity relations between the bipolar soft sets. Section 6 discusses an application of the rough bipolar soft sets in a problem of decision-making, illustrated by an example. The final section consists of the conclusions.

2 Preliminaries

In this section, some fundamental notions about rough sets and bipolar soft sets are given. We have V as a non-empty finite set, unless otherwise stated throughout the article. Consider E, a set of attributes or parameters. Let $P (V)$ be the collection of all subsets of V and φ ≠ A, B ⊆ E .

Definition 2.1. [24] A pair (ψ, A) is known as a soft set over V, where ψ is a set-valued mapping given by $ψ : A \to P (V)$ . In other words, a soft set over V gives a parameterized collection of subsets of the universe V . For e ∈ A, ψ (e) may be regarded as the set of e-approximate elements of V by the soft set (ψ, A).

Recall that a binary relation from V to W is a subset of V × W. A binary relation on V is a subset of V × V.

Definition 2.2. [32] The mapping S : E → P (V × W) is called a soft relation from V to W under E, where P (V × W) is the set of all relations from V to W.

Definition 2.3. [18] Suppose (S, E) is a soft set over V × W, then (S, E) is called a Soft Binary Relation (SB-relation) from V to W.

In other words, (S, E) is a parameterized family of binary relations from V to W.

Definition 2.4. [18] An SB-relation (S, E) on V is called a Soft Reflexive Relation on V if for each e ∈E, S (e) is a reflexive relation on V.

2.1 Rough Set

The conception of Rough Set was firstly proposed by Z. Pawlak in 1982 [26]. The theory of rough sets presents a new approach in mathematics to handle incomplete knowledge. The rough set theory establishes a firm foundation for knowledge identification in databases. Rough sets provide a powerful mathematical tool to identify arrangements obscured into data. It can be used for feature selection, feature extraction, data reduction, decision rule generation and pattern recognition.

Definition 2.5. [26] Consider a finite set V and $R$ an equivalence relation on V . Then $(V, R$ ) is said to be an Approximation Space. If K is a subset of V, then K may or may not be written as a union of some $R$ -equivalence classes in V. If K can be written as the union of some equivalence classes, then we say that K is definable otherwise it isnot definable. If K is not definable, then we can approximate it by two definable subsets called lower and upper approximations of K as the following: $\underline{R} (K) = \cup {[k]_{R} : [k]_{R} \subseteq K} and$ $\bar{R} (K) = \cup {[k]_{R} : [k]_{R} \cap K \neq φ}$ where $[k]_{R}$ stands for the equivalence class determined by equivalence relation $R$ containing object k ∈ V. A rough set is the pair $(\underline{R} (K), \bar{R} (K)) .$ The set $\bar{R} (K) - \underline{R} (K)$ is known as boundary region. Clearly, if $\underline{R} (K) = \bar{R} (K)$ then K is definable and $\bar{R} (K) - \underline{R} (K) = φ$

${POS}_{R} (K) = \underline{R} (K), R - positive region of K$ ${NEG}_{R} (K) = V - \bar{R} (K), R - negative region of K$ ${BND}_{R} (K) = \bar{R} (K) - \underline{R} (K), R - boundry region of K .$

$v \in {POS}_{R} (K)$ implies that V certainly contains the object v.

$v \in {NEG}_{R} (K)$ implies that V definitely does not contain v.

$v \in {BND}_{R} (K)$ implies that V may or may not contain v.

Definition 2.6. [28] A triplet (ψ, ξ : A) is known as a bipolar soft set over V, where ψ and ξ are the set-valued mappings given by, $ψ : A \to P (V)$ and $ξ : \neg A \to P (V)$ such that ψ (e)∩ ξ (¬ e) = ∅ for each e ∈ A . The set ¬A is the “not set of A”.

In fact, a bipolar soft set (ψ, ξ : A) over V gives two parameterized collections of subsets of V. For each e ∈ A, ψ (e) and ξ (¬ e) are considered as the set of e-approximate elements of bipolar soft set. Henceforth, we will use the representation A_(ψ,ξ) for a bipolar soft set (ψ, ξ : A) over V.

Definition 2.7. [28] If A_{(ψ₁,ξ₁)} and B_{(ψ₂,ξ₂)} are two bipolar soft sets over V then A_{(ψ₁,ξ₁)} is a bipolar soft subset of B_{(ψ₂,ξ₂)} over V, if

1) A ⊆ B.

2) ψ₁ (e) ⊆ ψ₂ (e) and ξ₂ (¬ e) ⊆ ξ₁ (¬ e) for each e ∈A, and

We denote it by A_{(ψ₁,ξ₁)} ⊆ B_{(ψ₂,ξ₂)} .

In this case B_{(ψ₂,ξ₂)} is called a bipolar soft superset of A_{(ψ₁,ξ₁)} .

Equality of two bipolar soft sets A_{(ψ₁,ξ₁)} and B_{(ψ₂,ξ₂)} is then defined if both are bipolar soft subsets of each other i.e. $\begin{matrix} A_{(ψ_{1}, ξ_{1})} = B_{(ψ_{2}, ξ_{2})} \Leftrightarrow A_{(ψ_{1}, ξ_{1})} \subseteq B_{(ψ_{2}, ξ_{2})} and \\ B_{(ψ_{2}, ξ_{2})} \subseteq A_{(ψ_{1}, ξ_{1})} . \end{matrix}$

Definition 2.8. [28] A relative null bipolar soft set over V is a bipolar soft set denoted by $A_{(Θ, V)},$ where

$Θ : A \to P (V), e \mapsto \emptyset$ ; and $V : \neg A \to P (V), \neg e \mapsto V .$

Definition 2.9. [28] A relative absolute bipolar soft set over V is a bipolar soft set denoted by $A_{(V, Θ)},$ where $V : A \to P (V), e \mapsto$ V and Θ :¬ A → P (V) , ¬ e ↦ ∅.

Definition 2.10. [28] Let A_(ψ,ξ) be a bipolar soft set over V. Then the complement of A_(ψ,ξ) is denoted by A_(ψ,ξ)^c and defined as A_(ψ,ξ)^c = A_(ψ^c,ξ^c) where, ψ^c and ξ^c are mappings given by ψ^c (e) = ξ (¬ e) and ξ^c (¬ e) = ψ (e) for each e ∈ A.

Definition 2.11. [28] Let A_{(ψ₁,ξ₁)} and B_{(ψ₂,ξ₂)} be two bipolar soft sets over V such that A ∩ B ≠ ∅ . Then their restricted intersection bipolar soft set is defined as a bipolar soft set $(A \cap B)_{(ψ_{1} \tilde{\cap} ψ_{2}, ξ_{1} \tilde{\cup} ξ_{2})}$ over V in which $ψ_{1} \cap ψ_{2} : (A \cap B) \to P (V),$ defined by $e \mapsto ψ_{1} (e) \cap ψ_{2} (e),$ and $ξ_{1} \cup ξ_{2} : \neg (A \cap B) \to P (V),$ defined by $\neg e \mapsto ξ_{1} (\neg e) \cup ξ_{2} (\neg e) .$ We can write A_{(ψ₁,ξ₁)} ∩ _rB_{(ψ₂,ξ₂)} = (A ∩ B) _{(ψ₁∩ψ₂,ξ₁∪ξ₂)} .

Definition 2.12.[28] Let A_{(ψ₁,ξ₁)} and B_{(ψ₂,ξ₂)} be two bipolar soft sets over V such that A ∩ B ≠ ∅ . Then their restricted union bipolar soft set is defined as a bipolar soft set (A ∩ B) _{(ψ₁∪ψ₂,ξ₁∩ξ₂)} over V in which $ψ_{1} \cup ψ_{2} : (A \cap B) \to P (V),$ defined by $e \mapsto ψ_{1} (e) \cup ψ_{2} (e),$ and $ξ_{1} \cap ξ_{2} : \neg (A \cap B) \to P (V),$ defined by $\neg e \mapsto ξ_{1} (\neg e) \cap ξ_{2} (\neg e) .$ We can write A_{(ψ₁,ξ₁)} ∪ _rB_{(ψ₂,ξ₂)} = (A ∩ B) _{(ψ₁∪ψ₂,ξ₁∩ξ₂)} .

3 Rough Bipolar soft sets

In this section, we define the roughness of a bipolar soft set by soft binary relations. For a nonempty set denoted by $BSS (W)$ , we mean the set of all bipolar soft subsets of W.

Definition 3.1. Suppose (S, A) is an SB-relation from V to W and $σ = (ψ, ξ : A) \in BSS (W) .$ Then we can define two bipolar soft sets over V, denoted by ${\underline{S}}^{σ} = ({\underline{S}}^{ψ}, {\underline{S}}^{ξ}, : A)$ and ${\bar{S}}^{σ} = ({\bar{S}}^{ψ}, {\bar{S}}^{ξ} : A),$ respectively, where ${\underline{S}}^{ψ}, {\bar{S}}^{ψ}$ are defined as: ${\underline{S}}^{ψ} (e) = {v \in V : φ \neq vS (e) \subseteq ψ (e)},$ ${\bar{S}}^{ψ} (e) = {v \in V : vS (e) \cap ψ (e) \neq φ}$ for all e ∈ A, and ${\underline{S}}^{ξ}, {\bar{S}}^{ξ}$ are defined as: ${\underline{S}}^{ξ} (\neg e) = {v \in V : vS (e) \cap ξ (\neg e) \neq φ},$ ${\bar{S}}^{ξ} (\neg e) = {v \in V : φ \neq vS (e) \subseteq ξ (\neg e)}$ for each ¬e ∈ ¬ A, where vS (e) = {w ∈ W : (v, w) ∈ S (e)} is called the afterset of v. If ${\underline{S}}^{σ} = {\bar{S}}^{σ},$ then, σ is called (S, A)-definable, otherwise σ is a Rough Bipolar Soft Set $(RBS)$ set over V with respect to aftersets.

Definition 3.2. Suppose (S, A) is an SB-relation from V to W and $σ = (ψ, ξ : A) \in BSS (V) .$ Then we define two bipolar soft sets over W, denoted by $^{σ} \underline{S} = (^{ψ} \underline{S},^{ξ} \underline{S}, : A)$ and $^{σ} \bar{S} = (^{ψ} \bar{S},^{ξ} \bar{S} : A),$ respectively, where $^{ψ} \underline{S},^{ψ} \bar{S}$ are defined as: $^{ψ} \underline{S} (e) = {w \in W : φ \neq S (e) w \subseteq ψ (e)},$ $^{ψ} \bar{S} (e) = {w \in W : S (e) w \cap ψ (e) \neq φ}$ for all e ∈ A, and $^{ξ} \underline{S},^{ξ} \bar{S}$ are defined as: $^{ξ} \underline{S} (\neg e) = {w \in W : S (e) w \cap ξ (\neg e) \neq φ},$ $^{ξ} \bar{S} (\neg e) = {w \in W : φ \neq S (e) w \subseteq ξ (\neg e)}$ for all ¬e ∈ ¬ A, where S (e) w = {v ∈ V : (v, w) ∈ S (e)} and is known as foreset of w. If $^{σ} \underline{S} =^{σ} \bar{S},$ then, σ is called (S, A)-definable, otherwise σ is a Rough Bipolar Soft Set $RBS$ set over W with respect to foresets.

