Modified rough bipolar soft sets

Abstract

Bipolar soft sets and rough sets are two different techniques to cope with uncertainty. A possible fusion of rough sets and bipolar soft sets is proposed by Karaaslan and Çağman. They introduced the notion of bipolar soft rough set. In this article, a new technique is being introduced to study roughness through bipolar soft sets. In this new technique of finding approximations of a set, flavour of both theories of bipolar soft set and rough set is retained. We call this new hybrid model modified rough bipolar soft set MRBS-set. Moreover, accuracy measure and roughness measure of modified rough bipolar soft sets are defined in MRBS-approximation space and its application in multi-criteria group decision making is presented.

Keywords

Rough set soft set bipolar soft set bipolar soft P-rough set modified rough bipolar soft set MRBS-approximations

1 Introduction

The classical (crisp) theory of sets is basic and also an essential mathematical tool. It deals with mathematical models for the class of problems that address exactness, precision, perfection, specificity and certainty. Characteristically, classical set theory is extensional.

Mostly, the real life problems inherently involves uncertainties, imprecision, inconsistency and ambiguity. Especially, such collections of problems arise in Economics, Ecology, Engineering, Environmental Sciences, Medical Sciences, Social Sciences and numerous different fields, which are highly dependent on the target of modelling uncertainties that can’t be solved using classical or current mathematical theories.

With the passage of time, numerous researchers, mathematicians and scientists are attempting to determine some appropriate tools and a number of mathematical theories to cope with these uncertainties, such as Probability Theory, Fuzzy Set Theory (Zadeh, 1965) [42], Rough Set Theory (Pawlak, 1982) [29], Interval Mathematical Theory (Gorzalzany, 1987), Vague Set Theory (Gau and Buchrer, 1993), Graph Theory, Automata Theory (Doostfatemeh and Kremer 2005; Li and Wang 2014), Decision-Making Theory (Roy and Maji 2007; Cagman et al. 2010; Feng et al. 2010) etc., are formulated to solve such problems, and have been found only partially successful. These theories reduced the distance between the classical mathematical designs and the vague real-world data.

However, all these theories have their inherent difficulties, perhaps because of inadequacy of the parameterization tools of the theories as mentioned in [26].

The fuzzy set theory [42] is a significant and successful mathematical tool which is found most suitable for managing uncertainties. However, it is short of providing a mechanism on how to fix a membership function, because the nature of the membership function is extremely individualistic. There are various types of fuzzy set extensions in the fuzzy set theory, for example, intuitionistic fuzzy sets, vague sets, interval-valued fuzzy sets and plenty of more.

Bipolar fuzzy set [45] is another extension of fuzzy set, given by Zhang in 1994. The membership degree of bipolar fuzzy sets is enlarged from the interval [0, 1] to [-1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1, 0) of an element indicates that the element somewhat satisfies the implicit counter-property.

In 1999, Molodtsov [26] proposed the notion of soft sets that can be seen as a completely new Mathematical tool for modelling uncertainties and vagueness; where a soft set is associated with adequate set of parameters and thus free from difficulties aforementioned. Soft sets are intended to capture and to defuse the conflicts among existing fuzzy set theories. Soft set theory is a consistent and unified theory implied implicitly by existing fuzzy theories. Thus soft set theory is a generalization of fuzzy set theory, that was proposed to deal with uncertainty in a non-parametric manner.

Unlike classical mathematics, where exact solution of a mathematical model is required, soft set theory instead requires an approximate description of an object as its initial point. The choice of adequate parameterization tools such as words, real number, functions etc., make soft set theory very convenient and easy to apply in practice. Applications of soft set theory can be seen in [2 , 27].

Recently, many researchers are engaged in the properties and applications of soft set theory. Maji et al. [21] defined new operations on soft sets and worked on theoretical study of soft sets in detail. Ali et al. [5] studied some new operations in soft set theory. Maji et al. [22] and Ali and Shabir [6] proposed the notion of fuzzy soft set. Shabir and Naz [35] introduced the concept of bipolar soft sets.

Fuzzy soft set theory [22] is very much interesting and useful for solving the day to day problems. It helps to take decision making in a critical situation. Applications of fuzzy soft set theory can also be seen in [8 , 15]. While the fuzzy bipolar soft sets [23] have ability to handle the uncertainty, as well as bipolarity of the information in many situations. Mardani et al. [24] reviewed fuzzy multicriteria decision making methods based on fuzzy set theory from 1994 to 2014. Regarding decision making methods based on rough set theory, many researchers put forward new procedures and techniques (for examples see [36, 37]). Zhan and Alcantud [44] put forward a multicriteria group decision making based on soft rough covering.

Abdullah et al. [1] introduced the notion of bipolar fuzzy soft sets by combining the soft sets and the bipolar fuzzy sets. Concept of bipolar soft set and its operations such as union, intersection and complement were first defined by Shabir and Naz [35]. Karaaslan and Karataş [16] redefined bipolar soft sets with a new approximation providing opportunity to study on topological structures of bipolar soft sets. Also Naz and Shabir [28] proposed the concept of fuzzy bipolar soft sets and investigated their algebraic structures.

Soft set theory [26] and rough set theory [29] are treated as successful mathematical tools to cope with uncertainty. Feng et al. [9] developed a link among these two theories and initiated the idea of new hybrid model of soft rough set, which could provide better approximations than Pawlak’s rough sets in some cases ([9], Example 4.7). In this model, they described the parametrize subset on the universe. As a result, a few strange situations have happened, like:

Upper approximation of a non-empty set may be empty.

Upper approximation of a subset X of the universe may not contain the set X.

These situations does not occur in Pawlak’s rough set theory. In order to get rid of these troublesome situations, Shabir et al. [33] redefined a version of soft rough sets known as modified soft rough sets (MRS-sets).

Karaaslan and Çağman [18] initiated the notion of bipolar soft rough set which is fusion of rough set and bipolar soft set, and discussed its applications in decision making. Structure of a bipolar soft set (f, g : A) consists of two maps f : A ⟶ P (U) and g : ⌉ A ⟶ P (U), which are known as positive soft set and negative soft set, respectively. Now we can see that soft set [26] is identical to positive soft set of a bipolar soft set (f, g : A). So same sort of strange situations also happened in bipolar soft rough sets [18]. Therefore concept of “Modified Rough Bipolar Soft sets” (MRBS-sets) have been presented in this article to get rid of these shortcomings.

The rest of this article is organized as follows. Section 2 is devoted to recall some basic and fundamental properties related to rough sets, soft sets, bipolar soft sets and bipolar soft rough sets. In section 3 we introduce the idea of modified rough bipolar soft sets (MRBS-sets). The notion is further explored by studying its structural properties. In section 4, based upon these properties, we define measure of accuracy and measure of roughness for modified rough bipolar soft set. In section 5, we propose a multi-criteria group decision making method based on modified rough bipolar soft sets to select optimal element among the alternatives. We then give an example of proposed decision making method to show that the method can be successfully applied to some real life problems. In section 6, we gave a comparative analysis of bipolar soft rough sets and MRBS-sets. At last, section 7 carries a few conclusions and presents some topics for future research.

2 Preliminaries

This section offers the essential definitions and basic results required in upcoming sections of the study. Throughout this paper, we will use U for an initial universe, E for set of parameters, A for a finite non-empty subset of the parameters set E and P (U) for the power set of U, unless stated otherwise.

Definition 2.1. [29] Let U be a non-empty finite universe, and R be an equivalence relation on U. Then the pair P = (U, R) is called a Pawlak approximation space.

If X ⊆ U, then X may or may not be written as a union of some equivalence classes of U. If X may be written as a union of some equivalence classes, then X is said to be R-definable; otherwise it is called R-undefinable. If X is R-undefinable then it can be approximated with the help of the following two definable (crisp) subsets:

$R_{*} (X) = {x \in U ∣ [x]_{R} \subseteq X},$ (2.1) $R^{*} (X) = {x \in U ∣ [x]_{R} \cap X \neq \emptyset} .$ (2.2)

Equations (2.1) and (2.2) are called lower and upper approximations of X with respect to the Pawlak approximation space P = (U, R) respectively, where the equivalence class [x] _R of an element x ∈ U is the set consists of all objects y ∈ U such that (x, y) ∈ R, that is, $[x]_{R} = {y \in U ∣ (x, y) \in R} .$

Moreover, the sets

$P {os}_{R} (X) = R_{*} (X),$

$B {nd}_{R} (X) = R^{*} (X) - R_{*} (X),$

$N {eg}_{R} (X) = (R^{*} (X))^{c},$

${\bar{E dg}}_{R} (X) = R^{*} (X) - X,$

${\underline{E dg}}_{R} (X) = X - R_{*} (X)$

are called as the R-positive region (R⁺ - region), R-boundary region (RB_nd - region), R-negative region (R^- - region), R- external edge (RE_ex) and R-internal edge (RE_in) of X, respectively, where (R^* (X)) ^c = U - R^* (X) .

Thus set X is said to be crisp (exact or definable with respect to R) if and only if R_* (X) = R^* (X); equivalently $B {nd}_{R} (X) = \emptyset$ . And set X is said to be rough (inexact or undefinable with respect to R) if and only if R_* (X) ≠ R^* (X); equivalently $B {nd}_{R} (X) \neq \emptyset$ .

Note that sometimes the pair (R_* (X), R^* (X)) ∈P (U) × P (U) is referred to the rough approximation of X with respect to R.

Definition 2.2. [26] Let U be a set of objects called the universe, A be a non-empty set of parameters (attributes or properties). Then a pair (f, A) is called a soft set over U, where f is a set- valued mapping given by f : A ⟶ P (U).

Therefore, a soft set over U gives a parameterized family of subsets of the universe U. For e ∈ A, f (e) may be considered as the set of e-approximate elements of U by the soft set (f, A). Thus $(f, A) = {f (e) \in P (U) ∣ e \in A \subseteq E} .$ Clearly, a soft set is not a classical set.

From now onwards, set of all soft sets over U will be represented by $SS (U)$ .

Example 2.3. Let U = {v₁, v₂, v₃, v₄, v₅} be the set of vehicles under consideration and A = {e₁, e₂, e₃} be the set of parameters of different features in vehicles, where the parameters e_i (i = 1, 2, 3) stands for “good”, “cheap” and “modern” respectively. If a mapping f : A ⟶ P (U) is given as; a set of vehicles {v₂, v₃, v₄} is good, a set of vehicles {v₁, v₃, v₄} is cheap, a set of vehicles {v₂, v₃, v₄, v₅} is modern, then the soft set (f, A) can be written as:

$f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {v_{2}, v_{3}, v_{4}} & if e = e_{1}, \\ {v_{1}, v_{3}, v_{4}} & if e = e_{2}, \\ {v_{2}, v_{3}, v_{4}, v_{5}} & if e = e_{3} . \end{matrix}$

Definition 2.4. [21] Let A be a set of parameters. Then, NOT set of parameters of A, denoted by ⌉A, and is defined by ⌉A = {¬ a : a ∈ A} where ¬a = notafor a ∈ A.

Example 2.5. If A = {e₁ = good, e₂ = cheap, e₃ = modern}, then the NOT set of A can be written as: $\begin{matrix} ⌉ A = {\neg e_{1} = notgood, \neg e_{2} = notcheap, \\ \neg e_{3} = notmodern} . \end{matrix}$

Definition 2.6. [35] A triplet (f, g : A) is said to be a bipolar soft set over the universe U, in which f and g are mappings, given by f : A ⟶ P (U) and g : ⌉ A ⟶ P (U) such that f (e)∩ g (¬ e) = ∅ for all e ∈ A.

Thus, a bipolar soft set over U gives two parameterized families of subsets of the universe U and the condition f (e)∩ g (¬ e) = ∅ for all e ∈ A, ¬e ∈ ⌉ A, is imposed as a consistency constraint. For each e ∈ A, ¬e ∈ ⌉ A, f (e) and g (¬ e) are seemed as the set of e-approximate elements and set of ¬e-approximate elements of bipolar soft set (f, g : A).

It is also found that the relationship between a complement function and the defining function of a soft set (f, A) becomes a special case for the defining functions of a bipolar soft set, that is, (f, f^c : A) is a bipolar soft set over U. The difference occurs due to the presence of uncertainty or hesitation or lack of knowledge in defining the membership function. We call this uncertainty as the approximation for the degree of hesitation. Thus the union of three approximations, that is, e-approximation, ¬e-approximation and approximation of hesitation is U.

Also, we observe that ∅ ⊆ U - {f (e) ∪ g (¬ e)} ⊆ U, for all, e ∈ A, ¬e ∈ ⌉ A. So, we may approximate the degree of hesitation in bipolar soft set (f, g : A) by an allied soft set (h, A) defined over U, where h (e) = U - {f (e) ∪ g (¬ e)}, for all e ∈ A, ¬e ∈ ⌉ A.

From now onwards, set of all bipolar soft sets over U will be represented by $BSS (U)$ .

Tabular representation of Bipolar Soft Sets. A bipolar soft set (f, g : A) over the universe U can be represented through a pair of binary tables, one for each of the function f and g, respectively. In both tables, rows are categorized via the objects of U and columns are categorized by means of parameters. We use the following keys for tables of f and g, respectively: $a_{ij} = {\begin{matrix} 1 & if x_{i} \in f (e_{j}) \\ 0 & if x_{i} \notin f (e_{j}) \end{matrix}$

$b_{ij} = {\begin{matrix} 1 & if x_{i} \in g (\neg e_{j}) \\ 0 & if x_{i} \notin g (\neg e_{j}), \end{matrix}$ where a_ij and b_ij are the ith entry of jth column of each table, respectively. We elaborate this concept in Example 2.17.

Definition 2.7. [18] Let $(f, g : A) \in BSS (U)$ . Then P = (U, (f, g : A)) is called a bipolar soft approximation space ( $BSA$ -space).

Definition 2.8. [18] Let P = (U, (f, g : A)) be a bipolar soft approximation space. Then the soft approximation spaces represented by P⁺ = (U, f) and P^- = (U, g) are said to be positive soft approximation space and negative soft approximation space of bipolar soft set (f, g : A), respectively.

Definition 2.9. [18] Let (f, g : A) be a bipolar soft set over the universe U. Then the mappings f : A ⟶ P (U) and g : ⌉ A ⟶ P (U) are said to be positive soft set and negative soft set of (f, g : A), respectively.

From now onwards, the complement of X in U will be denoted by X^c.

Definition 2.10. [18] Let $(f, g : A) \in BSS (U)$ , P = (U, (f, g : A)) be a $BSA$ -space and X ⊆ U. Based on P, the following four operators are defined: $\begin{matrix} {\underline{S}}_{P^{+}} (X) & = {u \in U : \exists e \in A, [u \in f (e) \subseteq X]}, \\ {\underline{S}}_{P^{-}} (X) & = {u \in U : \exists \neg e \in ⌉ A, [u \in g (\neg e), \\ g (\neg e) \cap X^{c} \neq \emptyset]}, \\ {\bar{S}}_{P^{+}} (X) & = {u \in U : \exists e \in A, [u \in f (e), f (e) \cap X \neq \emptyset]}, \\ {\bar{S}}_{P^{-}} (X) & = {u \in U : \exists \neg e \in ⌉ A, [u \in g (\neg e) \subseteq X^{c}]} \end{matrix}$ -6ptwhich are called soft P-lower positive approximation (SPL⁺ - approximation), soft P-lower negative approximation (SPL^- - approximation), soft P-upper positive approximation (SPU⁺ - approximation) and soft P-upper negative approximation (SPU^- - approximation) of X, respectively.

Remark 2.11.

In above definition, for X ⊆ U, we immediately have that ${\underline{S}}_{P^{+}} (X) \subseteq X$ and ${\bar{S}}_{P^{-}} (X) \subseteq X^{c}$ . However, ${\underline{S}}_{P^{-}} (X) \subseteq X^{c}$ and ${\bar{S}}_{P^{+}} (X) \subseteq X$ do not hold in general. Also, note that, it need not be ${\underline{S}}_{P^{+}} (X) \cap {\underline{S}}_{P^{-}} (X) = \emptyset$ .

In Definition (2.10), we see that ${\underline{S}}_{P^{+}} (X)$ and ${\bar{S}}_{P^{+}} (X)$ coincides with the Definition (3.1) given in [23]. That is, ${\underline{S}}_{P^{+}} (X) = {\underline{S}}_{P} (X)$ and ${\bar{S}}_{P^{+}} (X) = {\bar{S}}_{P} (X)$ .