Let $σ = (ψ, ξ : A) \in BSS (W) .$ Then, ψ (e) ∩ ξ (¬ e) = φ for each e ∈ A. Take the lower rough bipolar soft approximation ${\underline{S}}^{σ} = ({\underline{S}}^{ψ}, {\underline{S}}^{ξ} : A)$ of σ with respect to aftersets. Now if $v \in {\underline{S}}^{ψ} (e) \cap {\underline{S}}^{ξ} (\neg e)$ for some e ∈ A, then $v \in {\underline{S}}^{ψ} (e)$ and $v \in {\underline{S}}^{ξ} (\neg e) .$ This implies that φ ≠ vS (e) ⊆ ψ (e) and vS (e) ∩ ξ (¬ e) ≠ φ ⇒φ ≠ vS (e) ⊆ ψ (e) ⊆ ξ^c (¬ e) and vS (e) ∩ ξ (¬ e) ≠ φ, which is impossible because ψ (e) ∩ ξ (¬ e) = φ for each e ∈ A. Thus, ${\underline{S}}^{ψ} (e) \cap {\underline{S}}^{ξ} (\neg e) = φ$ and hence, ${\underline{S}}^{σ}$ becomes a bipolar soft set over V .

Similarly, we can show that ${\bar{S}}^{σ}$ is a bipolar soft set over V .

Now, if $σ = (φ, ξ : A) \in BSS (V)$ then $^{σ} \underline{S}$ and $^{σ} \bar{S}$ are bipolar soft sets over W .

Theorem 3.1. Let (S, A) be an SB-relation from V to W, that is $S : A \to P (V \times W)$ . For $σ_{1} = (ψ_{1}, ξ_{1} : A), σ_{2} = (ψ_{2}, ξ_{2} : A) \in BSS (W)$ , the following properties hold with respect to aftersets:

(1) $σ_{1} \subseteq σ_{2} \Rightarrow {\underline{S}}^{σ_{1}} \subseteq {\underline{S}}^{σ_{2}}$

(2) $σ_{1} \subseteq σ_{2} \Rightarrow {\bar{S}}^{σ_{1}} \subseteq {\bar{S}}^{σ_{2}}$

(3) ${\underline{S}}^{σ_{1}} \cap_{r} {\underline{S}}^{σ_{2}} = {\underline{S}}^{σ_{1} \cap_{r} σ_{2}}$

(4) ${\bar{S}}^{σ_{1}} \cap_{r} {\bar{S}}^{σ_{2}} \supseteq {\bar{S}}^{σ_{1} \cap_{r} σ_{2}}$

(5) ${\underline{S}}^{σ_{1}} \cup_{r} {\underline{S}}^{σ_{2}} \subseteq {\underline{S}}^{σ_{1} \cup_{r} σ_{2}}$

(6) ${\bar{S}}^{σ_{1}} \cup_{r} {\bar{S}}^{σ_{2}} = {\bar{S}}^{σ_{1} \cup_{r} σ_{2}} .$

Proof. (1) Suppose that σ₁ ⊆ σ₂. Then we have ψ₁ (e) ⊆ ψ₂ (e) and ξ₂ (¬ e) ⊆ ξ₁ (¬ e) for each e ∈A. Now, $\begin{matrix} {\underline{S}}^{ψ_{1}} (e) & = & {v \in V : φ \neq vS (e) \subseteq ψ_{1} (e)} \\ \subseteq & {v \in V : φ \neq vS (e) \subseteq ψ_{2} (e)} because ψ_{1} (e) \subseteq ψ_{2} (e) \\ = & {\underline{S}}^{ψ_{2}} (e) . \\ This implies that {\underline{S}}^{ψ_{1}} (e) & \subseteq & {\underline{S}}^{ψ_{2}} (e), for all e \in A . \\ {\underline{S}}^{ξ_{2}} (\neg e) & = & {v \in V : vS (e) \cap ξ_{2} (\neg e) \neq φ} \\ \subseteq & {v \in V : vS (e) \cap ξ_{1} (\neg e) \neq φ} because ξ_{2} (\neg e) \subseteq ξ_{1} (\neg e) \\ = & {\underline{S}}^{ξ_{1}} (\neg e) \\ This implies that {\underline{S}}^{ξ_{2}} (\neg e) & \subseteq & {\underline{S}}^{ξ_{1}} (\neg e), for all \neg e \in \neg A . \\ Hence {\underline{S}}^{σ_{1}} \subseteq {\underline{S}}^{σ_{2}} . \end{matrix}$ (2) Suppose that σ₁ ⊆ σ₂. Then we have ψ₁ (e) ⊆ ψ₂ (e) and ξ₂ (¬ e) ⊆ ξ₁ (¬ e) for each e ∈A. Now, $\begin{matrix} {\bar{S}}^{ψ_{1}} (e) & = & {v \in V : vS (e) \cap ψ_{1} (e) \neq φ} \\ \subseteq & {v \in V : vS (e) \cap ψ_{2} (e) \neq φ} because ψ_{1} (e) \subseteq ψ_{2} (e) \\ = & {\bar{S}}^{ψ_{2}} (e) \\ This implies that {\bar{S}}^{ψ_{1}} (e) & \subseteq & {\bar{S}}^{ψ_{2}} (e) for all e \in A . \\ Now, {\bar{S}}^{ξ_{2}} (\neg e) & = & {v \in V : φ \neq vS (e) \subseteq ξ_{2} (\neg e)} \\ \subseteq & {v \in V : φ \neq vS (e) \subseteq ξ_{1} (\neg e)} because ξ_{2} (\neg e) \subseteq ξ_{1} (\neg e) \\ = & {\bar{S}}^{ξ_{1}} (\neg e) \\ This implies that {\bar{S}}^{ξ_{2}} (\neg e) & \subseteq & {\bar{S}}^{ξ_{1}} (\neg e) for all \neg e \in \neg A . \\ Hence {\bar{S}}^{σ_{1}} \subseteq {\bar{S}}^{σ_{2}} . \end{matrix}$ (3) Obviously ${\underline{S}}^{σ_{1} \cap_{r} σ_{2}} = ({\underline{S}}^{ψ_{1} \cap ψ_{2}}, {\underline{S}}^{ξ_{1} \cup ξ_{2}} : A)$ and ${\underline{S}}^{σ_{1}} \cap_{r} {\underline{S}}^{σ_{2}} = ({\underline{S}}^{ψ_{1}} \cap {\underline{S}}^{ψ_{2}}, {\underline{S}}^{ξ_{1}} \cup {\underline{S}}^{ξ_{2}} : A) .$

Let $v \in {\underline{S}}^{ψ_{1} \cap ψ_{2}} (e) \Leftrightarrow φ \neq vS (e) \subseteq (ψ_{1} \cap ψ_{2}) (e) \Leftrightarrow φ \neq vS (e) \subseteq (ψ_{1} (e) \cap ψ_{2} (e)) \Leftrightarrow φ \neq vS (e) \subseteq ψ_{1} (e)$ and $φ \neq vS (e) \subseteq ψ_{2} (e) \Leftrightarrow v \in {\underline{S}}^{ψ_{1}} (e) and v \in {\underline{S}}^{ψ_{2}} (e) \Leftrightarrow v \in {\underline{S}}^{ψ_{1}} (e) \cap {\underline{S}}^{ψ_{2}} (e) \Leftrightarrow v \in ({\underline{S}}^{ψ_{1}} \cap {\underline{S}}^{ψ_{2}}) (e) .$ Thus, ${\underline{S}}^{ψ_{1} \cap ψ_{2}} (e) = ({\underline{S}}^{ψ_{1}} \cap {\underline{S}}^{ψ_{2}}) (e) .$

$Let v \in {\underline{S}}^{ξ_{1} \cup ξ_{2}} (\neg e) \Leftrightarrow vS (e) \cap (ξ_{1} \cup ξ_{2}) (\neg e) \neq φ \Leftrightarrow vS (e) \cap (ξ_{1} (\neg e) \cup ξ_{2} (\neg e)) \neq φ \Leftrightarrow vS (e) \cap ξ_{1} (\neg e) \neq φ$ or $vS (e) \cap ξ_{2} (\neg e) \neq φ \Leftrightarrow v \in {\underline{S}}^{ξ_{1}} (\neg e)$ or $v \in {\underline{S}}^{ξ_{2}} (\neg e) \Leftrightarrow v \in {\underline{S}}^{ξ_{1}} (\neg e) \cup {\underline{S}}^{ξ_{2}} (\neg e) \Leftrightarrow v \in ({\underline{S}}^{ξ_{1}} \cup {\underline{S}}^{ξ_{2}}) (\neg e) .$ Thus, ${\underline{S}}^{ξ_{1}} (\neg e) \cup {\underline{S}}^{ξ_{2}} (\neg e) = {\underline{S}}^{ψ_{1} \cup ξ_{2}} (\neg e) .$

Hence, ${\underline{S}}^{σ_{1}} \cap_{r} {\underline{S}}^{σ_{2}} = {\underline{S}}^{σ_{1} \cap_{r} σ_{2}} .$

(4) Obviously ${\bar{S}}^{σ_{1} \cap_{r} σ_{2}} = ({\bar{S}}^{ψ_{1} \cap ψ_{2}}, {\bar{S}}^{ξ_{1} \cup ξ_{2}} : A)$ and ${\bar{S}}^{σ_{1}} \cap_{r} \bar{S^{σ_{2}}} = ({\bar{S}}^{ψ_{1}} \cap {\bar{S}}^{ψ_{2}}, {\bar{S}}^{ξ_{1}} \cup {\bar{S}}^{ξ_{2}} : A) .$ $\begin{matrix} Let v \in {\bar{S}}^{ψ_{1} \cap ψ_{2}} (e) & \Rightarrow & vS (e) \cap (ψ_{1} (e) \cap ψ_{2} (e)) \neq φ \\ \Rightarrow & vS (e) \cap ψ_{1} (e) \neq φ and vS (e) \cap ψ_{2} (e) \neq φ \\ \Rightarrow & v \in {\bar{S}}^{ψ_{1}} (e) and v \in {\bar{S}}^{ψ_{2}} (e) \\ \Rightarrow & v \in {\bar{S}}^{ψ_{1}} (e) \cap {\bar{S}}^{ψ_{2}} (e) . \end{matrix}$ This implies that, ${\bar{S}}^{ψ_{1} \cap ψ_{2}} (e) \subseteq {\bar{S}}^{ψ_{1}} (e) \cap {\bar{S}}^{ψ_{2}} (e)$ . $\begin{matrix} Let v \in {\bar{S}}^{ξ_{1}} (\neg e) \cap {\bar{S}}^{ξ_{2}} (\neg e) & \Rightarrow & v \in {\bar{S}}^{ξ_{1}} (\neg e) and v \in {\bar{S}}^{ξ_{2}} (\neg e) \\ \Rightarrow & φ \neq vS (e) \subseteq ξ_{1} (\neg e) and φ \neq vS (e) \subseteq ξ_{2} (\neg e) \\ \Rightarrow & φ \neq vS (e) \subseteq ξ_{1} (\neg e) \cap ξ_{2} (\neg e) \\ \Rightarrow & v \in {\bar{S}}^{ξ_{1} \cap ξ_{2}} (\neg e) \\ \Rightarrow & {\bar{S}}^{ξ_{1}} (\neg e) \cap {\bar{S}}^{ξ_{2}} (\neg e) \subseteq {\bar{S}}^{ψ_{1} \cap ξ_{2}} (\neg e) . \end{matrix}$ $This implies that, {\bar{S}}^{σ_{1}} \cap_{r} {\bar{S}}^{σ_{2}} \supseteq {\bar{S}}^{σ_{1} \cap_{r} σ_{2}} .$

(5) $Obviously, {\underline{S}}^{σ_{1} \cup_{r} σ_{2}} = ({\underline{S}}^{ψ_{1} \cup ψ_{2}}, {\underline{S}}^{ξ_{1} \cap ξ_{2}} : A)$ and ${\underline{S}}^{σ_{1}} \cup_{r} {\underline{S}}^{σ_{2}} = ({\underline{S}}^{ψ_{1}} \cup {\underline{S}}^{ψ_{2}}, {\underline{S}}^{ξ_{1}} \cap {\underline{S}}^{ξ_{2}} : A) .$ $\begin{matrix} Let v \in ({\underline{S}}^{ψ_{1}} \cup {\underline{S}}^{ψ_{2}}) (e) & \Rightarrow & v \in ({\underline{S}}^{ψ_{1}} (e) \cup {\underline{S}}^{ψ_{2}} (e)) \\ \Rightarrow & v \in {\underline{S}}^{ψ_{1}} (e) or v \in {\underline{S}}^{ψ_{2}} (e) \\ \Rightarrow & φ \neq vS (e) \subseteq ψ_{1} (e) or φ \neq vS (e) \subseteq ψ_{2} (e) \\ \Rightarrow & φ \neq vS (e) \subseteq ψ_{1} (e) \cup ψ_{2} (e) \\ \Rightarrow & φ \neq vS (e) \subseteq (ψ_{1} \cup ψ_{2}) (e) \\ \Rightarrow & v \in {\underline{S}}^{ψ_{1} \cup ψ_{2}} (e) \\ \Rightarrow & ({\underline{S}}^{ψ_{1}} \cup {\underline{S}}^{ψ_{2}}) (e) \subseteq {\underline{S}}^{ψ_{1} \cup ψ_{2}} (e) . \\ Let v \in {\underline{S}}^{ξ_{1} \cap ξ_{2}} (\neg e) & \Rightarrow & vS (e) \cap (ξ_{1} \cap ξ_{2}) (\neg e) \neq φ \\ \Rightarrow & vS (e) \cap (ξ_{1} (\neg e) \cap ξ_{2} (\neg e)) \neq φ \\ \Rightarrow & vS (e) \cap ξ_{1} (\neg e) \neq φ and vS (e) \cap ξ_{2} (\neg e) \neq φ \\ \Rightarrow & v \in {\underline{S}}^{ξ_{1}} (\neg e) and v \in {\underline{S}}^{ξ_{2}} (\neg e) \\ \Rightarrow & v \in {\underline{S}}^{ξ_{1}} (\neg e) \cap {\underline{S}}^{ξ_{2}} (\neg e) \\ \Rightarrow & v \in ({\underline{S}}^{ξ_{1}} \cap {\underline{S}}^{ξ_{2}}) (\neg e) \\ \Rightarrow & {\underline{S}}^{ξ_{1} \cap ξ_{2}} (\neg e) \subseteq ({\underline{S}}^{ξ_{1}} \cap {\underline{S}}^{ξ_{2}}) (\neg e) . \end{matrix}$ $This implies that, {\underline{S}}^{σ_{1}} \cup_{r} {\underline{S}}^{σ_{2}} \subseteq {\underline{S}}^{σ_{1} \cup_{r} σ_{2}} .$

(6) $Obviously, {\bar{S}}^{σ_{1} \cup_{r} σ_{2}} = ({\bar{S}}^{ψ_{1} \cup ψ_{2}}, {\bar{S}}^{ξ_{1} \cap ξ_{2}} : A)$ and ${\bar{S}}^{σ_{1}} \cup_{r} {\bar{S}}^{σ_{2}} = ({\bar{S}}^{ψ_{1}} \cup {\bar{S}}^{ψ_{2}}, {\bar{S}}^{ξ_{1}} \cap {\bar{S}}^{ξ_{2}} : A) .$

Let $v \in ({\bar{S}}^{ψ_{1} \cup ψ_{2}}) (e) \Leftrightarrow vS (e) \cap (ψ_{1} \cup ψ_{2}) (e) \neq φ \Leftrightarrow vS (e) \cap (ψ_{1} (e) \cup ψ_{2} (e)) \neq φ \Leftrightarrow vS (e) \cap ψ_{1} (e) \neq φ or vS (e) \cap ψ_{2} (e) \neq φ \Leftrightarrow v \in {\bar{S}}^{ψ_{1}} (e) or v \in {\bar{S}}^{ψ_{2}} (e) \Leftrightarrow v \in {\bar{S}}^{ψ_{1}} (e) \cup {\bar{S}}^{ψ_{2}} (e) \Leftrightarrow v \in ({\bar{S}}^{ψ_{1}} \cup {\bar{S}}^{ψ_{2}}) (e) .$ Hence, ${\bar{S}}^{ψ_{1} \cup ψ_{2}} = {\bar{S}}^{ψ_{1}} \cup {\bar{S}}^{ψ_{2}} .$ Let v $\in ({\bar{S}}^{ξ_{1} \cap ξ_{2}}) (\neg e) \Leftrightarrow φ \neq vS (e) \subseteq (ξ_{1} \cap ξ_{2}) (\neg e) \Leftrightarrow φ \neq vS (e) \subseteq (ξ_{1} (\neg e) \cap ξ_{2} (\neg e)) \Leftrightarrow φ \neq vS (e) \subseteq ξ_{1} (\neg e) and φ \neq vS (e) \subseteq ξ_{2} (\neg e) \Leftrightarrow v \in {\bar{S}}^{ξ_{1}} (\neg e) and v \in {\bar{S}}^{ξ_{2}} (\neg e) \Leftrightarrow v \in {\bar{S}}^{ξ_{1}} (\neg e) \cap {\bar{S}}^{ξ_{2}} (\neg e) \Leftrightarrow v \in ({\bar{S}}^{ξ_{1}} \cap {\bar{S}}^{ξ_{2}}) (\neg e) .$ Hence, ${\bar{S}}^{ξ_{1} \cap ξ_{2}} = {\bar{S}}^{ξ_{1}} \cap {\bar{S}}^{ξ_{2}} .$

This implies that, ${\bar{S}}^{σ_{1}} \cup_{r} {\bar{S}}^{σ_{2}} = {\bar{S}}^{σ_{1} \cup_{r} σ_{2}} .$ □

The following example shows that the inclusions in parts (4) and (5) of the Theorem 3.1 may be proper.

Example 3.1. Consider W = {c₁, c₂, c₃, c₄, c₅} is a universe containing five computers. These computers are categorized by attributes age and colour represented by A = {e₁, e₂} . Let V = {old, new, white, black, used} be represented by {o, n, w, b, u} .

Define the relations $S : A \to P (V \times W)$ by $S (e_{1}) = {(u, c_{1}), (u, c_{2}), (o, c_{3}), (o, c_{4}), (n, c_{5})}$ $S (e_{2}) = {(w, c_{1}), (w, c_{2}), (b, c_{3}), (b, c_{4}) (b, c_{5})}$ Now, oS (e₁) = {c₃, c₄} , nS (e₁) = {c₅} , bS (e₁) = φ, wS (e₁) = φ and uS (e₁) = {c₁, c₂} .

Suppose $σ_{1} = (ψ_{1}, ξ_{1} : A), σ_{2} = (ψ_{2}, ξ_{2} : A) \in BSS (W),$ defined as:

ψ₁ (e₁) = {c₁, c₂, c₃} , ξ₁ (¬ e₁) = {c₄, c₅} and ψ₂ (e₁) = {c₂, c₄} , ξ₂ (¬ e₁) = {c₃, c₅} .

ψ₁ (e₂) = {c₁, c₃} , ξ₁ (¬ e₂) = {c₅} and ψ₂ (e₂) = {c₁, c₂} , ξ₂ (¬ e₂) = {c₃, c₄} .

Then, (ψ₁ ∪ ψ₂) (e₁) = ψ₁ (e₁) ∪ ψ₂ (e₁) = {c₁, c₂, c₃, c₄} and (ξ₁ ∩ ξ₂) (¬ e₁) = ξ₁ (¬ e₁) ∩ ξ₂ (¬ e₁) = {c₅} .

Now, ${\underline{S}}^{ψ_{1}} (e_{1}) = {u},$ ${\underline{S}}^{ψ_{2}} (e_{1}) = φ,$ ${\underline{S}}^{ξ_{1}} (\neg e_{1}) = {n, o},$ ${\underline{S}}^{ξ_{2}} (\neg e_{1}) = {n, o}$

$\Rightarrow ({\underline{S}}^{ψ_{1}} \cup {\underline{S}}^{ψ_{2}}) (e_{1}) = {u}$ and ( ${\underline{S}}^{ξ_{1}} \cap {\underline{S}}^{ξ_{2}}) (\neg e_{1}) = {n, o} .$ But, ${\underline{S}}^{ψ_{1} \cup ψ_{2}} (e_{1}) = {u, o}$ and ${\underline{S}}^{ξ_{1} \cap ξ_{2}} (\neg e_{1}) = {n} .$

This verifies the proper inclusion in (5), that is, ${\underline{S}}^{σ_{1}} \cup_{r} {\underline{S}}^{σ_{2}} \subseteq {\underline{S}}^{σ_{1} \cup_{r} σ_{2}} .$

Also, ${\bar{S}}^{ψ_{1}} (e_{1}) = {u, o},$ ${\bar{S}}^{ψ_{2}} (e_{1}) = {u, o},$ ${\bar{S}}^{ξ_{1}} (\neg e_{1}) = {n},$ ${\bar{S}}^{ξ_{2}} (\neg e) = {n} .$

$\Rightarrow ({\bar{S}}^{ψ_{1}} \cap {\bar{S}}^{ψ_{2}}) (e_{1}) = {u, o} and ({\bar{S}}^{ξ_{1}} \cup {\bar{S}}^{ξ_{2}}) (\neg e_{1}) = {n} .$ But, ${\bar{S}}^{ψ_{1} \cap ψ_{2}} (e_{1}) = {u}$ and ${\bar{S}}^{ξ_{1} \cap ξ_{2}} (\neg e_{1}) = {n, o} .$

This verifies the proper inclusion in (4), that is, ${\bar{S}}^{σ_{1} \cap_{r} σ_{2}} \subseteq {\bar{S}}^{σ_{1}} \cap_{r} {\bar{S}}^{σ_{2}} .$

Theorem 3.2. Let (S, A) be an SB-relation from V to W, that is $S : A \to P (V \times W)$ . For $σ_{1} = (ψ_{1}, ξ_{1} : A), σ_{2} = (ψ_{2}, ξ_{2} : A) \in BSS (V)$ , the following properties hold with respect to foresets:

(1) $σ_{1} \subseteq σ_{2} \Rightarrow^{σ_{1}} \underline{S} \subseteq^{σ_{2}} \underline{S}$

(2) $σ_{1} \subseteq σ_{2} \Rightarrow \bar{S} σ_{1} \subseteq^{σ_{2}} \bar{S}$

(3) $^{σ_{1}} \underline{S} \cap_{r}^{σ_{2}} \underline{S} =^{σ_{1} \cap_{r} σ_{2}} \underline{S}$