Example 2.12. To illustrate above remark, let $(f, g : A) \in BSS (U)$ , where U = {c₁, c₂, c₃, c₄, c₅}, A = {e₁, e₂, e₃, e₄} the set of parameters and ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} the NOT set of parameters of A. The mappings f and g are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {c_{2}, c_{3}} & if e = e_{1}, \\ {c_{2}, c_{5}} & if e = e_{2}, \\ {c_{3}} & if e = e_{3}, \\ {c_{2}, c_{3}, c_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {c_{4}, c_{5}} & if \neg e = \neg e_{1}, \\ {c_{3}, c_{4}} & if \neg e = \neg e_{2}, \\ {c_{2}, c_{4}} & if \neg e = \neg e_{3}, \\ {c_{4}} & if \neg e = \neg e_{4} . \end{matrix}$

For X = {c₁, c₂, c₃} ⊆ U, X^c = {c₄, c₅} we have ${\underline{S}}_{P^{+}} (X) = {c_{2}, c_{3}}$ , ${\underline{S}}_{P^{-}} (X) = {c_{2}, c_{3}, c_{4}, c_{5}}$ , ${\bar{S}}_{P^{+}} (X) = {c_{2}, c_{3}, c_{5}}$ and ${\bar{S}}_{P^{-}} (X) = {c_{4}, c_{5}}$ .

Clearly it can be seen that ${\underline{S}}_{P^{+}} (X) \subseteq X$ and ${\bar{S}}_{P^{-}} (X) \subseteq X^{c}$ . But ${\underline{S}}_{P^{-}} (X) ⊈ X^{c}$ and ${\bar{S}}_{P^{+}} (X)$ ⊈X. Also ${\underline{S}}_{P^{+}} (X) \cap {\underline{S}}_{P^{-}} (X) = {c_{2}, c_{3}} \cap {c_{2}, c_{3},$ c₄, c₅} = {c₂, c₃}≠ ∅.

Definition 2.13. [18] Let ${\underline{S}}_{P^{+}} (X)$ , ${\underline{S}}_{P^{-}} (X)$ , ${\bar{S}}_{P^{+}} (X)$ and ${\bar{S}}_{P^{-}} (X)$ be SPL⁺ - approximation, SPL^- - approximation, SPU⁺ - approximation and SPU^- - approximation of X, respectively. Then, the two pairs given as: $\begin{matrix} {\underline{BS}}_{P} (X) & = ({\underline{S}}_{P^{+}} (X), {\underline{S}}_{P^{-}} (X)), \\ {\bar{BS}}_{P} (X) & = ({\bar{S}}_{P^{+}} (X), {\bar{S}}_{P^{-}} (X)) \end{matrix}$ are called bipolar soft rough approximations of X with respect to $BSA$ -space P = (U, (f, g : A)). In general, ${\underline{BS}}_{P} (X)$ and ${\bar{BS}}_{P} (X)$ are called bipolar soft P-lower approximation and bipolar soft P-upper approximation of X (with respect to P), respectively.

Definition 2.14. [18] Let ${\underline{BS}}_{P} (X)$ be the bipolar soft P-lower approximation and ${\bar{BS}}_{P} (X)$ be the bipolar soft P-upper approximation of X ⊆ U. Then,

$BP {os}_{P} (X) = ({\underline{S}}_{P^{+}} (X), {\bar{S}}_{P^{-}} (X)),$

$BB {nd}_{P} (X) = {\bar{BS}}_{P} (X) - {\underline{BS}}_{P} (X) = ({\bar{S}}_{P^{+}} (X) \ {\underline{S}}_{P^{+}} (X), {\underline{S}}_{P^{-}} (X) \ {\bar{S}}_{P^{-}} (X)),$

$BN {eg}_{P} (X) = U - {\bar{BS}}_{P} (X) = (({\bar{S}}_{P^{+}} (X))^{c}$ , $({\underline{S}}_{P^{-}} (X))^{c}),$

are called the bipolar soft P-positive region (BSP⁺ - region), bipolar soft P-boundary region (BSPB_nd - region) and bipolar soft P-negative region (BSP^- - region), respectively.

Definition 2.15. [18] Let $(f, g : A) \in BSS (U)$ , P = (U, (f, g : A)) be a $BSA$ -space, ${\underline{BS}}_{P} (X)$ and ${\bar{BS}}_{P} (X)$ are bipolar soft rough approximations of X ⊆ U with respect to $BSA$ -space, respectively. If ${\underline{BS}}_{P} (X) \neq {\bar{BS}}_{P} (X)$ , then X is said to be bipolar soft P-rough set; otherwise X is called bipolar soft P-definable.

Example 2.16. Let U = {c₁, c₂, c₃, c₄, c₅} be the set of five cars under consideration and A = {e₁, e₂, e₃, e₄} = {cheap, comfort, high equipment, longtime warrant}. Then ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} = {expensive, uncomfortable, low equipment, not longtime warrant}. The bipolar soft set (f, g : A) describes the “requirements of the car” which Mr. X going to buy. Suppose that positive soft set and negative soft set of (f, g : A) are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {c_{2}, c_{3}} & if e = e_{1}, \\ {c_{2}, c_{5}} & if e = e_{2}, \\ {c_{3}} & if e = e_{3}, \\ {c_{2}, c_{3}, c_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {c_{4}, c_{5}} & if \neg e = \neg e_{1}, \\ {c_{3}, c_{4}} & if \neg e = \neg e_{2}, \\ {c_{2}, c_{4}} & if \neg e = \neg e_{3}, \\ {c_{4}} & if \neg e = \neg e_{4} . \end{matrix}$ For X = {c₁, c₂, c₃} ⊆ U, X^c = {c₄, c₅}. Now using Definition 2.10, we have

${\underline{S}}_{P^{+}} (X) = {c_{2}, c_{3}}$ and ${\underline{S}}_{P^{-}} (X) = {c_{2}, c_{3}, c_{4}, c_{5}}$ . Therefore ${\underline{BS}}_{P} (X) = ({\underline{S}}_{P^{+}} (X), {\underline{S}}_{P^{-}} (X)) = ({c_{2}, c_{3}}, {c_{2}, c_{3}, c_{4}, c_{5}}) .$ (2.3)

Similarly

${\bar{S}}_{P^{+}} (X) = {c_{2}, c_{3}, c_{5}}$ and ${\bar{S}}_{P^{-}} (X) = {c_{4}, c_{5}}$ . Therefore ${\bar{BS}}_{P} (X) = ({\bar{S}}_{P^{+}} (X), {\bar{S}}_{P^{-}} (X)) = ({c_{2}, c_{3}, c_{5}}, {c_{4}, c_{5}}) .$ (2.4) From Equations (2.3) and (2.4), we have ${\underline{BS}}_{P} (X) \neq {\bar{BS}}_{P} (X)$ and so X is a bipolar soft P-rough set.

Moreover, we can obtain BSP⁺ - region, BSPB_nd - region and BSP^- - region of X, respectively, as follows: $\begin{matrix} BP {os}_{P} (X) & = ({c_{2}, c_{3}}, {c_{4}, c_{5}}), \\ BB {nd}_{P} (X) & = ({c_{5}}, {c_{2}, c_{3}}), \\ BN {eg}_{P} (X) & = ({c_{1}, c_{4}}, {c_{1}}) . \end{matrix}$

Perhaps Definitions 2.10 does not fulfill the criteria of Pawlak’s rough sets. For example:

Upper approximation of a non-empty set may be empty.

Upper approximation of a subset of universe may not contain the set which does not occur in Pawlak’s rough set theory.

The following example explains the above observation:

Example 2.17. Consider the bipolar soft set (f, g : A) over the universe U, where U = {u₁, u₂, u₃, u₄, u₅, u₆}, A = {e₁, e₂, e₃, e₄} ⊆ E, then ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} . The tabular representation of bipolar soft set (f, g : A) is given in Tables (1. (a), 1. (b)) and the bipolar soft approximation space ( $BSA$ -space) is P = (U, (f, g : A)) .

From Table 1. (a) and 1. (b), it can be seen that u₄ does not have any of the mentioned properties in A as well as in ⌉A. For X = {u₃, u₄, u₅} ⊆ U, X^c = {u₁, u₂, u₆} . The soft P-lower positive, soft P-upper positive, soft P-lower negative and soft P-upper negative approximations of X are respectively given as: $\begin{matrix} {\underline{S}}_{P^{+}} (X) = {u_{3}}, {\bar{S}}_{P^{+}} (X) = {u_{1}, u_{2}, u_{3}, u_{5}}, \\ {\underline{S}}_{P^{-}} (X) = {u_{2}, u_{3}, u_{6}}, {\bar{S}}_{P^{-}} (X) = {} . \end{matrix}$ Thus the bipolar soft rough approximations of X are given as: ${\underline{BS}}_{P} (X) = ({u_{3}}, {u_{2}, u_{3}, u_{6}})$ ${\bar{BS}}_{P} (X) = ({u_{1}, u_{2}, u_{3}, u_{5}}, {}) .$ The object u₄ is not the member of any one of the soft P-lower positive approximation, soft P-upper positive approximation, soft P-lower negative approximation and soft P-upper negative approximation of X. But according to given information of X, u₄ should be a member of ${\bar{S}}_{P^{+}} (X)$ and ${\underline{S}}_{P^{-}} (X)$ . (Since, ${\underline{S}}_{P^{+}} (X) \subseteq X \subseteq {\bar{S}}_{P^{+}} (X)$ and ${\bar{S}}_{P^{-}} (X) \subseteq X \subseteq {\underline{S}}_{P^{-}} (X)$ ).

Table 1

Tabular representation of (f, g: A)

(a)
f	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
e ₁	1	0	0	0	0	1
e ₂	0	0	1	0	0	0
e ₃	0	0	0	0	0	0
e ₄	1	1	0	0	1	0
(b)
g	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
¬e₁	0	0	0	0	1	0
¬e₂	0	0	0	0	0	0
¬e₃	0	1	1	0	0	1
¬e₄	0	0	1	0	0	1

From Table 1. (a), it is clear that [u₁] = {u₁}, [u₂] = {u₂, u₅} = [u₅], [u₃] = {u₃}, [u₄] = {u₄} and [u₆] = {u₆}. Now according to Definition 2.1, we have R_* (X) = {u₃, u₄} and R^* (X) = {u₂, u₃, u₄, u₅}.

Also, u₁ is a member of ${\bar{S}}_{P^{+}} (X)$ . But there is no element in X which is equivalent to u₁, so its membership in ${\bar{S}}_{P^{+}} (X)$ is difficult to justify.

Similarly, from Table 1. (b), it is clear that [u₁] = [u₄] = {u₁, u₄}, [u₂] = {u₂}, [u₃] = [u₆] = {u₃, u₆}, and [u₅] = {u₅}. Now again according to Definition 2.1, we have R_* (X) = {u₅} and R^* (X) = {u₁, u₃, u₄, u₅, u₆}.

Also, we can see that u₂ is a member of ${\underline{S}}_{P^{-}} (X)$ . But there is no element in X which is equivalent to u₂, so its membership in ${\underline{S}}_{P^{-}} (X)$ is difficult to justify.

Another strange situation may happen, that is for a non-empty subset Y of U, ${\bar{S}}_{P^{+}} (Y)$ , ${\underline{S}}_{P^{+}} (Y)$ , ${\underline{S}}_{P^{-}} (Y)$ and ${\bar{S}}_{P^{-}} (Y)$ are empty sets. For this, let us assume ∅ ≠ Y = {u₄} ⊆ U. Then ${\underline{S}}_{P^{+}} (Y) = \emptyset = {\bar{S}}_{P^{+}} (Y)$ and ${\bar{S}}_{P^{-}} (X) = \emptyset = {\underline{S}}_{P^{-}} (X)$ . That is u₄ is an unfortunate element and will never be an element of ${\underline{S}}_{P^{+}} (X)$ , ${\bar{S}}_{P^{+}} (X)$ , ${\bar{S}}_{P^{-}} (X)$ or ${\underline{S}}_{P^{-}} (X)$ for any X ⊆ U.

In order to get rid of these strange situations, in [9] concept of full soft set was initiated, this is a strong condition. Moreover concepts of soft positive, soft negative and soft boundary regions of X are meaningful in the case of full soft sets. It is easy to see that the element u₄ in above example will be in negative region for any subset X of U.

Definition 2.18. Let (f, A) be a soft set over the universe U. Then (f, A) is said to be full soft set if $⋃_{e \in A} f (e) = U .$

Definition 2.19. A bipolar soft set (f, g : A) over the universe U is said to be full bipolar soft set if it satisfies the following two axioms:

⋃_e∈Af (e) = U

⋃_¬e∈⌉Ag (¬ e) = U.

Theorem 2.20. Let (f, g : A) be a bipolar soft set over the universe U, which is not full. Then ∃ at least one element u ∈ U such that $u \in BN {eg}_{P} (X)$ for all X ⊆ U, (where $BN {eg}_{P} (X)$ stands for bipolar soft P-boundary region).

Proof. Let (f, g : A) be a bipolar soft set over the universe U. Then P = (U, (f, g : A)) is $BSA$ -space. Since (f, g : A) is not full, therefore ⋃_e∈Af (e) or ⋃_¬e∈⌉Ag (¬ e) are properly contained in U. So ∃ some u ∈ U, such that u ∉ f (e) or u ∉ g (¬ e) for all e ∈ A, ¬e ∈ ⌉ A.

Now let u ∈ X, where X is a proper subset of the universe U. If possible let $u \in {\bar{S}}_{P^{+}} (X)$ , then by Definition (2.10) ∃ some e ∈ A such that u ∈ f (e) and f (e)∩ X ≠ ∅. Which is contradiction, since u ∉ f (e). Hence $u \notin {\bar{S}}_{P^{+}} (X)$ , and this implies that $u \in ({\bar{S}}_{P^{+}} (X))^{c}$ .

Also, if possible let $u \in {\underline{S}}_{P^{-}} (X)$ , then according to Definition (2.10) ∃ some ¬e ∈ ⌉ A such that u ∈ g (¬ e) and g (¬ e) ∩ X^c ≠ ∅. Which is contradiction, since u ∉ g (¬ e). Hence $u \notin {\underline{S}}_{P^{-}} (X)$ , and this implies that $u \in ({\underline{S}}_{P^{-}} (X))^{c}$ .

By a similar argument, we are able to show that when u ∉ X then $u \in ({\bar{S}}_{P^{+}} (X))^{c}$ and $u \in ({\underline{S}}_{P^{-}} (X))^{c}$ . Therefore $\begin{matrix} u \in (({\bar{S}}_{P^{+}} (X))^{c}, ({\underline{S}}_{P^{-}} (X))^{c}) = U - {\bar{BS}}_{P} (X) \\ = BN {eg}_{P} (X) for all X \subseteq U . □ \end{matrix}$

Corollary 2.21. If (f, g : A) is a full bipolar soft set over the universe U, then ∃ at least one element u ∈ U such that $u \notin BN {eg}_{P} (X)$ for all X ⊆ U .

Example 2.22. To illustrate the above theorem, we revisit Example 2.16 where U = {c₁, c₂, c₃, c₄, c₅} is the set of five cars under consideration and A = {e₁, e₂, e₃, e₄} = {cheap, comfort, high equipment, longtime warrant}, then ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} = {expensive, uncomfortable, low equipment, not longtime warrant}. The bipolar soft set (f, g : A) describes the “requirements of the car” which Mr. X going to buy. Suppose that positive soft set and negative soft set of (f, g : A) are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {c_{2}, c_{3}} & if e = e_{1}, \\ {c_{2}, c_{5}} & if e = e_{2}, \\ {c_{3}} & if e = e_{3}, \\ {c_{2}, c_{3}, c_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {c_{4}, c_{5}} & if \neg e = \neg e_{1}, \\ {c_{3}, c_{4}} & if \neg e = \neg e_{2}, \\ {c_{2}, c_{4}} & if \neg e = \neg e_{3}, \\ {c_{4}} & if \neg e = \neg e_{4} . \end{matrix}$ Since $\begin{matrix} ⋃_{e \in A} f (e) & = {c_{2}, c_{3}} \cup {c_{2}, c_{5}} \cup {c_{3}} \cup {c_{2}, c_{3}, c_{5}} \\ = {c_{2}, c_{3}, c_{5}} \neq U \end{matrix}$ and $\begin{matrix} ⋃_{\neg e \in ⌉ A} g (\neg e) & = {c_{4}, c_{5}} \cup {c_{3}, c_{4}} \cup {c_{2}, c_{4}} \cup {c_{4}} \\ = {c_{2}, c_{3}, c_{4}, c_{5}} \neq U . \end{matrix}$ Therefore (f, g : A) is not full bipolar soft set.