(4) $^{σ_{1}} \bar{S} \cap_{r} \bar{S}^{σ_{2}} \supseteq^{σ_{1} \cap_{r} σ_{2}} \bar{S}$

(5) $^{σ_{1}} \underline{S} \cup_{r}^{σ_{2}} \underline{S} \subseteq^{σ_{1} \cup_{r} σ_{2}} \underline{S}$

(6) $^{σ_{1}} \bar{S} \cup_{r}^{σ_{2}} \bar{S} =^{σ_{1} \cup_{r} σ_{2}} \bar{S} .$

Proof. The proof is similar to the proof of the Theorem 3.1. □

Theorem 3.3. Let (S, A) be an SB-relation from V to W, that is S : A → P (V × W). Let $σ = (ψ, ξ : A) \in BSS (W) .$ Then the following properties hold with respect to aftersets:

(1). ${\underline{S}}^{σ^{c}} = ({\bar{S}}^{σ})^{c}$

(2). ${\bar{S}}^{σ^{c}} = ({\underline{S}}^{σ})^{c} .$

Proof. 1).We know that σ^c = (ψ^c, ξ^c : A) where ψ^c (e) = ξ (¬ e) and ξ^c (¬ e) = ψ (e). Also, ${\underline{S}}^{σ} = ({\underline{S}}^{ψ}, {\underline{S}}^{ξ} : A)$ so $({\underline{S}}^{σ})^{c} = (({\underline{S}}^{ψ})^{c}, ({\underline{S}}^{ξ})^{c} : A)$ where $({\underline{S}}^{ψ})^{c} (e) = ({\underline{S}}^{ξ}) (\neg e)$ and $({\underline{S}}^{ξ})^{c} (\neg e) = ({\underline{S}}^{ψ}) (e) .$

We show that ${\underline{S}}^{ψ^{c}} = ({\bar{S}}^{ψ})^{c}$ and ${\underline{S}}^{ξ^{c}} = ({\bar{S}}^{ξ})^{c} .$

Now for e ∈A, we have

${\underline{S}}^{ψ^{c}} (e) = {v \in V : φ \neq vS (e) \subseteq ψ^{c} (e)} = {v \in V : φ \neq vS (e) \subseteq ξ (\neg e)} because ψ^{c} (e) = ξ (\neg e) = {\bar{S}}^{ξ} (\neg e) = ({\bar{S}}^{ψ})^{c} (e)$ .

${\underline{S}}^{ξ^{c}} (\neg e) = {v \in V : vS (e) \cap ξ^{c} (\neg e) \neq φ} = {v \in V : vS (e) \cap ψ (e) \neq φ} = {\bar{S}}^{ψ} (e) = ({\bar{S}}^{ξ})^{c} (\neg e) .$

Hence ${\underline{S}}^{σ^{c}} = ({\bar{S}}^{σ})^{c} .$

(2). To prove this, we show that ${\bar{S}}^{ψ^{c}} = ({\underline{S}}^{ψ})^{c}$ and ${\bar{S}}^{ξ^{c}} = ({\underline{S}}^{ξ})^{c} .$

Now for e ∈A, we have

${\bar{S}}^{ψ^{c}} (e) = {v \in V : vS (e) \cap ψ^{c} (e) \neq φ} = {v \in V : vS (e) \cap ξ (\neg e) \neq φ} = {\underline{S}}^{ξ} (\neg e) = ({\underline{S}}^{ψ})^{c} (e) .$

${\bar{S}}^{ξ^{c}} (\neg e) = {v \in V : φ \neq vS (e) \subseteq ξ^{c} (\neg e)} = {v \in V : φ \neq vS (e) \subseteq ψ (e)} = {\underline{S}}^{ψ} (e) = ({\underline{S}}^{ξ})^{c} (\neg e)$ .

Hence ${\bar{S}}^{σ^{c}} = ({\underline{S}}^{σ})^{c} .$ □

Theorem 3.4. Let (S, A) be an SB-relation from V to W, that is S : A → P (V × W). Let $σ = (ψ, ξ : A) \in BSS (V) .$ Then the following properties hold with respect to foresets:

(1) $^{σ^{c}} \underline{S} = {(^{σ} \bar{S})}^{c}$

(2) $^{σ^{c}} \bar{S} = (^{σ} \underline{S})^{c} .$

Proof. The proof is similar to the proof of the Theorem 3.3. □

Next we consider the soft relation on V. All the results proved above are true for the soft relations on V. If (S, A) is a soft reflexive relation on V then vS (e) and S (e) v are non-empty because v ∈ vS (e) and v ∈ S (e) v for all v ∈ V .

Theorem 3.5. Suppose $σ = (ψ, ξ : A) \in BSS (V)$ and (S, A) is a soft reflexive relation on V. Then the following properties hold with respect to aftersets:

(1) ${\underline{S}}^{σ} \subseteq σ$

(2) $σ \subseteq {\bar{S}}^{σ}$

(3) ${\underline{S}}^{Θ_{A}} = Θ_{A} = {\bar{S}}^{Θ_{A}}$

(4) ${\underline{S}}^{V_{A}} = V_{A} = {\bar{S}}^{V_{A}} .$

Proof. (1) To prove that ${\underline{S}}^{σ} \subseteq σ$ , we show that ${\underline{S}}^{ψ} (e) \subseteq ψ (e)$ and ${\underline{S}}^{ξ} (\neg e) \supseteq ξ (\neg e) .$

Let $v \in {\underline{S}}^{ψ} (e) .$ Then vS (e) ⊆ ψ (e) . But v ∈ vS (e) so v ∈ ψ (e) .

Hence, ${\underline{S}}^{ψ} (e) \subseteq ψ (e) .$

Let v ∈ ξ (¬ e) . Also v ∈ vS (e) , so v ∈ vS (e) ∩ ξ (¬ e) . This implies vS (e) ∩ ξ (¬ e) ≠ φ, that is $v \in {\underline{S}}^{ξ} (\neg e)$ . Thus ${\underline{S}}^{ξ} (\neg e) \supseteq ξ (\neg e) .$

Hence, ${\underline{S}}^{σ} \subseteq σ .$

(2) To prove $σ \subseteq {\bar{S}}^{σ},$ we show that $ψ (e) \subseteq {\bar{S}}^{ψ} (e)$ and ${\bar{S}}^{ξ} (\neg e) \subseteq ξ (\neg e) .$

Now, let v ∈ ψ (e) , also v ∈ vS (e) . This implies v ∈ vS (e) ∩ ψ (e) . This implies vS (e) ∩ ψ (e) ≠ φ, that is $v \in {\bar{S}}^{ψ} (e) .$ Thus, $ψ (e) \subseteq {\bar{S}}^{ψ} (e) .$

Now, let $v \in {\bar{S}}^{ξ} (\neg e) .$ This implies vS (e) ⊆ ξ (¬ e) . But v ∈ vS (e) , this implies v ∈ ξ (¬ e). Thus ${\bar{S}}^{ξ} (\neg e) \subseteq ξ (\neg e) .$

Hence $σ \subseteq {\bar{S}}^{σ} .$

(3) The null bipolar soft set $Θ_{A} = (Θ, V : A)$ has the lower and upper rough bipolar approximations denoted by ${\underline{S}}^{Θ_{A}} = ({\underline{S}}^{Θ}, {\underline{S}}^{V} : A)$ and ${\bar{S}}^{Θ_{A}} = ({\bar{S}}^{Θ}, {\bar{S}}^{V} : A)$ , respectively.

Now we have for e ∈A $\begin{matrix} {\underline{S}}^{Θ} (e) & = & {v \in V : vS (e) \subseteq Θ (e)} \\ = & {v \in V : vS (e) \subseteq φ} \\ = & φ = Θ (e), \\ {\bar{S}}^{Θ} (e) & = & {v \in V : vS (e) \cap Θ (e) \neq φ} \\ = & {v \in V : vS (e) \cap φ \neq φ} \\ = & φ = Θ (e) . \\ {\underline{S}}^{V} (\neg e) & = & {v \in V : vS (e) \cap V (\neg e) \neq φ} \\ = & {v \in V : vS (e) \cap V \neq φ} \\ = & V = V (\neg e), \\ {\bar{S}}^{V} (\neg e) & = & {v \in V : vS (e) \subseteq V (\neg e)} \\ = & {v \in V : vS (e) \subseteq V} \\ = & V = V (\neg e) . \end{matrix}$ Hence ${\underline{S}}^{Θ_{A}} = Θ_{A} = {\bar{S}}^{Θ_{A}} .$

(4) The absolute bipolar soft set $V_{A} = (V, Θ : A)$ has the lower and upper rough bipolar approximations denoted by ${\underline{S}}^{V_{A}} = ({\underline{S}}^{V}, {\underline{S}}^{Θ} : A)$ and ${\bar{S}}^{V_{A}} = ({\bar{S}}^{V}, {\bar{S}}^{Θ} : A)$ , respectively.

Now we have for e ∈A $\begin{matrix} {\underline{S}}^{V} (e) & = & {v \in V : vS (e) \subseteq V (e)} \\ = & {v \in V : vS (e) \subseteq V} \\ = & V = V (e), \\ {\bar{S}}^{V} (e) & = & {v \in V : vS (e) \cap V (\neg e) \neq φ} \\ = & {v \in V : vS (e) \cap V \neq φ} \\ = & V = V (e) . \\ {\underline{S}}^{Θ} (\neg e) & = & {v \in V : vS (e) \cap Θ (\neg e) \neq φ} \\ = & {v \in V : vS (e) \cap φ \neq φ} \\ = & φ = Θ (\neg e), \\ {\bar{S}}^{Θ} (\neg e) & = & {v \in V : vS (e) \subseteq Θ (\neg e)} \\ = & {v \in V : vS (e) \subseteq φ} \\ = & φ = Θ (\neg e) . \end{matrix}$ Hence ${\underline{S}}^{V_{A}} = V_{A} = {\bar{S}}^{V_{A}} .$ □

Theorem 3.6. Suppose $σ = (ψ, ξ : A) \in BSS (V)$ and (S, A) be a soft reflexive relation on V. Then the following assertions hold with respect to foresets:

(1) $^{σ} \underline{S} \subseteq σ$

(2) $σ \subseteq^{σ} \bar{S}$

(3) $^{Θ_{A}} \underline{S} = Θ_{A} =^{Θ_{A}} \bar{S}$

(4) $^{V_{A}} \underline{S} = V_{A} =^{V_{A}} \bar{S} .$

Proof. The proof is similar to the proof of the Theorem 3.5. □

Theorem 3.7. Let (S, A) be a soft reflexive relation on V and $σ = (ψ, ξ : A) \in BSS (V) .$ Let (T, A) be another soft reflexive relation on V such that, S ⊆ T . Then the following properties hold with respect to aftersets:

(1). ${\underline{T}}^{σ} \subseteq {\underline{S}}^{σ}$

(2). ${\bar{S}}^{σ} \subseteq {\bar{T}}^{σ} .$

Proof. (1) We show that, ${\underline{T}}^{ψ} (e) \subseteq {\underline{S}}^{ψ} (e)$ and ${\underline{S}}^{ξ} (\neg e) \subseteq {\underline{T}}^{ξ} (\neg e) .$ Let $v \in {\underline{T}}^{ψ} (e) \Rightarrow vT (e) \subseteq ψ (e) .$ Then vS (e) ⊆ ψ (e) , because, S (e) ⊆ T (e) $\Rightarrow v \in {\underline{S}}^{ψ} (e) .$ Thus, ${\underline{T}}^{ψ} (e) \subseteq {\underline{S}}^{ψ} (e),$

Also, let $v \in {\underline{S}}^{ξ} (\neg e) \Rightarrow vS (e) \cap ξ (\neg e) \neq φ .$ Then, vT (e) ∩ ξ (¬ e) ≠ φ, because, $S (e) \subseteq T (e) \Rightarrow v \in {\underline{T}}^{ξ} (\neg e) .$ Thus, ${\underline{S}}^{ξ} (\neg e) \subseteq {\underline{T}}^{ξ} (\neg e) .$ Hence ${\underline{T}}^{σ} \subseteq {\underline{S}}^{σ} .$

(2). To prove that ${\bar{S}}^{σ} \subseteq {\bar{T}}^{σ},$ we show that, ${\bar{S}}^{ψ} (e) \subseteq {\bar{T}}^{ψ} (e)$ and ${\bar{T}}^{ξ} (\neg e) \subseteq {\bar{S}}^{ξ} (\neg e) .$

Let $v \in {\bar{S}}^{ψ} (e) \Rightarrow vS (e) \cap ψ (e) \neq φ . Then vT (e) \cap ψ (e) \neq φ, because, S (e) \subseteq T (e) \Rightarrow v \in {\bar{T}}^{ψ} (e) .$ Thus, ${\bar{S}}^{ψ} (e) \subseteq {\bar{T}}^{ψ} (e) .$

Now, let $v \in {\bar{T}}^{ξ} (\neg e) \Rightarrow vT (e) \subseteq ξ (\neg e) .$ Then vS (e) ⊆ ξ (¬ e) , because, S (e) ⊆ T (e) , this implies, $v \in {\bar{S}}^{ξ} (\neg e) .$ Thus ${\bar{T}}^{ξ} (\neg e) \subseteq {\bar{S}}^{ξ} (\neg e) .$ Hence, ${\bar{S}}^{σ} \subseteq {\bar{T}}^{σ} .$ □

Theorem 3.8. Let (S, A) be a soft reflexive relation on V and $σ = (ψ, ξ : A) \in BSS (V) .$ Let (T,A) be another soft reflexive relation on V such that, S ⊆ T . Then the following properties hold with respect to foresets:

(1). $^{σ} \underline{T} \subseteq^{σ} \underline{S}$

(2). $^{σ} \bar{S} \subseteq^{σ} \bar{T} .$

Proof. The proof is similar to the proof of the Theorem 3.7. □

Definition 3.3. An SB-relation (S, A) on V is known as a soft symmetric relation on V if for each e ∈A, S(e) is a symmetric relation on V.

If (S, A) is a soft symmetric relation on V, then u ∈ vS (e) ⇒ v ∈ uS (e) and u ∈ S (e) v ⇒ v ∈ S (e) u .

Theorem 3.9. Let (S, A) be a soft symmetric relation on V. For a bipolar soft set σ = (ψ, ξ : A) on V, the following properties hold with respect to aftersets:

(1). ${\bar{S}}^{({\underline{S}}^{σ})} \subseteq σ$

(2). $σ \subseteq {\underline{S}}^{({\bar{S}}^{σ})}$

Proof. (1). To prove ${\bar{S}}^{({\underline{S}}^{σ})} \subseteq σ$ , we show that, ${\bar{S}}^{({\underline{S}}^{ψ} (e))} (e) \subseteq ψ (e)$ and $ξ (\neg e) \subseteq {\bar{S}}^{({\underline{S}}^{ξ} (\neg e))} (\neg e) .$

Let $v \in {\bar{S}}^{({\underline{S}}^{ψ} (e))} (e) \Rightarrow vS (e) \cap {\underline{S}}^{ψ} (e) \neq φ . So v_{1} \in vS (e) \cap {\underline{S}}^{ψ} (e) \Rightarrow v_{1} \in vS (e) and v_{1} \in {\underline{S}}^{ψ} (e) . Now v_{1} \in {\underline{S}}^{ψ} (e) \Rightarrow φ \neq v_{1} S (e) \subseteq ψ (e) . Also, v_{1} \in vS (e)$ and the relation is soft symmetric so, v ∈ v₁S (e) ⇒ v ∈ ψ (e) . Hence ${\bar{S}}^{({\underline{S}}^{ψ} (e))} \subseteq ψ (e) .$

Let $v \notin {\bar{S}}^{({\underline{S}}^{ξ} (\neg e))} (\neg e) \Rightarrow vS (e) ⊈ {\underline{S}}^{ξ} (\neg e) \Rightarrow \exists some v_{1} \in vS (e) such that v_{1} \notin {\underline{S}}^{ξ} (\neg e) \Rightarrow v_{1} S (e) \cap ξ (\neg e) = φ .$ There are three possibilities:

i) If ξ (¬ e) = φ then v ∉ ξ (¬ e) .

ii) If v₁S (e) = φ, then this is not possible because v₁ ∈ vS (e) ⇒ v ∈ v₁S (e) .

iii) If v₁S (e) ⊆ ξ^c (¬ e) while v ∈ v₁S (e) , then v ∈ ξ^c (¬ e) ⇒ v ∉ ξ (¬ e) .

Hence $ξ (\neg e) \subseteq {\bar{S}}^{({\underline{S}}^{ξ} (\neg e))} .$

Thus, ${\bar{S}}^{({\underline{S}}^{σ})} \subseteq σ .$

(2). To prove this $σ \subseteq {\underline{S}}^{({\bar{S}}^{σ})},$ we show $ψ (e) \subseteq {\underline{S}}^{({\bar{S}}^{ψ} (e))} (e)$ and ${\underline{S}}^{({\bar{S}}^{ξ} (\neg e))} (\neg e) \subseteq ξ (\neg e) .$

Let $v \notin {\underline{S}}^{({\bar{S}}^{ψ} (e))} (e) \Rightarrow vS (e) ⊈ {\bar{S}}^{ψ} (e) \Rightarrow \exists some v_{1} \in vS (e) such that v_{1} \notin {\bar{S}}^{ψ} (e) \Rightarrow v_{1} S (e) \cap ψ (e) = φ .$ Again, there are three possibilities:

i) If ψ (e) = φ then v ∉ ψ (e) .

ii) If v₁S (e) = φ, then this is not possible because v₁ ∈ vS (e) ⇒ v ∈ v₁S (e) .

iii) If v₁S (e) ⊆ ψ^c (e) while v ∈ v₁S (e) , then v ∈ ψ^c (e) ⇒ v ∉ ψ (e) .

Hence $ψ (e) \subseteq {\underline{S}}^{({\bar{S}}^{ψ} (e))} (e) .$

Let $v \in {\underline{S}}^{({\bar{S}}^{ξ (\neg e)})} (\neg e) \Rightarrow vS (e) \cap {\bar{S}}^{ξ} (\neg e) \neq φ .$ So $\exists v_{1} \in vS (e) \cap {\bar{S}}^{ξ} (\neg e) \Rightarrow v_{1} \in vS (e) and v_{1} \in {\bar{S}}^{ξ} (\neg e) . Now, if v_{1} \in {\bar{S}}^{ξ} (\neg e) then φ \neq v_{1} S (e) \subseteq ξ (\neg e) . But v_{1} \in vS (e) and the relation is symmetric so v \in v_{1} S (e) \Rightarrow v \in ξ (\neg e) .$

Hence ${\underline{S}}^{({\bar{S}}^{ξ} (\neg e))} \subseteq ξ (\neg e) .$

Thus, $σ \subseteq {\underline{S}}^{({\bar{S}}^{σ})} .$ □

Theorem 3.10. Let (S, A) be a soft symmetric relation on V. For a bipolar soft set σ = (ψ, ξ : A) on V, the following properties hold with respect to foresets:

(1) $^{({\underline{S}}^{σ})} \bar{S} \subseteq σ$

(2) $σ \subseteq^{({\bar{S}}^{σ})} \underline{S}$

Proof. The proof is similar to the proof of the Theorem 3.9. □

4 Bipolar soft topologies induced by soft reflexive relations

In this section, we will examine two kinds of bipolar soft topologies induced by soft reflexive relations. We consider all the bipolar soft sets with a parameter set A.

Definition 4.1. [29] A family $\tilde{τ} \subseteq BSS (V)$ is called a bipolar soft topology over V if it satisfies the following three conditions:

1) $Θ_{A}, V_{A} \in \tilde{τ}$ where, $Θ_{A} = (Θ, V : A)$ is a null bipolar soft set and $V_{A} = (V, Θ : A)$ is an absolute bipolar soft set.

2) For $σ_{1}, σ_{2} \in \tilde{τ} \Rightarrow σ_{1} \cap σ_{2} \in \tilde{τ} .$

3) For all $σ_{i} \in \tilde{τ}, i \in I \Rightarrow \cup_{i \in I} σ_{i} \in \tilde{τ} .$

Then the pair $(V, \tilde{τ} : A)$ is called a bipolar soft topological space over V. The elements of $\tilde{τ}$ are called bipolar soft open sets.

Remark 4.1. It is obvious that ${\underline{S}}^{σ} = σ$ if and only if ${\bar{S}}^{σ} = σ$ if and only if ${\underline{S}}^{σ} = {\bar{S}}^{σ}$ if and only if σ is (S, A)-definable.

Theorem 4.1 If (S, A) is a soft reflexive relation on V, then ${\tilde{τ}}_{σ} = {σ \in BSS (V) : {\underline{S}}^{σ} = σ}$ $({\tilde{τ}}_{σ} = {σ \in BSS (V) : {\bar{S}}^{σ} = σ})$ is a bipolar soft topology on V with respect to aftersets.

Proof. 1) Suppose $Θ_{A} = (Θ, V : A)$ is a null bipolar soft set and $V_{A} = (V, Θ : A)$ is an absolute bipolar soft set. By the Theorem 3.5, ${\underline{S}}^{Θ_{A}} = Θ_{A}$ and ${\underline{S}}^{V_{A}} = V_{A} .$ This implies $Θ_{A}, V_{A} \in {\tilde{τ}}_{σ} .$

2) Suppose $σ_{1}, σ_{2} \in {\tilde{τ}}_{σ}$ be two bipolar soft sets. This implies ${\underline{S}}^{σ_{1}} = σ_{1}$ and ${\underline{S}}^{σ_{2}} = σ_{2} .$ Now, by using the Theorem 3.1, ${\underline{S}}^{σ_{1} \cap σ_{2}} = {\underline{S}}^{σ_{1}} \cap {\underline{S}}^{σ_{2}} = σ_{1} \cap σ_{2} .$ This implies $σ_{1} \cap σ_{2} \in {\tilde{τ}}_{σ} .$

3) Suppose $σ_{i} \in {\tilde{τ}}_{σ}$ for i ∈ I . This implies ${\underline{S}}^{σ_{i}} = σ_{i}$ for i ∈ I .