For X = {c₁, c₂, c₃} ⊆ U, X^c = {c₄, c₅}. Now using Definition 2.10, we have

${\bar{S}}_{P^{+}} (X) = {c_{2}, c_{3}, c_{5}}$ and ${\bar{S}}_{P^{-}} (X) = {c_{4}, c_{5}}$ . Therefore ${\bar{BS}}_{P} (X) = ({\bar{S}}_{P^{+}} (X), {\bar{S}}_{P^{-}} (X)) = ({c_{2}, c_{3}, c_{5}}, {c_{4}, c_{5}}) .$ (2.6) From Equations (2.5) and (2.6), we have ${\underline{BS}}_{P} (X) \neq {\bar{BS}}_{P} (X)$ and so X is a bipolar soft P-rough set.

Moreover, we can obtain $BN {eg}_{P} (X)$ as follows: $BN {eg}_{P} (X) = ({c_{1}, c_{4}}, {c_{1}}) .$ (2.7) From Equation (2.7), we can easily see that c₁ ∉ ⋃ _e∈Af (e) and c₁ ∉ ⋃ _¬e∈⌉Ag (¬ e) but $c_{1} \in BN {eg}_{P} (X) .$

In the following example we interpret Theorem 2.19 in case of full bipolar soft set.

Example 2.23. Let U = {c₁, c₂, c₃, c₄, c₅} be the set of five cars under consideration and A = {e₁, e₂, e₃, e₄} = {cheap, comfort, high equipment, longtime warrant}. Then ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} = {expensive, uncomfortable, low equipment, not longtime warrant}. The bipolar soft set (f, g : A) describes the “requirements of the car” which Mr. X going to buy. Suppose that positive soft set and negative soft set of (f, g : A) are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {c_{1}, c_{2}, c_{3}} & if e = e_{1}, \\ {c_{2}, c_{5}} & if e = e_{2}, \\ {c_{4}} & if e = e_{3}, \\ {c_{2}, c_{3}, c_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {c_{4}, c_{5}} & if \neg e = \neg e_{1}, \\ {c_{1}} & if \neg e = \neg e_{2}, \\ {c_{2}, c_{3}} & if \neg e = \neg e_{3}, \\ {c_{4}} & if \neg e = \neg e_{4} . \end{matrix}$ Since $\begin{matrix} ⋃_{e \in A} f (e) & = {c_{1}, c_{2}, c_{3}} \cup {c_{2}, c_{5}} \cup {c_{4}} \cup {c_{2}, c_{3}, c_{5}} \\ = {c_{1}, c_{2}, c_{3}, c_{4}, c_{5}} = U \end{matrix}$ and $\begin{matrix} ⋃_{\neg e \in ⌉ A} g (\neg e) & = {c_{4}, c_{5}} \cup {c_{1}} \cup {c_{2}, c_{3}} \cup {c_{3}} \\ = {c_{1}, c_{2}, c_{3}, c_{4}, c_{5}} = U . \end{matrix}$ Therefore (f, g : A) is a full bipolar soft set.

For X = {c₁, c₂, c₃} ⊆ U, X^c = {c₄, c₅}. Now using Definition 2.10, to calculate ${\underline{S}}_{P^{+}} (X)$ , ${\underline{S}}_{P^{-}} (X)$ , ${\bar{S}}_{P^{+}} (X)$ and ${\bar{S}}_{P^{-}} (X)$ we have

${\underline{S}}_{P^{+}} (X) = {c_{1}, c_{2}, c_{3}}$ and ${\underline{S}}_{P^{-}} (X) = {c_{4}, c_{5}}$ . Therefore ${\underline{BS}}_{P} (X) = ({\underline{S}}_{P^{+}} (X), {\underline{S}}_{P^{-}} (X)) = ({c_{1}, c_{2}, c_{3}}, {c_{4}, c_{5}}) .$ (2.8) Similarly,

${\bar{S}}_{P^{+}} (X) = {c_{1}, c_{2}, c_{3}, c_{5}}$ and ${\bar{S}}_{P^{-}} (X) = {c_{4}, c_{5}}$ . Therefore ${\bar{BS}}_{P} (X) = ({\bar{S}}_{P^{+}} (X), {\bar{S}}_{P^{-}} (X)) = ({c_{1}, c_{2}, c_{3}, c_{5}}, {c_{4}, c_{5}}) .$ (2.9) From Equations (2.8) and (2.9), we have ${\underline{BS}}_{P} (X) \neq {\bar{BS}}_{P} (X)$ and so X is a bipolar soft P-rough set.

Moreover, we can obtain $BN {eg}_{P} (X)$ as follows: $BN {eg}_{P} (X) = (({\bar{S}}_{P^{+}} (X))^{c}, ({\underline{S}}_{P^{-}} (X))^{c}) = ({c_{4}}, {c_{1}, c_{2}, c_{3}}) .$ (2.10) From Equation (2.10), we can easily see that c₅ ∈ ⋃ _e∈Af (e) and c₅ ∈ ⋃ _¬e∈⌉Ag (¬ e) but $c_{5} \notin BN {eg}_{P} (X) .$

3 Modified rough bipolar soft sets (MRBS-SETS)

In section 2, we have seen that by applying the definition of lower rough approximation and upper rough approximation for a subset X of the universe U, some basic properties of rough sets may vanish and unfortunate elements which are in bipolar soft P-negative region of X ⊆ U, cannot be avoided. In this section, we strengthen the concept of bipolar soft rough set by defining modified rough bipolar soft set.

Definition 3.1. Let (f, g : A) be a bipolar soft set over the universe U, where f and g are the maps f : A ⟶ P (U) and g : ⌉ A ⟶ P (U). Let us define two other maps φ and ψ as: $φ : U ⟶ P (A)$ $φ (x) = {e : x \in f (e)}$ and $\begin{matrix} ψ & : U ⟶ P (⌉ A) \\ ψ (x) & = {\neg e : x \in g (\neg e)} for all x \in U . \end{matrix}$ Then the pair P = (U, (φ, ψ)) is called MRBS - approximation space (modified rough bipolar soft approximation space).

For any X ⊆ U, the lower modified bipolar pair and the upper modified bipolar pair with respect to P are defined in the following manner, respectively: $\begin{matrix} {\underline{MBS}}_{P} (X) & = ({\underline{X}}_{φ^{+}}, {\underline{X}}_{ψ^{-}}), \\ {\bar{MBS}}_{P} (X) & = ({\bar{X}}_{φ^{+}}, {\bar{X}}_{ψ^{-}}), \end{matrix}$ where $\begin{matrix} {\underline{X}}_{φ^{+}} & = {x \in X : φ (x) \neq φ (y) for all y \in X^{c}}, \\ {\bar{X}}_{φ^{+}} & = {x \in U : φ (x) = φ (y) for some y \in X}, \\ {\underline{X}}_{ψ^{-}} & = {x \in U : ψ (x) = ψ (y) for some y \in X}, \\ {\bar{X}}_{ψ^{-}} & = {x \in X : ψ (x) \neq ψ (y) for all y \in X^{c}} . \end{matrix}$ Here X^c = U - X. Generally ${\underline{X}}_{φ^{+}}$ , ${\bar{X}}_{φ^{+}}$ , ${\underline{X}}_{ψ^{-}}$ and ${\bar{X}}_{ψ^{-}}$ will be called φ - lower positive, φ - upper positive, ψ - lower negative and ψ - upper negative MRBS - approximations of X ⊆ U, respectively.

Definition 3.2. Let ${\underline{MBS}}_{P} (X)$ and ${\bar{MBS}}_{P} (X)$ be lower modified bipolar pair and upper modified bipolar pair of X ⊆ U with respect P = (U, (φ, ψ)), respectively. Then X is said to be MRBS - set with respect to P, if ${\underline{MBS}}_{P} (X) \neq {\bar{MBS}}_{P} (X)$ ; otherwise X is said to be MRBS - definable.

Moreover,

$M {Pos}_{P} (X) = ({\underline{X}}_{φ^{+}}, {\bar{X}}_{ψ^{-}}),$

$M {Bnd}_{P} (X) = {\bar{MBS}}_{P} (X) - {\underline{MBS}}_{P} (X) = ({\bar{X}}_{φ^{+}} \ {\underline{X}}_{φ^{+}}, {\underline{X}}_{ψ^{-}} \ {\bar{X}}_{ψ^{-}}),$

$M {Neg}_{P} (X) = U - {\bar{MBS}}_{P} (X) = (({\bar{X}}_{φ^{+}})^{c}, ({\underline{X}}_{ψ^{-}})^{c}),$

are called the modified bipolar soft P-positive region (MBSP⁺ - region), modified bipolar soft P-boundary region (MBSPB_nd - region) and modified bipolar soft P-negative region (MBSP^- - region), respectively.

Corollary 3.3. From above definition, we immediately have that X ⊆ U is a MRBS - definable set if and only if $M {Bnd}_{P} (X) = (\emptyset, \emptyset) .$

Example 3.4. As an illustration, let $(f, g : A) \in BSS (U)$ , where U = {u₁, u₂, u₃, u₄, u₅, u₆}, A = {e₁, e₂, e₃, e₄} ⊆ E and ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} . The mappings f and g are given as follows: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {u_{1}, u_{6}} & if e = e_{1}, \\ {u_{3}} & if e = e_{2}, \\ {} & if e = e_{3}, \\ {u_{1}, u_{2}, u_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {u_{3}} & if \neg e = \neg e_{1}, \\ {u_{5}} & if \neg e = \neg e_{2}, \\ {u_{2}, u_{3}, u_{6}} & if \neg e = \neg e_{3}, \\ {u_{3}} & if \neg e = \neg e_{4}, \end{matrix}$

From above mappings, we can get the tabular representation of the bipolar soft set (f, g : A) which is given in Tables (2. (a), 2. (b)) as:

For X = {u₃, u₄, u₅} ⊆ U, X^c = {u₁, u₂, u₆} . By using Definition 2.10, we have that:

${\underline{S}}_{P^{+}} (X) = {u_{3}}$ , ${\bar{S}}_{P^{+}} (X) = {u_{1}, u_{2}, u_{3}, u_{5}},$ ${\underline{S}}_{P^{-}} (X) = {u_{2}, u_{3}, u_{6}}$ , ${\bar{S}}_{P^{-}} (X) = {} .$

Now, let P = (U, (φ, ψ)) be the MRBS-approximation space. Then with the help of Table 2. (a) and 2. (b), we can define the two mappings φ and ψ as follows: $φ : U ⟶ P (A),$ $u \mapsto {\begin{matrix} {e_{1}, e_{4}} & if u = u_{1}, \\ {e_{4}} & if u = u_{2}, \\ {e_{2}} & if u = u_{3}, \\ {} & if u = u_{4}, \\ {e_{4}} & if u = u_{5}, \\ {e_{1}} & if u = u_{6}, \end{matrix}$ and $ψ : U ⟶ P (⌉ A),$ $u \mapsto {\begin{matrix} {} & if u = u_{1}, \\ {\neg e_{3}} & if u = u_{2}, \\ {\neg e_{1}, \neg e_{3}, \neg e_{4}} & if u = u_{3}, \\ {} & if u = u_{4}, \\ {\neg e_{2}} & if u = u_{5}, \\ {\neg e_{3}} & if u = u_{6} . \end{matrix}$ Using Definition 3.1, we find the modified rough bipolar soft approximations of X = {u₃, u₄, u₅} ⊆ U with respect to MBSR-space, $\begin{matrix} {\underline{X}}_{φ^{+}} & = {u_{3}, u_{4}}, \\ {\bar{X}}_{φ^{+}} & = {u_{2}, u_{3}, u_{4}, u_{5}}, \\ {\underline{X}}_{ψ^{-}} & = {u_{1}, u_{3}, u_{4}, u_{5}}, \\ {\bar{X}}_{ψ^{-}} & = {u_{3}, u_{5}} . \end{matrix}$ Thus, $\begin{matrix} {\underline{MBS}}_{P} (X) & = ({u_{3}, u_{4}}, {u_{1}, u_{3}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{P} (X) & = ({u_{2}, u_{3}, u_{4}, u_{5}}, {u_{3}, u_{5}}) . \end{matrix}$ Consequently, X is MRBS - set, since ${\underline{MBS}}_{P} (X) \neq {\bar{MBS}}_{P} (X) .$

Table 2
Tabular representation of (f, g: A)

(a)

f u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

e ₁ 1 0 0 0 0 1

e ₂ 0 0 1 0 0 0

e ₃ 0 0 0 0 0 0

e ₄ 1 1 0 0 1 0

(b)

g u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

¬e₁ 0 0 1 0 0 0

¬e₂ 0 0 0 0 1 0

¬e₃ 0 1 1 0 0 1

¬e₄ 0 0 1 0 0 0

(a)
e ₁	1	0	0	0	0	1
e ₂	0	0	1	0	0	0
e ₃	0	0	0	0	0	0
e ₄	1	1	0	0	1	0
(b)
g	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
¬e₁	0	0	1	0	0	0
¬e₂	0	0	0	0	1	0
¬e₃	0	1	1	0	0	1
¬e₄	0	0	1	0	0	0

Moreover, by direct computations we can obtain MBSP⁺ - region, MBSPB_nd - region and MBSP^- - region of X, respectively, as follows: $\begin{matrix} M {Pos}_{P} (X) & = ({u_{3}, u_{4}}, {u_{3}, u_{5}}), \\ M {Bnd}_{P} (X) & = ({u_{2}, u_{5}}, {u_{1}, u_{4}}), \\ M {Neg}_{P} (X) & = ({u_{1}, u_{6}}, {u_{2}, u_{6}}) . \end{matrix}$

Remark 3.5. The relation between ${\underline{MBS}}_{P} (X)$ and ${\bar{MBS}}_{P} (X)$ is ${\underline{X}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}}$ and ${\underline{X}}_{ψ^{-}} \supseteq {\bar{X}}_{ψ^{-}}$ .

Definition 3.6. Let (f, g : A) ∈ BSS(U), such that P = (U, (φ, ψ)) be a MBSR-approximation space and X, Y ⊆ U. Then,

${\underline{MBS}}_{P} (X) ⊑ {\underline{MBS}}_{P} (Y) \Leftrightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}} and {\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}}$ ,

${\bar{MBS}}_{P} (X) ⊑ {\bar{MBS}}_{P} (Y) \Leftrightarrow {\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}} and {\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}}$ .

Example 3.7. Consider the bipolar soft set (f, g : A) over the universe U in Example 3.4, where U = {u₁, u₂, u₃, u₄, u₅, u₆}. Let X, Y ⊆ U be such that X = {u₁, u₂} and Y = {u₁, u₂, u₄}. Clearly X ⊆ Y. Now using Definition 3.1, the modified rough bipolar soft approximations of X, Y ⊆ U with respect to MRBS-approximation space are $\begin{matrix} {\underline{X}}_{φ^{+}} & = {u_{1}}, \\ {\bar{X}}_{φ^{+}} & = {u_{1}, u_{2}, u_{5}}, \\ {\underline{X}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}, u_{6}}, \\ {\bar{X}}_{ψ^{-}} & = {}, \end{matrix}$ and $\begin{matrix} {\underline{Y}}_{φ^{+}} & = {u_{1}, u_{4}}, \\ {\bar{Y}}_{φ^{+}} & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ {\underline{Y}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}, u_{6}}, \\ {\bar{Y}}_{ψ^{-}} & = {u_{1}, u_{4}} . \end{matrix}$ Clearly $\begin{matrix} {u_{1}} & \subseteq {u_{1}, u_{4}} \Rightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}}, \\ {u_{1}, u_{2}, u_{4}, u_{6}} & \subseteq {u_{1}, u_{2}, u_{4}, u_{6}} \Rightarrow {\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}}, \\ {u_{1}, u_{2}, u_{5}} & \subseteq {u_{1}, u_{2}, u_{4}, u_{5}} \Rightarrow {\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}}, \\ {} & \subseteq {u_{1}, u_{4}} \Rightarrow {\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}} . \end{matrix}$ So, ${\underline{MBS}}_{P} (X) ⊑ {\underline{MBS}}_{P} (Y)$ and ${\bar{MBS}}_{P} (X) ⊑ {\bar{MBS}}_{P} (Y)$ .

Remark 3.8. It is easy to see that:

There is no link of containment exists in case of ${\bar{S}}_{P^{+}} (X)$ and ${\bar{X}}_{φ^{+}}$ .

There is no link of containment exists in case of ${\underline{S}}_{P^{-}} (X)$ and ${\underline{X}}_{ψ^{-}}$ .