To show $\cup_{i \in I} σ_{i} \in {\tilde{τ}}_{σ},$ it is enough to prove $\cup_{i \in I} σ_{i} \subseteq {\underline{S}}^{\cup_{i \in I} σ_{i}} .$ Since, σ_i ⊆ ∪ _i∈Iσ_i for i ∈ I, we have by the Theorem 3.1, ${\underline{S}}^{σ_{i}} \subseteq {\underline{S}}^{\cup_{i \in I} σ_{i}}$ for i ∈ I . This implies $\cup_{i \in I} {\underline{S}}^{σ_{i}} \subseteq {\underline{S}}^{\cup_{i \in I} σ_{i}} .$ This implies $\cup_{i \in I} {\underline{S}}^{σ_{i}} = \cup_{i \in I} σ_{i} \subseteq {\underline{S}}^{\cup_{i \in I} σ_{i}} .$ Therefore, ${\underline{S}}^{\cup_{i \in I} σ_{i}} = \cup_{i \in I} σ_{i} .$ Hence, $\cup_{i \in I} σ_{i} \in {\tilde{τ}}_{σ} .$ □

Theorem 4.2. If (S, A) is a soft reflexive relation on V, then ${\tilde{τ}}_{σ}^{'} = {σ \in BSS (V) :^{σ} \underline{S} = σ}$ $({\tilde{τ}}_{σ}^{'} = {σ \in BSS (V) :^{σ} \bar{S} = σ})$ is a bipolar soft topology over V with respect to foresets.

Proof. The proof is similar to the proof of the Theorem 4.1. □

5 Similarity relations associated with rough bipolar soft sets

In this section, a few similarity relations between two bipolar soft sets are defined based on their rough bipolar soft approximations and their properties are examined.

Definition 5.1. Suppose (S, A) is a soft reflexive relation on V and $σ_{1}, σ_{2} \in BSS (V) .$ Then we define the following notions.

• σ₁ ≃ _Aσ₂ if and only if ${\underline{S}}^{σ_{1}} = {\underline{S}}^{σ_{2}} .$

• σ₁_≂A σ₂ if and only if ${\bar{S}}^{σ_{1}} = {\bar{S}}^{σ_{2}} .$

• σ₁ ≈ _Aσ₂ if and only if ${\underline{S}}^{σ_{1}} = {\underline{S}}^{σ_{2}} and {\bar{S}}^{σ_{1}} = {\bar{S}}^{σ_{2}} .$

Definition 5.2. Suppose (S, A) is a soft reflexive relation on V and $σ_{1}, σ_{2} \in BSS (V) .$ Then we can define following notions.

• σ₁ ≃ _Fσ₂ if and only if

• σ₁_≂F σ₂ if and only if

• σ₁ ≈ _Fσ₂ if and only if

These soft binary relations are said to be the lower rough bipolar soft similarity relation, upper rough bipolar soft similarity relation and rough bipolar soft similarity relations concerning aftersets and foresets, respectively. Evidently, σ₁ and σ₂ are both lower and upper rough bipolar soft similar if and only if they are rough bipolar soft similar.

Proposition 5.1. The relation ≃_A, _≂A, ≈_A, ≃_F, _≂F and ≈_F are equivalence relations on V.

Proof. Straightforward. □

Theorem 5.1. Suppose (S, A) is a soft reflexive relation on V. Then the following assertions hold for all $σ_{i} = (ψ_{i}, ξ_{i} : A) \in BSS (V); i = 1, 2, 3, 4,$ with respect to aftersets:

1) σ₁_≂Aσ₂ if and only if σ₁_≂A (σ₁∪ _rσ₂) _≂Aσ₂ ;

2) σ₁_≂Aσ₂ and σ₃_≂Aσ₄ imply that (σ₁∪ _rσ₃) _≂A (σ₁ ∪ _rσ₄) ;

3) σ₁ ⊆ σ₂ and σ₂_≂AΘ imply that σ₁_≂AΘ ;

4) σ₁ ⊆ σ₂ and $σ_{1} ≂_{A} V$ imply that $σ_{2} ≂_{A} V .$

Proof. 1) Let σ₁_≂Aσ₂ . Then ${\bar{S}}^{σ_{1}} = {\bar{S}}^{σ_{2}} .$ From the Theorem 3.1(6), we have ${\bar{S}}^{σ_{1} \cup_{r} σ_{2}} = {\bar{S}}^{σ_{1}} \cup {\bar{S}}^{σ_{2}} = {\bar{S}}^{σ_{1}} = {\bar{S}}^{σ_{2}} .$ Thus, σ₁_≂A (σ₁ ∪ _rσ₂) _≂Aσ₂ .

Converse holds by the transitivity of the relation _≂A .

2) Let σ₁_≂Aσ₂ and σ₃_≂Aσ₄ . Then ${\bar{S}}^{σ_{1}} = {\bar{S}}^{σ_{2}}$ and ${\bar{S}}^{σ_{3}} = {\bar{S}}^{σ_{4}} .$ From the Theorem 3.1(6), we have ${\bar{S}}^{σ_{1} \cup_{r} σ_{3}} = {\bar{S}}^{σ_{1}} \cup {\bar{S}}^{σ_{3}} = {\bar{S}}^{σ_{2}} \cup {\bar{S}}^{σ_{4}} = {\bar{S}}^{σ_{2} \cup_{r} σ_{4}} .$ Thus, (σ₁ ∪ _rσ₃) _≂A (σ₂ ∪ _rσ₄) .

3) Given that, $σ_{2} ≂_{A} Θ \Rightarrow {\bar{S}}^{σ_{2}} = {\bar{S}}^{Θ} .$

Also, $σ_{1} \tilde{\subseteq} σ_{2} \Rightarrow {\bar{S}}^{σ_{1}} \subseteq {\bar{S}}^{σ_{2}} = {\bar{S}}^{Θ} .$ It follows that ${\bar{S}}^{σ_{1}} \subseteq {\bar{S}}^{Θ} .$ But ${\bar{S}}^{Θ} \subseteq {\bar{S}}^{σ_{1}}$ is always true, so ${\bar{S}}^{Θ} = {\bar{S}}^{σ_{1}} .$ This shows that σ₁≂Θ .

4) Given that, $σ_{1} ≂_{A} V \Rightarrow {\bar{S}}^{σ_{1}} = {\bar{S}}^{V} .$

Also, $σ_{1} \tilde{\subseteq} σ_{2} \Rightarrow {\bar{S}}^{σ_{1}} \subseteq {\bar{S}}^{σ_{2}} \Rightarrow {\bar{S}}^{V} = {\bar{S}}^{σ_{1}} \subseteq {\bar{S}}^{σ_{2}} .$ It follows that ${\bar{S}}^{V} \subseteq {\bar{S}}^{σ_{2}} .$ But ${\bar{S}}^{σ_{2}} \subseteq {\bar{S}}^{V}$ is always true, so ${\bar{S}}^{σ_{2}} = {\bar{S}}^{V} .$ This shows that $σ_{2} ≂_{A} V .$ □

Theorem 5.2. Suppose (S, A) is a soft reflexive relation on V. Then the following assertions hold for all $σ_{i} = (ψ_{i}, ξ_{i} : A) \in BSS (V); i = 1, 2, 3, 4,$ with respect to foresets:

1) σ₁_≂Fσ₂ if and only if σ₁_≂F (σ₁∪ _rσ₂) _≂Fσ₂ ;

2) σ₁_≂Fσ₂ and σ₃_≂Aσ₄ imply that (σ₁∪ _rσ₃) _≂F (σ₁ ∪ _rσ₄) ;

3) σ₁ ⊆ σ₂ and σ₂_≂FΘ imply that σ₁_≂FΘ ;

4) σ₁ ⊆ σ₂ and $σ_{1} ≂_{F} V$ imply that $σ_{2} ≂_{F} V .$

Proof. The proof is similar to the proof of the Theorem 5.1. □

Theorem 5.3 Suppose (S, A) is a soft reflexive relation on V. Then the following assertions hold for all $σ_{i} = (ψ_{i}, ξ_{i} : A) \in BSS (V); i = 1, 2, 3, 4,$ with respect to aftersets:

1) σ₁ ≃ _Aσ₂ if and only if σ₁≃ _A (σ₁ ∩ _rσ₂) ≃ _Aσ₂ ;

2) σ₁ ≃ _Aσ₂ and σ₃ ≃ _Aσ₄ imply that (σ₁∩ _rσ₃) ≃ _A (σ₁ ∩ _rσ₄) ;

3) σ₁ ⊆ σ₂ and σ₂ ≃ _AΘ imply that σ₁≃ _AΘ ;

4) σ₁ ⊆ σ₂ and $σ_{1} ≃_{A} V$ imply that $σ_{2} ≃_{A} V .$

Proof. 1) Let σ₁ ≃ _Aσ₂ . Then ${\underline{S}}^{σ_{1}} = {\underline{S}}^{σ_{2}} .$ From the Theorem 3.1(3), we have ${\underline{S}}^{σ_{1} \cap_{r} σ_{2}} = {\underline{S}}^{σ_{1}} \cap {\underline{S}}^{σ_{2}} = {\underline{S}}^{σ_{1}} = {\underline{S}}^{σ_{2}} .$ Thus, σ₁ ≃ _A (σ₁ ∩ _rσ₂) ≃ _Aσ₂ .

Converse hold by the transitivity of the relation ≃_A .

2) σ₁ ≃ _Aσ₂ and σ₃ ≃ _Aσ₄ . Then ${\underline{S}}^{σ_{1}} = {\underline{S}}^{σ_{2}}$ and ${\underline{S}}^{σ_{3}} = {\underline{S}}^{σ_{4}} .$ From the Theorem 3.1(3), we have ${\underline{S}}^{σ_{1} \cap_{r} σ_{3}} = {\underline{S}}^{σ_{1}} \cap {\underline{S}}^{σ_{3}} = {\underline{S}}^{σ_{2}} \cap {\underline{S}}^{σ_{4}} = {\underline{S}}^{σ_{2} \cap_{r} σ_{4}} .$ Thus, (σ₁ ∩ _rσ₃) ≃ _A (σ₂ ∩ _rσ₄) .

3) Given that, $σ_{2} ≃_{A} Θ \Rightarrow {\underline{S}}^{σ_{2}} = {\underline{S}}^{Θ} .$

Also, $σ_{1} \subseteq σ_{2} \Rightarrow {\underline{S}}^{σ_{1}} \subseteq {\underline{S}}^{σ_{2}} = {\underline{S}}^{Θ} .$ It follows that ${\underline{S}}^{σ_{1}} \subseteq {\underline{S}}^{Θ} .$ But ${\underline{S}}^{Θ} \subseteq {\underline{S}}^{σ_{1}}$ is always true so ${\underline{S}}^{Θ} = {\underline{S}}^{σ_{1}} .$ This shows that σ₁ ≃ Θ .