There is no link of containment exists in case of ${\underline{S}}_{P^{-}} (X)$ and ${\bar{X}}_{ψ^{-}}$ .

However ${\underline{S}}_{P^{+}} (X) \subseteq {\underline{X}}_{φ^{+}}$ .

To illustrate above remark, we revisit the Example 3.3, where U = {u₁, u₂, u₃, u₄, u₅, u₆}. Then for X = {u₃, u₄, u₅} ⊆ U, we have

${\underline{S}}_{P^{+}} (X) = {u_{3}}$ , ${\bar{S}}_{P^{+}} (X) = {u_{1}, u_{2}, u_{3}, u_{5}}$ and ${\underline{S}}_{P^{-}} (X) = {u_{2}, u_{3}, u_{6}}$ , ${\bar{S}}_{P^{-}} (X) = {} .$

Also we have,

${\underline{X}}_{φ^{+}} = {u_{3}, u_{4}},$ ${\bar{X}}_{φ^{+}} = {u_{2}, u_{3}, u_{4}, u_{5}}$ and ${\underline{X}}_{ψ^{-}} = {u_{1}, u_{3}, u_{4}, u_{5}},$ ${\bar{X}}_{ψ^{-}} = {u_{3}, u_{5}} .$

Then one can easily see that,

${\bar{S}}_{P^{+}} (X) ⊈ {\bar{X}}_{φ^{+}}$ or ${\bar{X}}_{φ^{+}} ⊈ {\bar{S}}_{P^{+}} (X) .$

Similarly,

${\underline{S}}_{P^{-}} (X) ⊈ {\underline{X}}_{ψ^{-}}$ or ${\underline{X}}_{ψ^{-}} ⊈ {\underline{S}}_{P^{-}} (X) .$

And also,

${\underline{S}}_{P^{-}} (X) ⊈ {\bar{X}}_{ψ^{-}}$ or ${\bar{X}}_{ψ^{-}} ⊈ {\underline{S}}_{P^{-}} (X) .$

Moreover, ${\bar{X}}_{ψ^{-}} = {u_{3}, u_{5}} \supseteq {u_{3}} = {\underline{S}}_{P^{+}} (X) .$

Proposition 3.9. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space and ${\underline{MBS}}_{P} (X) = ({\underline{X}}_{φ^{+}}, {\underline{X}}_{ψ^{-}}),$ ${\bar{MBS}}_{P} (X) = ({\bar{X}}_{φ^{+}}, {\bar{X}}_{ψ^{-}})$ be lower modified bipolar pair and upper modified bipolar pair, respectively. Then for any X, Y ⊆ U, we have

$X \subseteq Y \Rightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}},$

$X \subseteq Y \Rightarrow {\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}},$

$X \subseteq Y \Rightarrow {\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}},$

$X \subseteq Y \Rightarrow {\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}} .$

Proof.

Assume that X ⊆ Y and $x \in {\underline{X}}_{φ^{+}}$ . Then by Definition 3.1, we have φ (x) ≠ φ (y) for all y ∈ X^c. Since X ⊆ Y, so Y^c ⊆ X^c. Thus in particular, φ (x) ≠ φ (z) for all z ∈ Y^c. As x ∈ X and X ⊆ Y so x ∈ Y. Thus $x \in {\underline{Y}}_{φ^{+}}$ and hence we obtain ${\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}} .$

Assume that X ⊆ Y and $y \in {\bar{X}}_{φ^{+}}$ . Then by Definition 3.1, we have φ (y) = φ (z) for some z ∈ X. Since X ⊆ Y, so φ (y) = φ (z) for some z ∈ Y. Thus $x \in {\bar{Y}}_{φ^{+}}$ and hence ${\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}} .$

Assume that X ⊆ Y and $x \in {\underline{X}}_{ψ^{-}}$ . Then by Definition 3.1, we have ψ (x) = ψ (y) for some y ∈ X. Since X ⊆ Y, so ψ (x) = ψ (y) for some y ∈ Y. Thus $x \in {\underline{Y}}_{ψ^{-}}$ and hence ${\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}} .$

Suppose that X ⊆ Y and $x \in {\bar{X}}_{ψ^{-}}$ . Then by Definition 3.1, we have ψ (x) ≠ ψ (y) for all y ∈ X^c. As X ⊆ Y, so Y^c ⊆ X^c. So in particular, ψ (x) ≠ ψ (z) for all z ∈ Y^c. As x ∈ X and X ⊆ Y so x ∈ Y. Thus $x \in {\bar{Y}}_{ψ^{-}}$ and hence we get ${\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}} .$ □

Following is the example to illustrate the above proposition.

Example 3.10. Let $(f, g : A) \in BSS (U)$ , where U = {u₁, u₂, u₃, u₄, u₅, u₆}, A = {e₁, e₂, e₃, e₄} be the set of parameters and ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} be the NOT set of parameters of A. The mappings f and g are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {u_{1}, u_{6}} & if e = e_{1}, \\ {u_{3}} & if e = e_{2}, \\ {} & if e = e_{3}, \\ {u_{2}, u_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {u_{3}} & if \neg e = \neg e_{1}, \\ {u_{5}} & if \neg e = \neg e_{2}, \\ {u_{2}, u_{3}, u_{6}} & if \neg e = \neg e_{3}, \\ {u_{4}} & if \neg e = \neg e_{4} . \end{matrix}$ From above mappings, we can obtain the tabular representation of bipolar soft set (f, g : A) which is given in Table (3. (a), 3.(b)) as:

Now let P = (U, (φ, ψ)) be the MRBS-approximation space. Then with the help of Table 3. (a) and 3. (b), we can define the two mappings φ and ψ as follows: $φ : U ⟶ P (A),$ $u \mapsto {\begin{matrix} {e_{1}} & if u = u_{1}, \\ {e_{4}} & if u = u_{2}, \\ {e_{2}} & if u = u_{3}, \\ {} & if u = u_{4}, \\ {e_{4}} & if u = u_{5}, \\ {e_{1}} & if u = u_{6}, \end{matrix}$ and $ψ : U ⟶ P (⌉ A),$ $u \mapsto {\begin{matrix} {} & if u = u_{1}, \\ {\neg e_{3}} & if u = u_{2}, \\ {\neg e_{1}, \neg e_{3}} & if u = u_{3}, \\ {\neg e_{4}} & if u = u_{4}, \\ {\neg e_{2}} & if u = u_{5}, \\ {\neg e_{3}} & if u = u_{6} . \end{matrix}$ Now for X, Y ⊆ U such that X = {u₁, u₂} and Y = {u₁, u₂, u₄}. Then by direct computation we get $\begin{matrix} {\underline{X}}_{φ^{+}} & = {}, \\ {\bar{X}}_{φ^{+}} & = {u_{1}, u_{2}, u_{5}, u_{6}}, \\ {\underline{X}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{6}}, \\ {\bar{X}}_{ψ^{-}} & = {u_{1}} . \end{matrix}$ Similarly, $\begin{matrix} {\underline{Y}}_{φ^{+}} & = {u_{4}}, \\ {\bar{Y}}_{φ^{+}} & = {u_{1}, u_{2}, u_{4}, u_{5}, u_{6}}, \\ {\underline{Y}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}, u_{6}}, \\ {\bar{Y}}_{ψ^{-}} & = {u_{1}, u_{4}} . \end{matrix}$

Table 3

Tabular representation of (f, g: A)

(a)
f	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
e ₁	1	0	0	0	0	1
e ₂	0	0	1	0	0	0
e ₃	0	0	0	0	0	0
e ₄	0	1	0	0	1	0
(b)
g	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
¬e₁	0	0	1	0	0	0
¬e₂	0	0	0	0	1	0
¬e₃	0	1	1	0	0	1
¬e₄	0	0	0	1	0	0

Since X ⊆ Y, so one can easily see that

{} ⊆ {u₄} which shows that ${\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}}$ .

{u₁, u₂, u₅, u₆} ⊆ {u₁, u₂, u₄, u₅, u₆} which shows that ${\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}}$ .

{u₁, u₂, u₆} ⊆ {u₁, u₂, u₄, u₆} which shows that ${\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}}$ .

{u₁} ⊆ {u₁, u₄} which shows that ${\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}}$ .

Proposition 3.11. If φ (x) = φ (y) for some x, y ∈ U, then for any X ∈ P (U), either both $x, y \in {\bar{X}}_{φ^{+}}$ or $x, y \notin {\bar{X}}_{φ^{+}}$ .

Proof. Obvious.□

Proposition 3.12. If ψ (x) = ψ (y) for some x, y ∈ U, then for any X ∈ P (U), either both $x, y \in {\underline{X}}_{ψ^{-}}$ or $x, y \notin {\underline{X}}_{ψ^{-}}$ .

Proof. Obvious.□

Theorem 3.13. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space. Then for any X, Y ⊆ U, we have the following axioms:

${\underline{X}}_{φ^{+}} \subseteq X \subseteq {\bar{X}}_{φ^{+}},$

${\underline{\emptyset}}_{φ^{+}} = \emptyset = {\bar{\emptyset}}_{φ^{+}},$

${\underline{U}}_{φ^{+}} = U = {\bar{U}}_{φ^{+}},$

${\underline{(X \cap Y)}}_{φ^{+}} = {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}},$

${\bar{(X \cap Y)}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}},$

${\bar{(X \cup Y)}}_{φ^{+}} = {\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}},$

${\underline{(X \cup Y)}}_{φ^{+}} \supseteq {\underline{X}}_{φ^{+}} \cup {\underline{Y}}_{φ^{+}},$

${\bar{X}}_{φ^{+}} = ({\underline{(X^{c})}}_{φ^{+}})^{c},$

${\underline{X}}_{φ^{+}} = ({\bar{(X^{c})}}_{φ^{+}})^{c}$ .

Proof.

By definition, ${\underline{X}}_{φ^{+}} \subseteq X$ . For the next inclusion, let x ∈ X ⊆ U. Then φ (x) = φ (x) ; x ∈ X. This implies that $x \in {\bar{X}}_{φ^{+}}$ . Hence ${\underline{X}}_{φ^{+}} \subseteq X \subseteq {\bar{X}}_{φ^{+}}$ .

This is the direct consequence of the definitions of φ - lower positive and φ - upper positive MRBS - approximations (as given in Definition 3.1).

Since, ${\underline{U}}_{φ^{+}} = {x \in U : φ (x) \neq φ (y)$ for all y ∈ U^c = ∅} = {x : x ∈ U} = U . Also, ${\bar{U}}_{φ^{+}} = {x \in U : φ (x) = φ (y)$ for some y ∈ U} = {x : x ∈ U} = U .

Hence, ${\underline{U}}_{φ^{+}} = U = {\bar{U}}_{φ^{+}}$ .

Let $x \in {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}} .$ Then $x \in {\underline{X}}_{φ^{+}}$ and $x \in {\underline{Y}}_{φ^{+}}$ . So by definition, we have x ∈ X : φ (x) ≠ φ (y) for all y ∈ X^c, and x ∈ Y : φ (x) ≠ φ (z) for all z ∈ Y^c. It follows that x ∈ X ∩ Y : φ (x) ≠ φ (y) for all y ∈ X^c ∪ Y^c = (X ∩ Y) ^c. So $x \in {\underline{(X \cap Y)}}_{φ^{+}}$ and thus ${\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}} \subseteq {\underline{(X \cap Y)}}_{φ^{+}}$ .

For reverse inclusion, since we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (1) of Proposition 3.9 that ${\underline{(X \cap Y)}}_{φ^{+}} \subseteq {\underline{X}}_{φ^{+}}$ and ${\underline{(X \cap Y)}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}} .$ Thus ${\underline{(X \cap Y)}}_{φ^{+}} \subseteq {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}} .$

Consequently we get, ${\underline{(X \cap Y)}}_{φ^{+}} = {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}} .$

Since we know that, X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so we deduced from part (2) of Proposition 3.9 that ${\bar{(X \cap Y)}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}}$ and ${\bar{(X \cap Y)}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}} .$ Therefore it follows that ${\bar{(X \cap Y)}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}}$ .

Suppose $x \in {\bar{(X \cup Y)}}_{φ^{+}}$ . Then by definition, we have x ∈ U : φ (x) = φ (y) for some y ∈ (X ∪ Y). It follows that x ∈ U : φ (x) = φ (y) for some y ∈ X or x ∈ U : φ (x) = φ (y) for some y ∈ Y. Thus $x \in {\bar{X}}_{φ^{+}}$ or $x \in {\bar{Y}}_{φ^{+}}$ . So, $x \in {\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}}$ and hence, ${\bar{(X \cup Y)}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}}$ .

For reverse inclusion, since X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so it follows from part (2) of Proposition 3.9 that ${\bar{X}}_{φ^{+}} \subseteq {\bar{(X \cup Y)}}_{φ^{+}}$ and ${\bar{Y}}_{φ^{+}} \subseteq {\bar{(X \cup Y)}}_{φ^{+}} .$ So, it implies that ${\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}} \subseteq {\bar{(X \cup Y)}}_{φ^{+}} .$

Consequently we get, ${\bar{(X \cup Y)}}_{φ^{+}} = {\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}} .$

Since we know that, X ∪ Y ⊇ X and X ∪ Y ⊇ Y, so we deduced from part (1) of Proposition 3.9 that ${\underline{(X \cup Y)}}_{φ^{+}} \supseteq {\underline{X}}_{φ^{+}}$ and ${\underline{(X \cup Y)}}_{φ^{+}} \supseteq {\underline{Y}}_{φ^{+}}$ . It follows that ${\underline{(X \cup Y)}}_{φ^{+}} \supseteq {\underline{X}}_{φ^{+}} \cup {\underline{Y}}_{φ^{+}} .$

Suppose that $x \in ({\underline{(X^{c})}}_{φ^{+}})^{c} = U - {\underline{(X^{c})}}_{φ^{+}}$ . Then x ∈ U and $x \notin {\underline{(X^{c})}}_{φ^{+}}$ . This implies that φ (x) = φ (y) for some y ∈ (X^c) ^c = X. Thus, x ∈ U : φ (x) = φ (y) for some y ∈ X and so, $x \in {\bar{X}}_{φ^{+}}$ . Consequently $({\underline{(X^{c})}}_{φ^{+}})^{c} \subseteq {\bar{X}}_{φ^{+}}$ .

Conversely, let $x \in {\bar{X}}_{φ^{+}}$ . Then x ∈ U : φ (x) = φ (y) for some y ∈ X, so that $x \notin {\underline{(X^{c})}}_{φ^{+}}$ that is, $x \in U - {\underline{(X^{c})}}_{φ^{+}} = ({\underline{(X^{c})}}_{φ^{+}})^{c}$ . Thus $x \in ({\underline{(X^{c})}}_{φ^{+}})^{c}$ . Consequently ${\bar{X}}_{φ^{+}} \subseteq ({\underline{(X^{c})}}_{φ^{+}})^{c}$ .

Hence, ${\bar{X}}_{φ^{+}} = ({\underline{(X^{c})}}_{φ^{+}})^{c} .$

In part (8), taking X^c instead of X we get ${\bar{(X^{c})}}_{φ^{+}} = ({\underline{X}}_{φ^{+}})^{c}$ . This implies that $({\bar{(X^{c})}}_{φ^{+}})^{c} = {\underline{X}}_{φ^{+}}$ .

Hence, ${\underline{X}}_{φ^{+}} = ({\bar{(X^{c})}}_{φ^{+}})^{c} .$ □

The following example shows that, the inclusions in parts (5) and (7) in above theorem might be strict.

Example 3.14. Let $(f, g : A) \in BSS (U)$ , where U = {u₁, u₂, u₃, u₄, u₅, u₆} and set of parameters A = {e₁, e₂, e₃, e₄}. Then NOT set of parameters of A is ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄}. The mappings f and g are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}} & if e = e_{1}, \\ {u_{1}, u_{3}, u_{4}, u_{6}} & if e = e_{2}, \\ {u_{1}, u_{4}, u_{6}} & if e = e_{3}, \\ {u_{2}, u_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {} & if \neg e = \neg e_{1}, \\ {u_{2}, u_{5}} & if \neg e = \neg e_{2}, \\ {u_{3}, u_{5}} & if \neg e = \neg e_{3}, \\ {u_{1}, u_{4}} & if \neg e = \neg e_{4} . \end{matrix}$ From above mappings f and g, we can obtain the tabular representation of bipolar soft set (f, g : A) which is given in Table (4. (a), 4.(b)) as:

Now, let P = (U, (φ, ψ)) be the MRBS-approximation space. Then with the help of above pair of tables, we can define the mappings φ and ψ as follows: $φ : U ⟶ P (A),$ $u \mapsto {\begin{matrix} {e_{1}, e_{2}, e_{3}} & if u = u_{1}, \\ {e_{1}, e_{4}} & if u = u_{2}, \\ {e_{1}, e_{2}} & if u = u_{3}, \\ {e_{1}, e_{2}, e_{3}} & if u = u_{4}, \\ {e_{1}, e_{4}} & if u = u_{5}, \\ {e_{1}, e_{2}, e_{3}} & if u = u_{6}, \end{matrix}$ and $ψ : U ⟶ P (⌉ A),$ $u \mapsto {\begin{matrix} {\neg e_{4}} & if u = u_{1}, \\ {\neg e_{2}} & if u = u_{2}, \\ {\neg e_{3}} & if u = u_{3}, \\ {\neg e_{4}} & if u = u_{4}, \\ {\neg e_{2}, \neg e_{3}} & if u = u_{5}, \\ {} & if u = u_{6} . \end{matrix}$ If we consider X = {u₁, u₂} and Y = {u₂, u₄} then X ∩ Y = {u₂}. Now by direct computation we get $\begin{matrix} {\bar{X}}_{φ^{+}} & = {u_{1}, u_{2}, u_{4}, u_{5}, u_{6}}, \\ {\bar{Y}}_{φ^{+}} & = {u_{1}, u_{2}, u_{4}, u_{5}, u_{6}}, \\ {\bar{(X \cap Y)}}_{φ^{+}} & = {u_{2}, u_{5}} . \end{matrix}$ Clearly, ${\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}} = {u_{1}, u_{2}, u_{4}, u_{5}, u_{6}} \supset {u_{2}, u_{5}} = {\bar{(X \cap Y)}}_{φ^{+}}$ . That is, ${\bar{(X \cap Y)}}_{φ^{+}} \subset {\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}}$ , which indicates the inclusion in part (5) of Theorem 3.13 may hold strictly.

Now, assuming X = {u₂, u₃} and Y = {u₃, u₅}, then X ∪ Y = {u₂, u₃, u₅}. So by direct computation we obtain $\begin{matrix} {\underline{X}}_{φ^{+}} & = {u_{3}}, \\ {\underline{Y}}_{φ^{+}} & = {u_{3}}, \\ {\underline{(X \cup Y)}}_{φ^{+}} & = {u_{2}, u_{3}, u_{5}} . \end{matrix}$ Clearly, ${\underline{X}}_{φ^{+}} \cup {\underline{Y}}_{φ^{+}} = {u_{3}} \subset {u_{2}, u_{3}, u_{5}} = {\underline{(X \cup Y)}}_{φ^{+}}$ . That is, ${\underline{(X \cup Y)}}_{φ^{+}} \supset {\underline{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}}$ , which concludes the inclusion in part (7) of Theorem 3.13 might be strict.

Theorem 3.15. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space. Then for any X, Y ⊆ U, we have the following axioms:

${\bar{X}}_{ψ^{-}} \subseteq X \subseteq {\underline{X}}_{ψ^{-}},$

${\underline{\emptyset}}_{ψ^{-}} = \emptyset = {\bar{\emptyset}}_{ψ^{-}},$

${\underline{U}}_{ψ^{-}} = U = {\bar{U}}_{ψ^{-}},$

${\underline{(X \cap Y)}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}},$

${\bar{(X \cap Y)}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}},$

${\bar{(X \cup Y)}}_{ψ^{-}} \supseteq {\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}},$

${\underline{(X \cup Y)}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}},$

${\bar{X}}_{ψ^{-}} = ({\underline{(X^{c})}}_{ψ^{-}})^{c},$

${\underline{X}}_{ψ^{-}} = ({\bar{(X^{c})}}_{ψ^{-}})^{c}$ .

Proof.

By definition, ${\bar{X}}_{ψ^{-}} \subseteq X$ . For the other inclusion, let x ∈ X ⊆ U. Then ψ (x) = ψ (x) ; x ∈ X. This implies that $x \in {\underline{X}}_{ψ^{-}}$ . Hence ${\bar{X}}_{ψ^{-}} \subseteq X \subseteq {\underline{X}}_{ψ^{-}}$ .

This is the direct consequence of the definitions of ψ - lower negative and ψ - upper negative MRBS - approximations (as given in Definition 3.1).

Since, ${\underline{U}}_{ψ^{-}} = {x \in U : ψ (x) = ψ (y)$ for some y ∈ U} = {x : x ∈ U} = U. Similarly, ${\bar{U}}_{ψ^{-}} = {x \in U : ψ (x) \neq ψ (y)$ for all y ∈ U^c = ∅} = {x : x ∈ U} = U.

Hence, ${\underline{U}}_{ψ^{-}} = U = {\bar{U}}_{ψ^{-}}$ .

Since we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (3) of Proposition 3.9 that ${\underline{(X \cap Y)}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}}$ and ${\underline{(X \cap Y)}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}} .$ Therefore it follows that ${\underline{(X \cap Y)}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}} .$

Let $x \in {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}} .$ Then $x \in {\bar{X}}_{ψ^{-}}$ and $x \in {\bar{Y}}_{ψ^{-}}$ . So by definition, it follows that x ∈ X : ψ (x) ≠ ψ (y) for all y ∈ X^c, and x ∈ Y : ψ (x) ≠ ψ (y) for all y ∈ Y^c. It follows that x ∈ (X ∩ Y) : ψ (x) ≠ ψ (y) for all y ∈ X^c ∪ Y^c = (X ∩ Y) ^c. So $x \in {\bar{(X \cap Y)}}_{ψ^{-}}$ and thus ${\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}} \subseteq {\bar{(X \cap Y)}}_{ψ^{-}}$ .

For the reverse inclusion, as we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (4) of Proposition 3.9 that ${\bar{(X \cap Y)}}_{ψ^{-}} \subseteq {\bar{X}}_{ψ^{-}}$ and ${\bar{(X \cap Y)}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}} .$ So it follows that ${\bar{(X \cap Y)}}_{ψ^{-}} \subseteq {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}} .$

Hence we get, ${\bar{(X \cap Y)}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}} .$

Since we know that X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so it follows from part (4) of Proposition 3.9 that ${\bar{X}}_{ψ^{-}} \subseteq {\bar{(X \cup Y)}}_{ψ^{-}}$ and ${\bar{Y}}_{ψ^{-}} \subseteq {\bar{(X \cup Y)}}_{ψ^{-}} .$ Therefore it follows that ${\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}} \subseteq {\bar{(X \cup Y)}}_{ψ^{-}} .$

Let $x \in {\underline{(X \cup Y)}}_{ψ^{-}}$ . Then by definition, it follows that x ∈ U : ψ (x) = ψ (y) for some y ∈ (X ∪ Y). It implies that x ∈ U : ψ (x) = ψ (y) for some y ∈ X or x ∈ U : ψ (x) = ψ (y) for some y ∈ Y. Thus $x \in {\underline{X}}_{ψ^{-}}$ or $x \in {\underline{Y}}_{ψ^{-}}$ . Thus, $x \in {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}}$ and so ${\underline{(X \cup Y)}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}}$ .

For the reverse inclusion, since we know that X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so we deduce from part (3) of Proposition 3.9 that ${\underline{X}}_{ψ^{-}} \subseteq {\underline{(X \cup Y)}}_{ψ^{-}}$ and ${\underline{Y}}_{ψ^{-}} \subseteq {\underline{(X \cup Y)}}_{ψ^{-}} .$ So it follows that ${\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}} \subseteq {\underline{(X \cup Y)}}_{ψ^{-}} .$

Consequently we get, ${\underline{(X \cup Y)}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}} .$

Let $x \in ({\underline{(X^{c})}}_{ψ^{-}})^{c} = U - {\underline{(X^{c})}}_{ψ^{-}}$ . Then $x \in U and x \notin {\underline{(X^{c})}}_{ψ^{-}}$ . This implies that ψ (x) ≠ ψ (y) for all y∈ Xc. Since x ∈ U and X ⊆ U so, x ∈ X. Thus x ∈ X : ψ (x) ≠ ψ (y) for all y∈ Xc and so, $x \in {\bar{X}}_{ψ^{-}}$ . Consequently $({\underline{(X^{c})}}_{ψ^{-}})^{c} \subseteq {\bar{X}}_{ψ^{-}} .$

Conversely, let $x \in {\bar{X}}_{ψ^{-}}$ .Then x ∈ X : ψ (x) ≠ ψ (y) for all y∈ Xc, so that $x \notin {\underline{(X^{c})}}_{ψ^{-}}$ that is, $x \in U - {\underline{(X^{c})}}_{ψ^{-}} = ({\underline{(X^{c})}}_{ψ^{-}})^{c}$ . Thus $x \in ({\underline{(X^{c})}}_{ψ^{-}})^{c}$ . Consequently, ${\bar{X}}_{ψ^{-}} \subseteq ({\underline{(X^{c})}}_{ψ^{-}})^{c} .$

Hence ${\bar{X}}_{ψ^{-}} = ({\underline{(X^{c})}}_{ψ^{-}})^{c} .$

Taking X^c instead of X in part (8), we get ${\bar{(X^{c})}}_{ψ^{-}} = ({\underline{X}}_{ψ^{-}})^{c}$ . This implies that $({\bar{(X^{c})}}_{ψ^{-}})^{c} = {\underline{X}}_{ψ^{-}}$ .

Hence, ${\underline{X}}_{ψ^{-}} = ({\bar{(X^{c})}}_{ψ^{-}})^{c}$ .□

The following example shows that, the inclusions in parts (4) and (6) in above theorem may hold strictly.

Example 3.16. Consider the bipolar soft set (f, g : A) over the universe U given in Example 3.14, where U = {u₁, u₂, u₃, u₄, u₅, u₆}. Let X, Y ⊆ U be such that X = {u₁, u₂} and Y = {u₂, u₄} then X ∩ Y = {u₂} and X ∪ Y = {u₁, u₂, u₄}. By direct computation we get $\begin{matrix} {\underline{X}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}}, \\ {\underline{Y}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}}, \\ {\underline{(X \cap Y)}}_{ψ^{-}} & = {u_{2}} . \end{matrix}$ Clearly, ${\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}} = {u_{1}, u_{2}, u_{4}} \supset {u_{2}} = {\underline{(X \cap Y)}}_{ψ^{-}}$ . That is, ${\underline{(X \cap Y)}}_{ψ^{-}} \subset {\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}}$ , which indicates that inclusion in part (4) of Theorem 3.15 may hold strictly. Similarly, $\begin{matrix} {\bar{X}}_{ψ^{-}} & = {u_{2}}, \\ {\bar{Y}}_{ψ^{-}} & = {u_{2}}, \\ {\bar{(X \cup Y)}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}} . \end{matrix}$ Clearly, ${\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}} = {u_{2}} \subset {u_{1}, u_{2}, u_{4}} = {\bar{(X \cup Y)}}_{ψ^{-}}$ . That is, ${\bar{(X \cup Y)}}_{ψ^{-}} \supset {\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}}$ , which concludes that inclusion in part (6) of Theorem 3.15 might be strict.

Theorem 3.17. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space. Then for any X ⊆ U, we have

${\bar{({\underline{X}}_{φ^{+}})}}_{φ^{+}} = {\underline{X}}_{φ^{+}},$

${\underline{({\underline{X}}_{φ^{+}})}}_{φ^{+}} = {\underline{X}}_{φ^{+}},$

${\underline{({\bar{X}}_{φ^{+}})}}_{φ^{+}} = {\bar{X}}_{φ^{+}},$

$\bar{({\bar{X}}_{φ^{+}})} φ^{+} = {\bar{X}}_{φ^{+}},$

Proof.

From part (1) of Theorem 3.13, it follows that ${\underline{X}}_{φ^{+}} \subseteq {\bar{({\underline{X}}_{φ^{+}})}}_{φ^{+}}$ .

For the reverse inclusion, let $x \in {\bar{({\underline{X}}_{φ^{+}})}}_{φ^{+}}$ . Then we have x ∈ U : φ (x) = φ (y) for some $y \in {\underline{X}}_{φ^{+}}$ . By definition of φ - lower positive MRBS-approximation of X, we have y ∈ X : φ (y) ≠ φ (z) for all z ∈ X^c. As φ (x) = φ (y), so we get φ (x) ≠ φ (z) for all z ∈ X^c. This shows that x ∉ X^c. Thus x ∈ X and as ${\underline{X}}_{φ^{+}} \subseteq X$ so, $x \in {\underline{X}}_{φ^{+}}$ . Thus ${\bar{({\underline{X}}_{φ^{+}})}}_{φ^{+}} \subseteq {\underline{X}}_{φ^{+}}$ .

Hence, ${\bar{({\underline{X}}_{φ^{+}})}}_{φ^{+}} = {\underline{X}}_{φ^{+}} .$

By definition of φ - lower positive MRBS-approximation, we have $\begin{matrix} {\underline{({\underline{X}}_{φ^{+}})}}_{φ^{+}} = {x \in {\underline{X}}_{φ^{+}} : φ (x) \neq φ (y) \\ for all y \in ({\underline{X}}_{φ^{+}})^{c}} . \end{matrix}$ From part (1) of Theorem 3.13, we know that ${\underline{X}}_{φ^{+}} \subseteq X$ so, $X^{c} \subseteq ({\underline{X}}_{φ^{+}})^{c}$ . Thus, in particular, we have $\begin{matrix} {\underline{({\underline{X}}_{φ^{+}})}}_{φ^{+}} = {x \in {\underline{X}}_{φ^{+}} : φ (x) \neq φ (y) \\ for all y \in X^{c}} = {\underline{X}}_{φ^{+}} . \end{matrix}$

Hence, ${\underline{({\underline{X}}_{φ^{+}})}}_{φ^{+}} = {\underline{X}}_{φ^{+}} .$

From part (1) of Theorem 3.13, it follows that ${\underline{({\bar{X}}_{φ^{+}})}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}}$ .

For the reverse inclusion, if $y \notin {\underline{({\bar{X}}_{φ^{+}})}}_{φ^{+}}$ then by definition, either $y \notin {\bar{X}}_{φ^{+}}$ or φ (y) = φ (z) for some $z \in ({\bar{X}}_{φ^{+}})^{c}$ . If $y \notin {\bar{X}}_{φ^{+}}$ then we get our required result. In the second case, φ (y) = φ (z) for some $z \in ({\bar{X}}_{φ^{+}})^{c}$ , so that $z \notin {\bar{X}}_{φ^{+}}$ that is, φ (z) ≠ φ (w) for all w ∈ X. But since φ (y) = φ (z) so, φ (y) ≠ φ (w) for all w ∈ X = (X^c) ^c. Therefore, $y \in {\underline{(X^{c})}}_{φ^{+}}$ that is, $y \notin ({\underline{(X^{c})}}_{φ^{+}})^{c} = {\bar{X}}_{φ^{+}}$ , by part (8) of Theorem 3.13. That is $y \notin {\bar{X}}_{φ^{+}}$ . Thus ${\bar{X}}_{φ^{+}} \subseteq {\underline{(\bar{X} φ^{+})}}_{φ^{+}}$ .

Hence, ${\underline{({\bar{X}}_{φ^{+}})}}_{φ^{+}} = {\bar{X}}_{φ^{+}} .$

From part (1) of Theorem 3.13, it follows that ${\bar{X}}_{φ^{+}} \subseteq {\bar{({\bar{X}}_{φ^{+}})}}_{φ^{+}}$ .

For the reverse inclusion, let $x \in {\bar{({\bar{X}}_{φ^{+}})}}_{φ^{+}}$ . Then by definition, x ∈ U : φ (x) = φ (y) for some $y \in {\bar{X}}_{φ^{+}}$ . By definition of φ - upper positive MRBS-approximation of X, we have φ (y) = φ (z) for some z ∈ X. As φ (x) = φ (y), so we get φ (x) = φ (z) for some z ∈ X. So $x \in {\bar{X}}_{φ^{+}}$ . Thus $\bar{({\bar{X}}_{φ^{+}})} φ^{+} \subseteq {\bar{X}}_{φ^{+}}$ . Hence we get, ${\bar{({\bar{X}}_{φ^{+}})}}_{φ^{+}} = {\bar{X}}_{φ^{+}}$ .□

Theorem 3.18. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space. Then for any X ⊆ U, we have

${\underline{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}},$

${\bar{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}},$

${\underline{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}},$

${\bar{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}} .$

Proof.