4) Given that, $σ_{1} ≃_{A} V \Rightarrow {\underline{S}}^{σ_{1}} = {\underline{S}}^{V} .$

Also, $σ_{1} \subseteq σ_{2} \Rightarrow {\underline{S}}^{σ_{1}} \subseteq {\underline{S}}^{σ_{2}} \Rightarrow {\underline{S}}^{V} = {\underline{S}}^{σ_{1}} \subseteq {\underline{S}}^{σ_{2}} .$ It follows that ${\underline{S}}^{V} \subseteq {\underline{S}}^{σ_{2}} .$ But ${\bar{S}}^{σ_{2}} \subseteq {\underline{S}}^{V}$ is always true so ${\underline{S}}^{σ_{2}} = {\underline{S}}^{V} .$ This shows that $σ_{2} ≃_{A} V .$ □

Theorem 5.4. Suppose (S, A) is a soft reflexive relation on V. Then the following assertions hold for all $σ_{i} = (ψ_{i}, ξ_{i} : A) \in BSS (V); i = 1, 2,$ with respect to foresets:

1) σ₁ ≃ _Fσ₂ if and only if σ₁≃ _A (σ₁ ∩ _rσ₂) ≃ _Fσ₂ ;

2) σ₁ ≃ _Fσ₂ and σ₃ ≃ _Aσ₄ imply that (σ₁∩ _rσ₃) ≃ _F (σ₁ ∩ _rσ₄) ;

3) σ₁ ⊆ σ₂ and σ₂ ≃ _FΘ imply that σ₁≃ _FΘ ;

4) σ₁ ⊆ σ₂ and $σ_{1} ≃_{F} V$ imply that $σ_{2} ≃_{F} V .$

Proof. The proof is similar to the proof of the Theorem 5.3. □

Theorem 5.5. Suppose (S, A) is a soft reflexive relation on V. Then the following assertions hold for all $σ_{i} = (ψ_{i}, ξ_{i} : A) \in BSS (V); i = 1, 2, 3,$ with respect to aftersets:

1) σ₁ ≈ _Aσ₂ if and only if σ₁_≂A (σ₁ ∪ _rσ₂) _≂Aσ₂ and σ₁≃ _A (σ₁ ∩ _rσ₂) ≃ _Aσ₂ ;

2) σ₁ ⊆ σ₂ and σ₂ ≈ _AΘ imply that σ₁≈ _AΘ ;

3) σ₁ ⊆ σ₂ and $σ_{1} \approx_{A} V$ imply that $σ_{2} \approx_{A} V .$

Proof. The proof is similar to the proof of Theorems 5.1, and 5.3. □

Theorem 5.6. Suppose (S, A) is a soft reflexive relation on V. Then the following assertion hold for all $σ_{i} = (ψ_{i}, ξ_{i} : A) \in BSS (V); i = 1, 2, 3,$ with respect to aftersets:

1) σ₁ ≈ _Aσ₂ if and only if σ₁_≂F (σ₁ ∪ _rσ₂) _≂Fσ₂ and σ₁≃ _F (σ₁ ∩ _rσ₂) ≃ _Fσ₂ ;

2) σ₁ ⊆ σ₂ and σ₂ ≈ _FΘ imply that σ₁≈ _FΘ ;

3) σ₁ ⊆ σ₂ and $σ_{1} \approx_{F} V$ imply that $σ_{2} \approx_{F} V .$

Proof. The proof is similar to the proof of Theorems 5.2, and 5.4.□

6 Application in decision-making problems

In this section, we will present the decision-making methods with soft binary relations based on rough bipolar soft set theory. Decision-making is a significant field to be conferred in almost all types of data analysis. We propose an algorithm for the most favourable object in V by applying the concept of rough bipolar soft approximations of bipolar soft sets. Let V = {v_j : 1 ≤ j ≤ n} and E = {e_i : 1 ≤ i ≤ m} be the collection of objects and parameters, respectively.

We obtain two bipolar soft sets ${\underline{S}}^{σ}$ and ${\bar{S}}^{σ}$ which are very close with respect to aftersets by the soft lower and upper approximation of the bipolar soft set σ . Therefore, choice value α_i is redefined with respect to aftersets as follows: $α_{i} = \sum_{j = 1}^{n} {\underline{d}}_{ij} + \sum_{j = 1}^{n} {\bar{d}}_{ij}$ In decision-making problem, the maximum choice value α_i is the optimum decision for the object v_i ∈ V and the minimum choice value α_i is the worst decision for the object v_i ∈ V .

Algorithm 6.1. Here we present an algorithm for the approach to a decision-making problem with respect to aftersets. The decision algorithm is as follows:

1) Input the lower soft approximation ${\underline{S}}^{σ}$ and upper soft approximation ${\bar{S}}^{σ}$ of a bipolar soft set σ with respect to aftersets.

2) Corresponding to each v_i ∈ V, we calculate ${\underline{d}}_{ij}$ which is 1 if $v_{i} \in {\underline{S}}^{ψ} (e_{j}),$ -1 if $v_{i} \in {\underline{S}}^{ξ} (\neg e_{j})$ and 0 otherwise. Similarly, we calculate ${\bar{d}}_{ij}$ which is 1 if $v_{i} \in {\bar{S}}^{ψ} (e_{j}),$ -1 if $v_{i} \in {\bar{S}}^{ξ} (\neg e_{j})$ and 0 otherwise.

3) Compute the choice value $α_{i} = \sum_{j = 1}^{n} {\underline{d}}_{ij} + \sum_{j = 1}^{n} {\bar{d}}_{ij}$ with respect to aftersets.

4) The best decision is v_k ∈ V if $α_{k} = \max_{i} α_{i}, i = 1, 2, 3, . . ., | V | .$

5) The worst decision is v_k ∈ V if $α_{k} = \min_{i} α_{i}, i = 1, 2, 3, . . ., | V | .$

6) If k has more than one values, then we can equally choose anyone of v_k.

Algorithm 6.2. Here we present an algorithm for the approach to a decision-making problem with respect to foresets. The decision algorithm is as follows:

1) Input the lower soft approximation $^{σ} \underline{S}$ and upper soft approximation $^{σ} \bar{S}$ of a bipolar soft set σ with respect to foresets.

2) Corresponding to each w_i ∈ W, we calculate ${\underline{d^{'}}}_{ij}$ which is 1 if $w_{i} \in^{ψ} \underline{S} (e_{j}),$ -1 if $w_{i} \in^{ξ} \underline{S} (\neg e_{j})$ and 0 otherwise. Similarly, we calculate ${\bar{d^{'}}}_{ij}$ which is 1 if $w_{i} \in^{ψ} \bar{S} (e_{j}),$ -1 if $w_{i} \in^{ξ} \bar{S} (\neg e_{j})$ and 0 otherwise.

3) Compute the choice value $α_{i}^{'} = \sum_{j = 1}^{n} {\underline{d^{'}}}_{ij} + \sum_{j = 1}^{n} {\bar{d^{'}}}_{ij}$ with respect to foresets.

4) The best decision is w_k ∈ W if $α_{k}^{'} = \max_{i} α_{i}^{'}, i = 1, 2, 3, . . ., | W | .$

5) The worst decision is w_k ∈ W if $α_{k}^{'} = \min_{i} α_{i}^{'}, i = 1, 2, 3, . . ., | W | .$

6) If k has more than one values, then we can equally choose anyone of w_k.

To get a decision under the complete picture with foresets and aftersets, it is recommended that both algorithms should be used to make a final decision. The final score will be determined by taking the sum of scores obtained through each algorithm and will be significant for a final choice.

6.1 An application of the decision-making approach

By an example in this subsection, an application of decision-making approach is given.

Example 6.1 Assume that Mr. Z likes to purchase a car for his personal usage. Let V = {the set of all colours of cars} = {c₁, c₂, c₃, c₄, c₅} and W = {the set of all models of cars} = {m₁, m₂, m₃, m₄, m₅, m₆} and A = {e₁, e₂, e₃} = {the set of showrooms in the city } be the set of attributes.

Define $S : A \to P (V \times W)$ by

S (e₁) = {(c₁, m₁) , (c₁, m₂) , (c₂, m₃) , (c₂, m₄) ,

(c₄, m₂) , (c₄, m₄) , (c₄, m₆) , (c₅, m₁) , (c₅, m₅)} ,

S (e₂) = {(c₁, m₄) , (c₁, m₆) , (c₃, m₂) , (c₃, m₄) ,

(c₄, m₆) , (c₅, m₁) , (c₅, m₃) , (c₅, m₆)} and

S (e₃) = {(c₁, m₂) , (c₂, m₂) , (c₂, m₆) , (c₃, m₁) ,

(c₃, m₂) , (c₃, m₆) , (c₄, m₁) , (c₄, m₂) , (c₄, m₄) ,

(c₄, m₆)} where S (e_i) shows the relation between colours and models of cars in colour c_i available on showroom e_i for 1 ≤ i ≤ 3 . Then,

c₁S (e₁) = {m₁, m₂} , c₂S (e₁) = {m₃, m₄} , c₃S (e₁) = φ, c₄S (e₁) = {m₂, m₄, m₆} , c₅S (e₁) = {m₁, m₅} and

c₁S (e₂) = {m₄, m₆} , c₂S (e₂) = φ, c₃S (e₂) = {m₂, m₄} , c₄S (e₂) = {m₆} , c₅S (e₂) = {m₁, m₃, m₆} and

c₁S (e₃) = {m₂} , c₂S (e₃) = {m₂, m₆} , c₃S (e₃) = {m₁, m₂, m₆} , c₄S (e₃) = {m₁, m₂, m₄, m₆} , c₅S (e₃) = φ .

where c_iS (e_j) shows the models of all cars of colour c_i available on showroom e_j .

And

S (e₁) m₁ = {c₁, c₅} , S (e₁) m₂ = {c₁, c₄} , S (e₁) m₃ = {c₂} , S (e₁) m₄ = {c₂} , S (e₁) m₅ = {c₅} , S (e₁) m₆ = {c₄} , and

S (e₂) m₁ = {c₅} , S (e₂) m₂ = {c₃} , S (e₂) m₃ = {c₅} , S (e₂) m₄ = {c₁, c₃} , S (e₂) m₅ = φ, S (e₂) m₆ = {c₁, c₄, c₅} , and

S (e₃) m₁ = {c₃} , S (e₃) m₂ = {c₁, c₂, c₃, c₄} , S (e₃) m₃ = φ, S (e₃) m₄ = {c₄} , S (e₃) m₅ = φ, S (e₃) m₆ = {c₂, c₃, c₄} .

where S (e_j) m_i shows the colour c_i of all cars of model m_i available on showroom e_j .

Let $σ = (ψ, ξ : A) \in BSS (W),$ where ψ and ξ are given by:

$ψ : A \to P (W),$ which shows the liking of the models given by Mr. Z such that

ψ (e₁) = {m₁, m₂, m₅} , ψ (e₂) = {m₂, m₄} and ψ (e₃) = {m₁, m₂, m₆} .

And $ξ : \neg A \to P (W),$ which shows the disliking of the models given by Mr. Z such that

ξ (¬ e₁) = {m₃, m₆} , ξ (¬ e₂) = {m₁, m₃, m₆} and ξ (¬ e₃) = {m₃, m₄, m₅} .

Consider the Table 1 which is obtained after applying the Algorithm 6.1.