From part (1) of Theorem 3.15, it follows that ${\bar{X}}_{ψ^{-}} \subseteq {\underline{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} .$

For the reverse inclusion, let $x \in {\underline{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} .$ Then we have x ∈ U : ψ (x) = ψ (y) for some $y \in {\bar{X}}_{ψ^{-}} .$ By definition of ψ - upper negative MRBS-approximation of X, we have y ∈ X : ψ (y) ≠ ψ (z) for all z ∈ X^c . As ψ (x) = ψ (y), so we get ψ (x) ≠ ψ (z) for all z ∈ X^c . This shows that x ∉ X^c. Thus x ∈ X and so, $x \in {\bar{X}}_{ψ^{-}} .$ This implies that, ${\underline{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} \subseteq {\bar{X}}_{ψ^{-}} .$

Hence we get, ${\underline{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}} .$

By definition of ψ - upper negative MRBS-approximation, it follows that $\begin{matrix} {\bar{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} = {x \in {\bar{X}}_{ψ^{-}} : ψ (x) \neq ψ (y) \\ forall y \in ({\bar{X}}_{ψ^{-}})^{c}} . \end{matrix}$ Now by part (1) of Theorem 3.15, we know that ${\bar{X}}_{ψ^{-}} \subseteq X$ , so $X^{c} \subseteq ({\bar{X}}_{ψ^{-}})^{c} .$ Thus, in particular, we have $\begin{matrix} {\bar{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} = {x \in {\bar{X}}_{ψ^{-}} : ψ (x) \neq ψ (y) \\ forall y \in X^{c}} = {\bar{X}}_{ψ^{-}} . \end{matrix}$

Hence we get, ${\bar{({\bar{X}}_{ψ^{-}})}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}}$ .

From part (1) of Theorem 3.15, it follows that ${\underline{X}}_{ψ^{-}} \subseteq {\underline{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} .$

For the reverse inclusion, let $x \in {\underline{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} .$ Then by definition of ψ - lower negative MRBS-approximation, we have x ∈ U: ψ (x) = ψ (y) for some $y \in {\underline{X}}_{ψ^{-}} .$ Again by definition of ψ - lower negative MBSR-approximation, it follows that y ∈ U : ψ (y) = ψ (z) for some z ∈ X . As ψ (x) = ψ (y), so we get ψ (x) = ψ (z) for some z ∈ X . This shows that x ∈ X. Since $X \subseteq {\underline{X}}_{ψ^{-}}$ and x ∈ X so, $x \in {\underline{X}}_{ψ^{-}}$ . This implies that, ${\underline{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}} .$

Hence we obtain, ${\underline{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}} .$

From part (1) of Theorem 3.15, it follows that ${\bar{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}}$ .

For the reverse inclusion, if $y \notin {\bar{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}}$ then by definition, we have two possibilities, either $y \notin {\underline{X}}_{ψ^{-}}$ or ψ (y) = ψ (z) for some $z \in ({\underline{X}}_{ψ^{-}})^{c}$ . In the first case, if $y \notin {\underline{X}}_{ψ^{-}}$ then we get our required result. In the later case, ψ (y) = ψ (z) for some $z \in ({\underline{X}}_{ψ^{-}})^{c}$ , so that $z \notin {\underline{X}}_{ψ^{-}}$ that is, ψ (z) ≠ ψ (w) for all w ∈ X. But since ψ (y) = ψ (z), so ψ (y) ≠ ψ (w) for all w ∈ X = (X^c) ^c. Therefore, $y \in {\bar{(X^{c})}}_{ψ^{-}}$ that is, $y \notin ({\bar{(X^{c})}}_{ψ^{-}})^{c} = {\underline{X}}_{ψ^{-}}$ , by part (9) of Theorem 3.15. That is $y \notin {\underline{X}}_{ψ^{-}}$ . Thus ${\underline{X}}_{ψ^{-}} \subseteq {\bar{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}}$ .

Hence, ${\bar{({\underline{X}}_{ψ^{-}})}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}}$ .□

Remark 3.19. From parts (1) to (4) of Theorems 3.17 and 3.18, it can be seen that ${\underline{X}}_{φ^{+}}$ , ${\bar{X}}_{φ^{+}}$ , ${\underline{X}}_{ψ^{-}}$ and ${\bar{X}}_{ψ^{-}}$ are definable sets in MBSR-approximation space (U, (φ, ψ)). Furthermore their φ - lower positive, φ - upper positive, ψ - lower negative and ψ - upper negative approximations with respect to (U, (φ, ψ)) are invariant.

Now we discuss some basic set theoretical operations of modified bipolar pairs such as union, intersection and compliment.

Definition 3.20. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space and X, Y ⊆ U. Then union of lower modified bipolar pair and upper modified bipolar pair of set X and Y are defined as, respectively,

${\underline{MBS}}_{P} (X) ⋓ {\underline{MBS}}_{P} (Y) = ({\underline{X}}_{φ^{+}} \cup {\underline{Y}}_{φ^{+}}, {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}})$ ,

${\bar{MBS}}_{P} (X) ⋓ {\bar{MBS}}_{P} (Y) = ({\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}}, {\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}})$ .

Definition 3.21. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MRBS-approximation space and X, Y ⊆ U. Then intersection of lower modified bipolar pair and upper modified bipolar pair of set X and Y are defined as, respectively,

${\underline{MBS}}_{P} (X) ⋒ {\underline{MBS}}_{P} (Y) = ({\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}}, {\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}})$ ,

${\bar{MBS}}_{P} (X) ⋒ {\bar{MBS}}_{P} (Y) = ({\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}}, {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}})$ .

Definition 3.22. Let $(f, g : A) \in BSS (U)$ , such that P = (U, (φ, ψ)) be a MBSR-approximation space and X ⊆ U. Then compliment of lower modified bipolar pair and upper modified bipolar pair of set X is defined as, respectively,

${\underline{MBS}}_{P}^{c} (X) = ({\underline{X}}_{ψ^{-}}, {\underline{X}}_{φ^{+}}),$

${\bar{MBS}}_{P}^{c} (X) = ({\bar{X}}_{ψ^{-}}, {\bar{X}}_{φ^{+}}) .$

Example 3.23. As an illustrate, let $(f, g : A) \in BSS (U)$ , where U = {u₁, u₂, u₃, u₄, u₅, u₆} be the set of six universities under consideration and A = {e₁, e₂, e₃, e₄} = {good faculty, good research opportunities, low rates of tuition, for good security} be the set of parameters (features) of universities, then NOT set of parameters of A is ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} = {not good faculty, bad research opportunities, high rates of tuition, for bad security}. The bipolar soft set (f, g : A) describes the “requirements of the university” in which Mr. X going to take admission. Suppose that positive soft set and negative soft set of (f, g : A) are given as:

$f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {u_{1}, u_{6}} & if e = e_{1}, \\ {u_{3}} & if e = e_{2}, \\ {} & if e = e_{3}, \\ {u_{2}, u_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {u_{3}} & if \neg e = \neg e_{1}, \\ {u_{5}} & if \neg e = \neg e_{2}, \\ {u_{2}, u_{3}, u_{6}} & if \neg e = \neg e_{3}, \\ {u_{4}} & if \neg e = \neg e_{4} . \end{matrix}$ Now, for X = {u₃, u₄, u₅} ⊆ U and Y = {u₁, u₂, u₄} ⊆ U, the modified rough bipolar approximations of X and Yare given as: $\begin{matrix} {\underline{X}}_{φ^{+}} & = {u_{3}, u_{4}}, \\ {\bar{X}}_{φ^{+}} & = {u_{2}, u_{3}, u_{4}, u_{5}}, \\ {\underline{X}}_{ψ^{-}} & = {u_{3}, u_{4}, u_{5}}, \\ {\bar{X}}_{ψ^{-}} & = {u_{3}, u_{4}, u_{5}}, \\ {\underline{Y}}_{φ^{+}} & = {u_{4}}, \\ {\bar{Y}}_{φ^{+}} & = {u_{1}, u_{2}, u_{4}, u_{5}, u_{6}}, \\ {\underline{Y}}_{ψ^{-}} & = {u_{1}, u_{2}, u_{4}, u_{6}}, \\ {\bar{Y}}_{ψ^{-}} & = {u_{1}, u_{4}} . \end{matrix}$ So, the lower and upper modified bipolar pairs of X and Y are: $\begin{matrix} {\underline{MBS}}_{P} (X) & = ({u_{3}, u_{4}}, {u_{3}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{P} (X) & = ({u_{2}, u_{3}, u_{4}, u_{5}}, {u_{3}, u_{4}, u_{5}}), \\ {\underline{MBS}}_{P} (Y) & = ({u_{4}}, {u_{1}, u_{2}, u_{4}, u_{6}}), \\ {\bar{MBS}}_{P} (Y) & = ({u_{1}, u_{2}, u_{4}, u_{5}, u_{6}}, {u_{1}, u_{4}}) . \end{matrix}$ Thus the union of lower and upper modified bipolar pair is given by: $\begin{matrix} {\underline{MBS}}_{P} (X) ⋓ {\underline{MBS}}_{P} (Y) = ({u_{3}, u_{4}}, {u_{1}, u_{2}, u_{3}, \\ u_{4}, u_{5}, u_{6}}), \\ {\bar{MBS}}_{P} (X) ⋓ {\bar{MBS}}_{P} (Y) = ({u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}}, \\ {u_{1}, u_{3}, u_{4}, u_{5}}) . \end{matrix}$ Similarly the intersection of lower and upper modified bipolar pair is given by:

$\begin{matrix} {\underline{MBS}}_{P} (X) ⋒ {\underline{MBS}}_{P} (Y) & = ({u_{4}}, {u_{4}}), \\ {\bar{MBS}}_{P} (X) ⋒ {\bar{MBS}}_{P} (Y) & = ({u_{2}, u_{4}, u_{5}}, {u_{4}}) . \end{matrix}$ Moreover, the compliment of lower and upper modified bipolar pair of set X is given by: $\begin{matrix} {\underline{MBS}}_{P}^{c} (X) & = ({u_{3}, u_{4}, u_{5}}, {u_{3}, u_{4}}), \\ {\bar{MBS}}_{P}^{c} (X) & = ({u_{3}, u_{4}, u_{5}}, {u_{2}, u_{3}, u_{4}, u_{5}}) . \end{matrix}$

Proposition 3.24. (bfInvolution Property) Let $(f, g : A) \in BSS (U)$ be such that P = (U, (φ, ψ)) be a MRBS-approximation space. Then for X ⊆ U, we have

$({\underline{MBS}}_{P}^{c} (X))^{c} = {\underline{MBS}}_{P} (X),$

$({\bar{MBS}}_{P}^{c} (X))^{c} = {\bar{MBS}}_{P} (X) .$

Proof. The proof can be made via the use of definition of compliment of lower modified bipolar pair and upper modified bipolar pair of the set X.□

Proposition 3.25. (bfDe Morgan’s Laws) Let $(f, g : A) \in BSS (U)$ be such that P = (U, (φ, ψ)) be a MRBS-approximation space. Then for X, Y ⊆ U, we have

$({\underline{MBS}}_{P} (X) ⋓ {\underline{MBS}}_{P} (Y))^{c} = {\underline{MBS}}_{P}^{c} (X) ⋒ {\underline{MBS}}_{P}^{c} (Y),$

$({\bar{MBS}}_{P} (X) ⋓ {\bar{MBS}}_{P} (Y))^{c} = {\bar{MBS}}_{P}^{c} (X) ⋒ {\bar{MBS}}_{P}^{c} (Y),$

$({\underline{MBS}}_{P} (X) ⋒ {\underline{MBS}}_{P} (Y))^{c} = {\underline{MBS}}_{P}^{c} (X) ⋓ {\underline{MBS}}_{P}^{c} (Y),$

$({\bar{MBS}}_{P} (X) ⋒ {\bar{MBS}}_{P} (Y))^{c} = {\bar{MBS}}_{P}^{c} (X) ⋓ {\bar{MBS}}_{P}^{c} (Y) .$

Proof. The proof can be made via the use of definition of union and intersection of lower modified bipolar pair and upper modified bipolar pair of the set X.□

Proposition 3.26. Let $(f, g : A) \in BSS (U)$ and P = (U, (φ, ψ)) be a MRBS-approximation space. Then for X, Y ⊆ U, we have

${\underline{MBS}}_{P} (X) ⊑ {\underline{MBS}}_{P} (Y) \Leftrightarrow {\underline{MBS}}_{P}^{c} (Y) ⊑ {\underline{MBS}}_{P}^{c} (X),$

${\bar{MBS}}_{P} (X) ⊑ {\bar{MBS}}_{P} (Y) \Leftrightarrow {\bar{MBS}}_{P}^{c} (Y) ⊑ {\bar{MBS}}_{P}^{c} (X) .$

Proof. Obvious.□

4 Accuracy and roughness measures for MRBS-sets

In order to express the quality of an approximation some accuracy measures are important. In [29] Pawlak suggested two numerical measures for characterizing the imprecision of rough set approximations, which can help us to get a perception, that how accurate is the information related with some equivalence relation for a specific classification.

In this section, we introduce accuracy measure (degree of accuracy) and roughness measure (degree of roughness) for modified rough bipolar soft sets (MRBS-sets) and study some of their basic properties.

According to Pawlak [29], accuracy measure η_R (X) of a subset X of the universe U with respect to Pawlak approximation space P = (U, R) is the ratio of cardinality of lower approximation R_* (X) to the cardinality of upper approximation R^* (X), that is, $η_{R} (X) = \frac{| R_{*} (X) |}{| R^{*} (X) |} .$ Similarly on the basis of accuracy measure, the roughness measure (degree of roughness) is defined as: $ρ_{R} (X) = 1 - η_{R} (X) .$ Following the similar technique we have.

Definition 4.1. Assume that $(f, g : A) \in BSS (U)$ , $P = (U, (f, g : A))$ a $BSA$ -space and $X \subseteq U$ . Then the measure of accuracy for bipolar soft rough set with respect to X represented by $A (X)$ and is defined by an ordered pair: $A (X) = (A^{X^{+}}, A^{X^{-}}),$ where $A^{X^{+}} = \frac{| {\underline{S}}_{P^{+}} (X) |}{| {\bar{S}}_{P^{+}} (X) |}$ and $A^{X^{-}} = \frac{| {\bar{S}}_{P^{-}} (X) |}{| {\underline{S}}_{P^{-}} (X) |} .$

Here $| {\underline{S}}_{P^{+}} (X) |$ , $| {\bar{S}}_{P^{+}} (X) |$ , $| {\bar{S}}_{P^{-}} (X) |$ and $| {\underline{S}}_{P^{-}} (X) |$ denote the cardinality of the sets ${\underline{S}}_{P^{+}} (X)$ , ${\bar{S}}_{P^{+}} (X)$ , ${\bar{S}}_{P^{-}} (X)$ and ${\underline{S}}_{P^{-}} (X)$ , respectively.

Similarly, the measure of roughness for bipolar soft rough set with respect to X represented by $R (X)$ and is defined as: $\begin{matrix} R (X) & = (1, 1) - A (X) = (1, 1) - (A^{X^{+}}, A^{X^{-}}) \\ = (1 - A^{X^{+}}, 1 - A^{X^{-}}) . \end{matrix}$

Definition 4.2. Let (f, g : A) be a bipolar soft set over the universe U and P = (U, (φ, ψ)) be a MRBS-approximation space. For any non-empty subset X of U, measure of accuracy for modified rough bipolar soft set (MRBS-set) with respect to X represented by $A (X)$ is defined by an ordered pair: $A (X) = (A^{X^{+}}, A^{X^{-}})$ where $A^{X^{+}} = \frac{| {\underline{X}}_{φ^{+}} |}{| {\bar{X}}_{φ^{+}} |}$ and $A^{X^{-}} = \frac{| {\bar{X}}_{ψ^{-}} |}{| {\underline{X}}_{ψ^{-}} |} .$

Here $| {\underline{X}}_{φ^{+}} |$ , $| {\bar{X}}_{φ^{+}} |$ , $| {\bar{X}}_{ψ^{-}} |$ and $| {\underline{X}}_{ψ^{-}} |$ denote the cardinality of the sets ${\underline{X}}_{φ^{+}}$ , ${\bar{X}}_{φ^{+}}$ , ${\bar{X}}_{ψ^{-}}$ and ${\underline{X}}_{ψ^{-}}$ , respectively.

Similarly, measure of roughness for modified rough bipolar soft set (MRBS-set) with respect to X represented by $R (X)$ is defined as: $\begin{matrix} R (X) & = (1, 1) - A (X) = (1, 1) - (A^{X^{+}}, A^{X^{-}}) \\ = (1 - A^{X^{+}}, 1 - A^{X^{-}}) . \end{matrix}$

Obviously, $0 \leq A^{X^{+}} \leq 1$ and $0 \leq A^{X^{-}} \leq 1$ for any subset X of U .

Conventionally for an empty set ∅, we define $A (\emptyset) = (1, 1)$ and $R (\emptyset) = (0, 0)$ . Also it can be seen that $A (X) = (1, 1) if and only if X = U$ .

Following example is presented to interpret the above definition:

Example 4.3. We revisit the Example 2.16, where U = {c₁, c₂, c₃, c₄, c₅} is the set of five cars under consideration and A = {e₁, e₂, e₃, e₄} = {cheap, comfort, high equipment, longtime warrant} and ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄} = {expensive, uncomfortable, low equipment, not longtime warrant}. The bipolar soft sets (f, g : A) describe the “requirements of the car” which Mr. X going to buy. Suppose that positive soft set and negative soft set of (f, g : A) are given as: $f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {c_{2}, c_{3}} & if e = e_{1}, \\ {c_{2}, c_{5}} & if e = e_{2}, \\ {c_{3}} & if e = e_{3}, \\ {c_{2}, c_{3}, c_{5}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $\neg e \mapsto {\begin{matrix} {c_{4}, c_{5}} & if \neg e = \neg e_{1}, \\ {c_{3}, c_{4}} & if \neg e = \neg e_{2}, \\ {c_{2}, c_{4}} & if \neg e = \neg e_{3}, \\ {c_{4}} & if \neg e = \neg e_{4} . \end{matrix}$

From above two mappings, we can obtain the tabular representation of bipolar soft set (f, g : A) which is given in Table 5. (a) and 5.(b) as:

Table 4
Tabular representation of (f, g: A)

(a)

f u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

e ₁ 1 1 1 1 1 1

e ₂ 1 0 1 1 0 1

e ₃ 1 0 0 1 0 1

e ₄ 0 1 0 0 1 0

(b)

g u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

¬e₁ 0 0 0 0 0 0

¬e₂ 0 1 0 0 1 0

¬e₃ 0 0 1 0 1 0

¬e₄ 1 0 0 1 0 0

(a)
e ₁	1	1	1	1	1	1
e ₂	1	0	1	1	0	1
e ₃	1	0	0	1	0	1
e ₄	0	1	0	0	1	0
(b)
g	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
¬e₁	0	0	0	0	0	0
¬e₂	0	1	0	0	1	0
¬e₃	0	0	1	0	1	0
¬e₄	1	0	0	1	0	0

Table 5

Tabular representation of (f, g: A)

(a)
f	c ₁	c ₂	c ₃	c ₄	c ₅
e ₁	0	1	1	0	0
e ₂	0	1	0	0	1
e ₃	0	0	1	0	0
e ₄	0	1	1	0	1
(b)
g	c ₁	c ₂	c ₃	c ₄	c ₅
¬e₁	0	0	0	1	1
¬e₂	0	0	1	1	0
¬e₃	0	1	0	1	0
¬e₄	0	0	0	1	0

Now assume that P = (U, (φ, ψ)) is a MRBS-approximation space. Then with the help of above pair of tables, we can construct the two mappings as follows: $φ : U ⟶ P (A),$ $c \mapsto {\begin{matrix} {} & if c = c_{1}, \\ {e_{1}, e_{2}, e_{4}} & if c = c_{2}, \\ {e_{1}, e_{3}, e_{4}} & if c = c_{3}, \\ {} & if c = c_{4}, \\ {e_{2}, e_{4}} & if c = c_{5}, \end{matrix}$ and $ψ : U ⟶ P (⌉ A),$ $c \mapsto {\begin{matrix} {} & if c = c_{1}, \\ {\neg e_{3}} & if c = c_{2}, \\ {\neg e_{2}} & if c = c_{3}, \\ {\neg e_{1}, \neg e_{2}, \neg e_{3}, \neg e_{4}} & if c = c_{4}, \\ {\neg e_{1}} & if c = c_{5} . \end{matrix}$ Using Definition 3.1, the modified rough bipolar soft approximations of X = {c₁, c₂, c₃} ⊆ U with respect to MRBS-space are $\begin{matrix} {\underline{X}}_{φ^{+}} & = {c_{2}, c_{3}}, \\ {\bar{X}}_{φ^{+}} & = {c_{1}, c_{2}, c_{3}, c_{4}}, \\ {\underline{X}}_{ψ^{-}} & = {c_{1}, c_{2}, c_{3}}, \\ {\bar{X}}_{ψ^{-}} & = {c_{1}, c_{2}, c_{3}} . \end{matrix}$ Since $| {\underline{X}}_{φ^{+}} | = 2$ , $| {\bar{X}}_{φ^{+}} | = 4$ , $| {\bar{X}}_{ψ^{-}} | = 3$ , $| {\underline{X}}_{ψ^{-}} | = 3$ .

So measure of accuracy and measure of roughness for modified rough bipolar soft set (MRBS-set) with respect to X are respectively given as: $A (X) = (A^{X^{+}}, A^{X^{-}}) = (2 / 4, 3 / 3) = (0.5, 1)$ and $R (X) = (1, 1) - A (X) = (1, 1) - (0.5, 1) = (0.5, 0) .$ Hence, the positive membership map φ⁺ of MRBS-approximation space P = (U, (φ, ψ)) describes the objects of the universe U accurately upto the degree 0.5. While the negative membership map ψ^- of MRBS-approximation space P = (U, (φ, ψ)) describes the objects of the universe U accurately upto the degree 1.

5 Multi-criteria group decision making based on modified rough bipolar soft sets

In this section, we construct a multi-criteria group decision making technique based on the modified rough bipolar soft sets.

Let U = {u₁, u₂, …, u_n} be the finite universe of objects, A = {e₁, e₂, …, e_m} be the set of all possible parameters and $(f, g : A) \in BSS (U)$ . Suppose that H = {p₁, p₂, …, p_k} is a set of expert persons, X₁, X₂, …, X_k are non-empty subsets of U, represent results of primary evaluations of expert persons p₁, p₂, …, p_k, respectively and $T_{1}, T_{2}, \dots, T_{r} \in BSS (U)$ are the actual result that previously obtained for problems in different places or different times.

Definition 5.1. Let ${\underline{MBS}}_{T_{q}} (X_{j}) = ({\underline{X_{j}}}_{φ^{+}}, {\underline{X_{j}}}_{ψ^{-}})$ and ${\bar{MBS}}_{T_{q}} (X_{j}) = ({\bar{X_{j}}}_{φ^{+}}, {\bar{X_{j}}}_{ψ^{-}})$ be lower and upper modified bipolar pairs of X_j ; (j = 1, 2, …, k) related to T_q ; (q = 1, 2, …, r). Then, $\begin{matrix} \underline{mb} & = [\begin{matrix} {{\underline{X_{1}}}_{φ^{+}}^{1^{+}}, {\underline{X_{1}}}_{ψ^{-}}^{1^{-}}} & {{\underline{X_{2}}}_{φ^{+}}^{1^{+}}, {\underline{X_{2}}}_{ψ^{-}}^{1^{-}}} & \dots & {{\underline{X_{k}}}_{φ^{+}}^{1^{+}}, {\underline{X_{k}}}_{ψ^{-}}^{1^{-}}} \\ {{\underline{X_{1}}}_{φ^{+}}^{2^{+}}, {\underline{X_{1}}}_{ψ^{-}}^{2^{-}}} & {{\underline{X_{2}}}_{φ^{+}}^{2^{+}}, {\underline{X_{2}}}_{ψ^{-}}^{2^{-}}} & \dots & {{\underline{X_{k}}}_{φ^{+}}^{2^{+}}, {\underline{X_{k}}}_{ψ^{-}}^{2^{-}}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {{\underline{X_{1}}}_{φ^{+}}^{r^{+}}, {\underline{X_{1}}}_{ψ^{-}}^{r^{-}}} & {{\underline{X_{2}}}_{φ^{+}}^{r^{+}}, {\underline{X_{2}}}_{ψ^{-}}^{r^{-}}} & \dots & {{\underline{X_{k}}}_{φ^{+}}^{r^{+}}, {\underline{X_{k}}}_{ψ^{-}}^{r^{-}}} \end{matrix}], \\ \bar{mb} & = [\begin{matrix} {{\bar{X_{1}}}_{φ^{+}}^{1^{+}}, {\bar{X_{1}}}_{ψ^{-}}^{1^{-}}} & {{\bar{X_{2}}}_{φ^{+}}^{1^{+}}, {\bar{X_{2}}}_{ψ^{-}}^{1^{-}}} & \dots & {{\bar{X_{k}}}_{φ^{+}}^{1^{+}}, {\bar{X_{k}}}_{ψ^{-}}^{1^{-}}} \\ {{\bar{X_{1}}}_{φ^{+}}^{2^{+}}, {\bar{X_{1}}}_{ψ^{-}}^{2^{-}}} & {{\bar{X_{2}}}_{φ^{+}}^{2^{+}}, {\bar{X_{2}}}_{ψ^{-}}^{2^{-}}} & \dots & {{\bar{X_{k}}}_{φ^{+}}^{2^{+}}, {\bar{X_{k}}}_{ψ^{-}}^{2^{-}}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {{\bar{X_{1}}}_{φ^{+}}^{r^{+}}, {\bar{X_{1}}}_{ψ^{-}}^{r^{-}}} & {{\bar{X_{2}}}_{φ^{+}}^{r^{+}}, {\bar{X_{2}}}_{ψ^{-}}^{r^{-}}} & \dots & {{\bar{X_{k}}}_{φ^{+}}^{r^{+}}, {\bar{X_{k}}}_{ψ^{-}}^{r^{-}}} \end{matrix}] \end{matrix}$ are said to be modified bipolar soft lower approximation matrix and modified bipolar soft upper approximation matrix, respectively. Here $\begin{matrix} {\underline{X_{j}}}_{φ^{+}}^{q^{+}} & = ({\underline{u_{1 j}}}_{φ^{+}}^{q^{+}}, {\underline{u_{2 j}}}_{φ^{+}}^{q^{+}}, \dots, {\underline{u_{nj}}}_{φ^{+}}^{q^{+}}), \\ {\underline{X_{j}}}_{ψ^{-}}^{q^{-}} & = ({\underline{u_{1 j}}}_{ψ^{-}}^{q^{-}}, {\underline{u_{2 j}}}_{ψ^{-}}^{q^{-}}, \dots, {\underline{u_{nj}}}_{ψ^{-}}^{q^{-}}), \\ {\bar{X_{j}}}_{φ^{+}}^{q^{+}} & = ({\bar{u_{1 j}}}_{φ^{+}}^{q^{+}}, {\bar{u_{2 j}}}_{φ^{+}}^{q^{+}}, \dots, {\bar{u_{nj}}}_{φ^{+}}^{q^{+}}), \\ {\bar{X_{j}}}_{ψ^{-}}^{q^{-}} & = ({\bar{u_{1 j}}}_{ψ^{-}}^{q^{-}}, {\bar{u_{2 j}}}_{ψ^{-}}^{q^{-}}, \dots, {\bar{u_{nj}}}_{ψ^{-}}^{q^{-}}) . \end{matrix}$

Where, $\begin{matrix} {\underline{u_{ij}}}_{φ^{+}}^{q^{+}} & = {\begin{matrix} 1 & if u_{i} \in {\underline{X_{j}}}_{φ^{+}}, \\ 0 & if u_{i} \notin {\underline{X_{j}}}_{φ^{+}} . \end{matrix} \\ {\underline{u_{ij}}}_{ψ^{-}}^{q^{-}} & = {\begin{matrix} - 1 / 2 & if u_{i} \in {\underline{X_{j}}}_{ψ^{-}}, \\ 0 & if u_{i} \notin {\underline{X_{j}}}_{ψ^{-}} . \end{matrix} \\ {\bar{u_{ij}}}_{φ^{+}}^{q^{+}} & = {\begin{matrix} 1 / 2 & if u_{i} \in {\bar{X_{j}}}_{φ^{+}}, \\ 0 & if u_{i} \notin {\bar{X_{j}}}_{φ^{+}} . \end{matrix} \\ {\bar{u_{ij}}}_{ψ^{-}}^{q^{-}} & = {\begin{matrix} - 1 & if u_{i} \in {\bar{X_{j}}}_{ψ^{-}}, \\ 0 & if u_{i} \notin {\bar{X_{j}}}_{ψ^{-}} . \end{matrix} \end{matrix}$

Definition 5.2. Let $\underline{mb}$ and $\bar{mb}$ be modified bipolar soft lower and upper approximation matrices with respect to ${\underline{MBS}}_{T_{q}} (X_{j}) = ({\underline{X_{j}}}_{φ^{+}}, {\underline{X_{j}}}_{ψ^{-}})$ and ${\bar{MBS}}_{T_{q}} (X_{j}) = ({\bar{X_{j}}}_{φ^{+}}, {\bar{X_{j}}}_{ψ^{-}})$ , where j = 1, 2, …, k and q = 1, 2, …, r. Then modified bipolar soft lower approximation vector (represented by $\underline{V}$ ) and modified bipolar soft upper approximation vector (represented by $\bar{V}$ ) are respectively defined as: $\underline{V} = \sum_{j = 1}^{k} \sum_{q = 1}^{r} ({\underline{X_{j}}}_{φ^{+}}^{q^{+}} \oplus {\underline{X_{j}}}_{ψ^{-}}^{q^{-}})$ (5.11) $\bar{V} = \sum_{j = 1}^{k} \sum_{q = 1}^{r} ({\bar{X_{j}}}_{φ^{+}}^{q^{+}} \oplus {\bar{X_{j}}}_{ψ^{-}}^{q^{-}}) .$ (5.12) Here the operation ∑ and ⊕ represent the vector addition.

Definition 5.3. Let $\underline{V}$ and $\bar{V}$ be modified bipolar soft T_q-lower approximation vector and modified bipolar soft T_q-upper approximation vector, respectively. Then $V = \underline{V} \oplus \bar{V} = (v_{1}, v_{2}, \dots, v_{n})$ (5.13) is called the decision vector.

Definition 5.4. Let $\underline{V} \oplus \bar{V} = (v_{1}, v_{2}, \dots, v_{n})$ be the decision vector. Then each v_i is said to be a weighted number of u_i ∈ U. An element u_i ∈ U is called an optimal element of the universe U if its weighted number is maximum of v_i for all i = 1, 2, …, n. If there are more than one optimal elements of the universe U, then select any one of them.

Proposed algorithm

Let $(f, g : A) \in BSS (U)$ , where U = {u₁, u₂, …, u_n} be the finite universe of certain objects and A = {e₁, e₂, …, e_m} be the set of all possible parameters. Consider H = {p₁, p₂, …, p_k} a set of expert persons, X₁, X₂, …, X_k are non-empty subsets of U, represent results of primary evaluations of expert persons p₁, p₂, …, p_k, respectively and $T_{1}, T_{2}, \dots, T_{r} \in BSS (U)$ are the actual result that previously obtained for problems in different places or different times.

Step 1: Take primary evaluations X₁, X₂, …, X_k of experts persons p₁, p₂, …, p_k.

Step 2: construct T₁, T₂, …, T_r bipolar soft sets using the real results.

Step 3: Compute ${\underline{MBS}}_{T_{q}} (X_{j})$ and ${\bar{MBS}}_{T_{q}} (X_{j})$ for each j = 1, 2, …, k and q = 1, 2, …, r.

Step 4: Construct modified bipolar soft lower approximation matrix $\underline{mb}$ and modified bipolar soft upper approximation matrix $\bar{mb}$ .

Step 5: Compute modified bipolar soft lower approximation vector $\underline{V}$ and modified bipolar soft upper approximation vector $\bar{V}$ .

Step 6: Compute decision vector $V = \underline{V} \oplus \bar{V}$ .

Step 7: Find $max_{i} v_{i}$ ; i = 1, 2, …, n.

Illustrative example

Suppose there are three investment experts p₁, p₂ and p₃ who wants to make an investment in a certain company. Let U = {u₁, u₂, u₃, u₄, u₅}, where u₁ = Automobile company, u₂ = Food company, u₃ = Computer company, u₄ = Arms company and u₅ = Medicine company. Consider A = {e₁, e₂, e₃} is the set of parameters (characteristics of companies), where e₁ = appreciation, e₂ = economical growth and e₃ = annual profit. Then NOT set of parameters of A is ⌉A = {¬ e₁ = depreciation, ¬ e₂ = economical decay, ¬ e₃ = annual loss}.