Table 1
The results of decision algorithm with respect to aftersets

c ₁ c ₂ c ₃ c ₄ c ₅

${\underline{d}}_{i 1}$ 1 -1 0 -1 1

${\bar{d}}_{i 1}$ 1 0 0 1 1

${\underline{d}}_{i 2}$ -1 0 1 -1 -1

${\bar{d}}_{i 2}$ 1 0 1 0 -1

${\underline{d}}_{i 3}$ 1 1 1 -1 0

${\bar{d}}_{i 3}$ 1 1 1 1 0

Choice value α_i 4 1 4 -1 0

	c ₁	c ₂	c ₃	c ₄	c ₅
${\underline{d}}_{i 1}$	1	-1	0	-1	1
${\bar{d}}_{i 1}$	1	0	0	1	1
${\underline{d}}_{i 2}$	-1	0	1	-1	-1
${\bar{d}}_{i 2}$	1	0	1	0	-1
${\underline{d}}_{i 3}$	1	1	1	-1	0
${\bar{d}}_{i 3}$	1	1	1	1	0
Choice value α_i	4	1	4	-1	0

Here the choice value $α_{i} = \sum_{j = 1}^{3} {\underline{d}}_{ij} + \sum_{j = 1}^{3} {\bar{d}}_{ij}$ is calculated with respect to aftersets. The car of colour c₁ and c₃ scores the maximum choice value α_k = 4 = α₁ = α₃, and decision is in favour of the car of colour c₁ and c₃ for selection. Moreover, the cars of colour c₄ is totally ignored. Hence Mr. Z will choose the car c₁ and c₃ for his personal use.

Also, let $σ = (ψ, ξ : A) \in BSS (V),$ where ψ and ξ are given by:

$ψ : A \to P (V),$ which shows the liking of the colours give by Mr. Z such that

ψ (e₁) = {c₁, c₂, c₅} , ψ (e₂) = {c₂, c₄} and ψ (e₃) = {c₁, c₅} .

And $ξ : \neg A \to P (V),$ which shows the disliking of the colours give by Mr. Z such that

ξ (¬ e₁) = {c₃, c₄} , ξ (¬ e₂) = {c₁, c₃} and ξ (¬ e₃) = {c₂, c₃, c₄} .

Consider the Table 2 which is obtained after applying the Algorithm 6.2.

Table 2

The results of decision algorithm with respect to foresets

	m ₁	m ₂	m ₃	m ₄	m ₅	m ₆
${\underline{d}}_{i 1}^{'}$	1	-1	1	0	1	-1
${\bar{d}}_{i 1}^{'}$	1	1	1	1	1	-1
${\underline{d}}_{i 2}^{'}$	0	-1	0	-1	0	-1
${\bar{d}}_{i 2}^{'}$	0	-1	0	-1	0	1
${\underline{d}}_{i 3}^{'}$	-1	-1	0	-1	0	-1
${\bar{d}}_{i 3}^{'}$	-1	1	0	-1	0	-1
Choice value $α_{i}^{'}$	0	-2	2	-3	2	-4

Here the choice value $α_{i}^{'} = \sum_{j = 1}^{3} {\underline{d}}_{ij}^{'} + \sum_{j = 1}^{3} {\bar{d}}_{ij}^{'}$ is calculated with respect to foresets. The car of model m₃ and m₅ scores the maximum choice value $α_{k}^{'} = 2 = α_{3}^{'} = α_{5}^{'}$ and decision is in the favour of car of models m₃ and m₅ for selection. Moreover the cars of model m₆ are totally ignored. Now, for getting a complete picture of the most likely decision we aggregate the approximations obtained from both algorithms for the chosen value of each pair as $S (c_{i}, m_{j}) = α_{i} + α_{j}^{'}$ and get the following table of scores:

Table 3

The Aggregated Result

$S$	m ₁	m ₂	m ₃	m ₄	m ₅	m ₆
c ₁	4	2	6	1	6	0
c ₂	1	-1	3	-2	3	-3
c ₃	4	2	6	1	6	0
c ₄	-1	-3	1	-4	1	-5
c ₅	0	-2	2	-3	2	-4

As we can see from the Table 3 that the best scores are of pairs (c₁, m₃), (c₁, m₅), (c₃, m₃) and (c₃, m₅) but none is available in any of the showrooms in the city. So next best choices are pairs (c₁, m₁), (c₃, m₁) which are available in showrooms e₁ and e₃ respectively. Mr. Z can choose car model m₁ in either color c₁ or c₃ from the respective showroom.

7 Conclusions

In this analysis, we applied Pawlak’s idea of rough sets to the bipolar soft sets and introduced the rough bipolar soft sets by describing a rough approximation of a bipolar soft set in a generalized soft approximation space. This research could be seen as an extension of [18]. We studied their structural properties and explored how the soft binary relation affects the rough approximations of a bipolar soft set. Furthermore two kinds of bipolar soft topologies induced by soft binary relation are studied. Finally, an application of the rough bipolar soft sets presented in a decision-making problem, and an algorithm proposed for that application. In future, we would like to extend this study in more general frameworks e.g. with T-soft equality relation [5] and Partial belong relation [3 , 9] within the scenario of bipolar soft sets. A general topological sense as in more recent literature by Göçür [11, 12] and Al-shami [6] may also be applied to study roughness in bipolar soft sets. We can also apply these methods to two-sided matching decision -making problems or group decision-making following [33 –35]. This work provides a theoretical framework for a new idea of Bipolar soft rough sets. In future, we aim at working out comparative analysis of our proposed decision-making technique with existing methods.

Footnotes

Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions, which greatly improved the quality of this article.

References

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computers and Mathematics with Applications 57(9) (2009), 1547–1553.

Ali

M.I.

, A note on soft sets, rough soft sets and fuzzy soft sets, Applied Soft Computing 11(4) (2011), 3329–3332.

Al-shami

T.M.

, Partial soft separation axioms and soft compact spaces, Filomat 32(13) (2018), 4755–4771.

Al-shami

T.M.

and El-Shafei

M.E.

, Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone, Soft Computing 24 (2020), 5377–5387.

Al-shami

T.M.

and El-Shafei

M.E.

, T-soft equality relation, Turkish Journal of Mathematics 44(4) (2020), 1427–1441.

Al-shami

T.M.

, Bipolar soft sets: relations between them and ordinary points and their applications, Complexity Article ID 6621854, (2021). https://doi.org/10.1155/2021/6621854.

Aygün

and Kamací

, Some Generalized Operations in Soft Set Theory and Their Role in Similarity and Decision Making, Journal of Intelligent and Fuzzy Systems 36(6) (2019), 6537–6547.

Çağman

, Karataş

and Enginoğlu

, Soft topology, Computers and Mathematics with Applications 62 (2011), 351–358.

El-Shafei

M.E.

and Al-shami

T.M.

, Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem, Computational and Applied Mathematics 39 (2020). doi.org/10.1007/s40314-020-01161-3.

10.

Feng

, Ali

M.I.

and Shabir

, Soft relations applied to semigroups, Filomat 27(7) (2013), 1183–1196.

11.

Göçür

, Monad Metrizable Space, Mathematics 8(11) (2020), 1891. https://doi.org/10.3390/math8111891

12.

Göçür

, Amply soft set and its topologies: AS and PAS topologies, AIMS Mathematics 6(4) (2021), 3121–3141.

13.

Enginoğlu

, Çağman

, Karataş

and Aydín

, On Soft Topology, El-Cezeri Journal of Science and Engineering 2(3) (2015), 23–38.

14.

Hazra

, Majumdar

and Samanta

S.K.

, Soft Topology, Fuzzy Information and Engineering 4 (2012), 105–115.

15.

Kamací

, Selectivity analysis of parameters in soft set and its effect on decision-Making, International Journal of Machine Learning and Cybernetics 11(2) (2020), 313–324.

16.

Kamací

and Petchimuthu

, Bipolar N-soft set theory with applications, Soft Computing 24(22) (2020), 16727–16743.

17.

Kanwal

R.S.

and Shabir

, Approximation of ideals in semigroups by soft relations, Journal of Intelligent and Fuzzy Systems (Preprint) (2018), 1–14.

18.

Kanwal

R.S.

and Shabir

, Approximation of soft set by soft relations, (submitted).

19.

Karaaslan

, Bipolar Soft Rough Relations, Communications de la Faculté,des Sciences de ĺ Université d’;Ankara. Séries A1. Mathematics and Statistic 65(1) (2016), 105–126.

20.

Karaaslan

and Çağman

, Bipolar soft rough sets and their applications in decision-making, Afrika Matematika 29 (2018), 823–839.

21.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision-making problem, Computers and Mathematics with Applications 44(8-9) (2002), 1077–1083.

22.

Maji

P.K.

, Biswas

and Roy

, Soft set theory, Computers and Mathematics with Applications 45(4-5) (2003), 555–562.

23.

Malik

and Shabir

, Rough fuzzy Rough bipolar soft sets and application in decision-making problems, Soft Computing 23(5) (2019), 1603–1614.

24.

Molodtsov

, Soft set theory–first results, Computers and Mathematics with Applications 37(4-5) (1999), 19–31.

25.

Naz

and Shabir

, On fuzzy bipolar soft sets, their algebraic structures and applications, Journal of Intelligent and Fuzzy Systems 26(4) (2014), 1645–1656.

26.

Pawlak

, Rough sets, International Journal of Computer and Information Sciences 11(5) (1982), 341–356.

27.

Shabir

and Naz

, On soft topological spaces, Computers and Mathematics with Applications 61 (2011), 1786–1799.

28.

Shabir

and Naz

, On bipolar soft sets, (2013). arXiv preprint arXiv:1303.1344.

29.

Shabir

and Bakhtawar

, Bipolar soft connected, bipolar soft disconnected and bipolar soft compact spaces, Songklanakarin Journal of Science and Technology 39(3) (2017), 359–371.

30.

Shabir

, Kanwal

R.S.

and Ali

M.I.

, Reduction of an information system, Soft Computing 24 (2019), 10801–10813.

31.

Shabir

and Gul

, Modified rough bipolar soft sets, Journal of Intelligent and Fuzzy Systems 39(3) (2020), 4259–4283.

32.

Zhang

and Yuan

, Soft Relation and Fuzzy Soft Relation, Fuzzy Information and Engineering and Operations Research and Management, Advances in Intelligent Systems and Computing 211 (2014), Springer, Berlin, Heidelberg.

33.

Zhang

, Kou

, Yu

and Gao

, Consistency improvement for fuzzy preference relations with self-confidence: An application in two-sided matching decision-Making, Journal of the Operational Research Society (2020). https://doi.org/10.1080/01605682.2020.1748529

34.

Zhang

, Gao

and Li

, Consensus reaching for social network group decision-making by considering leadership and bounded confidence, Knowledge-Based Systems, 204, (2020). https://doi.org/10.1016/j.knosys.2020.106240

35.

Zhang

, Gao

and Yu

, Two-sided matching decision-making with multi-granular hesitant fuzzy linguistic term sets and incomplete criteria weight information, Expert Systems with Applications (2021), 168. https://doi.org/10.1016/j.eswa.2020.114311

Rough approximations of bipolar soft sets by soft relations and their application in decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Rough Set

3 Rough Bipolar soft sets

4 Bipolar soft topologies induced by soft reflexive relations

5 Similarity relations associated with rough bipolar soft sets

6 Application in decision-making problems

6.1 An application of the decision-making approach

Table 1 The results of decision algorithm with respect to aftersets c 1 c 2 c 3 c 4 c 5 d _ i 1 1 -1 0 -1 1 d ¯ i 1 1 0 0 1 1 d _ i 2 -1 0 1 -1 -1 d ¯ i 2 1 0 1 0 -1 d _ i 3 1 1 1 -1 0 d ¯ i 3 1 1 1 1 0 Choice value α i 4 1 4 -1 0

Footnotes

Acknowledgment

References