Step 1: Primary evaluations of experts persons p₁, p₂ and p₃ are: $\begin{matrix} X_{1} = {u_{1}, u_{2}, u_{5}}, X_{2} = {u_{1}, u_{3}, u_{5}} and \\ X_{3} = {u_{2}, u_{4}, u_{5}} . \end{matrix}$

Step 2: Real results in three different periods are represented as bipolar soft sets T₁, T₂ and T₃ as follows: $\begin{matrix} T_{1}^{+} : & A ⟶ P (U), & T_{1}^{-} : & ⌉ A ⟶ P (U) \\ e \mapsto & {\begin{matrix} {u_{1}} & if e = e_{1}, \\ {u_{1}, u_{5}} & if e = e_{2}, \\ {u_{4}, u_{5}} & if e = e_{3}, \end{matrix} & e \mapsto & {\begin{matrix} {u_{3}, u_{5}} & if e = \neg e_{1}, \\ {u_{3}} & if e = \neg e_{2}, \\ {u_{1}, u_{3}} & if e = \neg e_{3} . \end{matrix} \end{matrix}$ $\begin{matrix} T_{2}^{+} : & A ⟶ P (U), & T_{2}^{-} : & ⌉ A ⟶ P (U) \\ e \mapsto & {\begin{matrix} {u_{2}} & if e = e_{1}, \\ {u_{2}, u_{4}} & if e = e_{2}, \\ {u_{3}, u_{4}} & if e = e_{3}, \end{matrix} & e \mapsto & {\begin{matrix} {u_{1}, u_{4}} & if e = \neg e_{1}, \\ {u_{5}} & if e = \neg e_{2}, \\ {u_{1}, u_{5}} & if e = \neg e_{3} . \end{matrix} \end{matrix}$ and $\begin{matrix} T_{3}^{+} : & A ⟶ P (U), & T_{3}^{-} : & ⌉ A ⟶ P (U) \\ e \mapsto & {\begin{matrix} {u_{3}, u_{5}} & if e = e_{1}, \\ {u_{2}} & if e = e_{2}, \\ {u_{2}, u_{5}} & if e = e_{3}, \end{matrix} & e \mapsto & {\begin{matrix} {u_{1}, u_{2}} & if e = \neg e_{1}, \\ {u_{4}} & if e = \neg e_{2}, \\ {u_{1}, u_{3}} & if e = \neg e_{3} . \end{matrix} \end{matrix}$

Step 3: $\begin{matrix} {\underline{MBS}}_{T_{1}} (X_{1}) = ({u_{1}, u_{5}}, {u_{1}, u_{2}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{T_{1}} (X_{1}) = ({u_{1}, u_{2}, u_{3}, u_{5}}, {u_{1}, u_{5}}) . \\ {\underline{MBS}}_{T_{1}} (X_{2}) = ({u_{1}, u_{5}}, {u_{1}, u_{3}, u_{5}}), \\ {\bar{MBS}}_{T_{1}} (X_{2}) = ({u_{1}, u_{2}, u_{3}, u_{5}}, {u_{1}, u_{3}, u_{5}}) . \\ {\underline{MBS}}_{T_{1}} (X_{3}) = ({u_{4}, u_{5}}, {u_{2}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{T_{1}} (X_{3}) = ({u_{2}, u_{3}, u_{4}, u_{5}}, {u_{2}, u_{4}, u_{5}}) . \end{matrix}$ Similarly, $\begin{matrix} {\underline{MBS}}_{T_{2}} (X_{1}) = ({u_{1}, u_{2}, u_{5}}, {u_{1}, u_{5}}), \\ {\bar{MBS}}_{T_{2}} (X_{1}) = ({u_{1}, u_{2}, u_{5}}, {u_{1}, u_{2}, u_{3}, u_{5}}) . \\ {\underline{MBS}}_{T_{2}} (X_{2}) = ({u_{1}, u_{3}, u_{5}}, {u_{1}, u_{2}, u_{3}, u_{5}}), \\ {\bar{MBS}}_{T_{2}} (X_{2}) = ({u_{1}, u_{3}, u_{5}}, {u_{1}, u_{5}}) . \\ {\underline{MBS}}_{T_{2}} (X_{3}) = ({u_{2}, u_{4}}, {u_{2}, u_{3}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{T_{2}} (X_{3}) = ({u_{1}, u_{2}, u_{4}, u_{5}}, {u_{4}, u_{5}}) . \end{matrix}$ Also, $\begin{matrix} {\underline{MBS}}_{T_{3}} (X_{1}) = ({u_{2}, u_{5}}, {u_{2}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{T_{3}} (X_{1}) = ({u_{1}, u_{2}, u_{4}, u_{5}}, {u_{2}, u_{4}, u_{5}}) . \\ {\underline{MBS}}_{T_{3}} (X_{2}) = ({u_{3}, u_{5}}, {u_{1}, u_{3}, u_{5}}), \\ {\bar{MBS}}_{T_{3}} (X_{2}) = ({u_{1}, u_{3}, u_{4}, u_{5}}, {u_{1}, u_{3}, u_{5}}) . \\ {\underline{MBS}}_{T_{3}} (X_{3}) = ({u_{2}, u_{5}}, {u_{2}, u_{4}, u_{5}}), \\ {\bar{MBS}}_{T_{3}} (X_{3}) = ({u_{1}, u_{2}, u_{4}, u_{5}}, {u_{2}, u_{4}, u_{5}}) . \end{matrix}$

Step 4: Modified bipolar soft lower approximation matrix $\underline{mb}$ and modified bipolar soft upper approximation matrix $\bar{mb}$ are obtained as follows: $\begin{matrix} \underline{mb} & = [\begin{matrix} {(1, 0, 0, 0, 1), (- . 5, - . 5, 0, - . 5, - . 5)} & {(1, 0, 0, 0, 1), (- . 5, 0, - . 5, 0, - . 5)} & {(0, 0, 0, 1, 1), (0, - . 5, 0, - . 5, - . 5)} \\ {(1, 1, 0, 0, 1), (- . 5, 0, 0, 0, - . 5)} & {(1, 0, 1, 0, 1), (- . 5, - . 5, - . 5, 0, - . 5)} & {(0, 1, 0, 1, 0), (0, - . 5, - . 5, - . 5, - . 5)} \\ {(0, 1, 0, 0, 1), (0, - . 5, 0, - . 5, - . 5)} & {(0, 0, 1, 0, 1), (- . 5, 0, - . 5, 0, - . 5)} & {(0, 1, 0, 0, 1), (0, - . 5, 0, - . 5, - . 5)} \end{matrix}], \\ \bar{mb} & = [\begin{matrix} {(. 5, . 5, . 5, 0, . 5), (- 1, 0, 0, 0, - 1)} & {(. 5, . 5, . 5, 0, . 5), (- 1, 0, - 1, 0, - 1)} & {(0, . 5, . 5, . 5, . 5), (0, - 1, 0, - 1, - 1)} \\ {(. 5, . 5, 0, 0, . 5), (- 1, - 1, - 1, 0, - 1)} & {(. 5, 0, . 5, 0, . 5), (- 1, 0, 0, 0, - 1)} & {(. 5, . 5, 0, . 5, . 5), (0, 0, 0, - 1, - 1)} \\ {(. 5, . 5, 0, . 5, . 5), (0, - 1, 0, - 1, - 1)} & {(. 5, 0, . 5, . 5, . 5), (- 1, 0, - 1, 0, - 1)} & {(. 5, . 5, 0, . 5, . 5), (0, - 1, 0, - 1, - 1)} \end{matrix}] . \end{matrix}$

Step 5: Using equations (5.11) and (5.12), modified bipolar soft lower approximation vector $\underline{V}$ and modified bipolar soft upper approximation vector $\bar{V}$ can be calculated as follows: $\begin{matrix} \underline{V} & = (1.5, 1, - 1, - 0.5, 1.5), \\ \bar{V} & = (- 1, - 0.5, 0.5, 1.5, 4.5) . \end{matrix}$

Step 6: Decision vector is obtained as: $V = \underline{V} \oplus \bar{V} = (0.5, 0.5, - 0.5, 1, 6)$ .

Step 7: As $max_{i \in I_{n}} u_{i} = u_{5} = 6$ . So u₅ is the optimal element and u₃ is the worst element.

6 Comparison of bipolar soft rough sets and (MRBS-sets)

In this section, we provide some theoretical comparisons of bipolar soft rough sets and modified rough bipolar soft sets.

In order to prove that ${\underline{S}}_{P^{+}} (X \cap Y) = {\underline{S}}_{P^{+}} (X) \cap {\underline{S}}_{P^{+}} (Y),$ there is strong condition on positive soft set to be intersection complete. However in Proving ${\underline{(X \cap Y)}}_{φ^{+}} = {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}}$ no such condition is required. Similarly, to prove ${\underline{S}}_{P^{+}} (U) = U = {\bar{S}}_{P^{+}} (U)$ , the bipolar soft set (f, g: A) must be full bipolar soft set. However in proving ${\underline{U}}_{φ^{+}} = U = {\bar{U}}_{φ^{+}}$ no such condition is required. Also in order to prove ${\underline{S}}_{P^{+}} (X) \cap {\underline{S}}_{P^{-}} (X) = \emptyset$ and ${\bar{S}}_{P^{+}} (X) \cap {\bar{S}}_{P^{-}} (X) = \emptyset$ in [18] Karaaslan and Çağman employed a strong condition on bipolar soft set (f, g : A) to be semi-intersection. However there is no such condition is required to prove ${\underline{X}}_{φ^{+}} \cap {\underline{X}}_{ψ^{-}} = \emptyset$ and ${\bar{X}}_{φ^{+}} \cap {\bar{X}}_{ψ^{-}} = \emptyset$ .

Moreover, the measure of accuracy of modified rough bipolar soft sets can be greater than the measure of accuracy of bipolar soft rough sets in some cases.

Example 6.1. As an illustrate, let $(f, g : A) \in BSS (U)$ , where U = {u₁, u₂, u₃, u₄, u₅, u₆}, A =

{e₁, e₂, e₃, e₄} and ⌉A = {¬ e₁, ¬ e₂, ¬ e₃, ¬ e₄}. Suppose that positive soft set and negative soft set of (f, g : A) are given as:

$f : A ⟶ P (U),$ $e \mapsto {\begin{matrix} {u_{5}} & if e = e_{1}, \\ {u_{1}, u_{2}} & if e = e_{2}, \\ {} & if e = e_{3}, \\ {u_{1}, u_{2}, u_{3}, u_{6}} & if e = e_{4}, \end{matrix}$ and $g : ⌉ A ⟶ P (U),$ $e \mapsto {\begin{matrix} {u_{1}, u_{2}} & if e = \neg e_{1}, \\ {u_{3}, u_{4}, u_{5}} & if e = \neg e_{2}, \\ {u_{1}, u_{3}} & if e = \neg e_{3}, \\ {u_{4}} & if e = \neg e_{4} . \end{matrix}$

Let $X = {u_{1}, u_{4}} \subseteq U$ , X^c = {u₂, u₃, u₅, u₆} . The bipolar soft rough approximations of X are given as: $\begin{matrix} {\underline{S}}_{P^{+}} (X) & = \emptyset, \\ {\bar{S}}_{P^{+}} (X) & = {u_{1}, u_{2}, u_{3}, u_{6}}, \\ {\underline{S}}_{P^{-}} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ {\bar{S}}_{P^{-}} (X) & = \emptyset . \end{matrix}$ That is, $| {\underline{S}}_{P^{+}} (X) | = 0$ , $| {\bar{S}}_{P^{+}} (X) | = 4$ , $| {\underline{S}}_{P^{-}} (X) | = 5$ , $| {\bar{S}}_{P^{-}} (X) | = 0$ .

So therefore, measure of accuracy and measure of roughness for $(f, g : A) \in BSS (U)$ with respect to X are respectively given as: $\begin{matrix} A (X) = (A^{X^{+}}, A^{X^{-}}) & = (\frac{| {\underline{S}}_{P^{+}} (X) |}{| {\bar{S}}_{P^{+}} (X) |}, \frac{| {\bar{S}}_{P^{-}} (X) |}{| {\underline{S}}_{P^{-}} (X) |}) \\ = (0 / 4, 0 / 5) = (0, 0) \end{matrix}$ and $R (X) = (1, 1) - A (X) = (1, 1) - (0, 0) = (1, 1) .$

From above mappings f and g, we can obtain the tabular representation of bipolar soft set (f, g : A) which is given in Table (7. (a), 7.(b)) as:

Now let P = (U, (φ, ψ)) be the MRBS-approximation space. Then with the help of Table 7. (a) and 7. (b), we can define the two mappings φ and ψ as follows: $φ : U ⟶ P (A),$ $u \mapsto {\begin{matrix} {e_{2}, e_{4}} & if u = u_{1}, \\ {e_{2}, e_{4}} & if u = u_{2}, \\ {e_{4}} & if u = u_{3}, \\ {} & if u = u_{4}, \\ {e_{1}} & if u = u_{5}, \\ {e_{4}} & if u = u_{6}, \end{matrix}$ and $ψ : U ⟶ P (⌉ A),$ $u \mapsto {\begin{matrix} {\neg e_{1}, \neg e_{3}} & if u = u_{1}, \\ {\neg e_{1}} & if u = u_{2}, \\ {\neg e_{2}, \neg e_{3}} & if u = u_{3}, \\ {\neg e_{2}, \neg e_{4}} & if u = u_{4}, \\ {\neg e_{2}} & if u = u_{5}, \\ {} & if u = u_{6} . \end{matrix}$ Using Definition 3.1, the modified rough bipolar soft approximations of X = {u₁, u₄} ⊆ U with respect to MRBS-space are $\begin{matrix} {\underline{X}}_{φ^{+}} & = {u_{4}}, \\ {\bar{X}}_{φ^{+}} & = {u_{1}, u_{2}, u_{4}}, \\ {\underline{X}}_{ψ^{-}} & = {u_{1}, u_{3}, u_{4}}, \\ {\bar{X}}_{ψ^{-}} & = {u_{4}} . \end{matrix}$ Since $| {\underline{X}}_{φ^{+}} | = 1$ , $| {\bar{X}}_{φ^{+}} | = 3$ , $| {\underline{X}}_{ψ^{-}} | = 3$ , $| {\bar{X}}_{ψ^{-}} | = 1$ .

Table 6
Comparison of Bipolar Soft Rough Sets and (MRBS-sets)

Bipolar soft rough sets Modified rough bipolar soft sets

${\underline{BS}}_{P} (X) ⊑ {\underline{BS}}_{P} (Y) \Leftrightarrow {\underline{S}}_{P^{+}} (X) \subseteq {\underline{S}}_{P^{+}} (Y) and {\underline{S}}_{P^{-}} (X) \supseteq {\underline{S}}_{P^{-}} (Y)$ ${\underline{MBS}}_{P} (X) ⊑ {\underline{MBS}}_{P} (Y) \Leftrightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}} and {\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}}$

${\bar{BS}}_{P} (X) ⊑ {\bar{BS}}_{P} (Y) \Leftrightarrow {\bar{S}}_{P^{+}} (X) \subseteq {\bar{S}}_{P^{+}} (Y) and {\bar{S}}_{P^{-}} (X) \supseteq {\bar{S}}_{P^{-}} (Y)$ ${\bar{MBS}}_{P} (X) ⊑ {\bar{MBS}}_{P} (Y) \Leftrightarrow {\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}} and {\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}}$

$X \subseteq Y \Rightarrow {\underline{S}}_{P^{+}} (X) \subseteq {\underline{S}}_{P^{+}} (Y),$ ${\bar{S}}_{P^{+}} (X) \subseteq {\bar{S}}_{P^{+}} (Y),$ ${\underline{S}}_{P^{-}} (X) \supseteq {\underline{S}}_{P^{-}} (Y)$ and ${\bar{S}}_{P^{-}} (X) \supseteq {\bar{S}}_{P^{-}} (Y) .$ $X \subseteq Y \Rightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}},$ ${\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}},$ ${\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}},$ and ${\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}} .$

${\underline{S}}_{P^{+}} (X) \subseteq {\bar{S}}_{P^{+}} (X)$ , ${\underline{X}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}},$

${\bar{S}}_{P^{-}} (X) \subseteq {\underline{S}}_{P^{-}} (X)$ , ${\bar{X}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}},$

${\underline{S}}_{P^{+}} (\emptyset) = \emptyset = {\bar{S}}_{P^{+}} (\emptyset),$ ${\underline{\emptyset}}_{φ^{+}} = \emptyset = {\bar{\emptyset}}_{φ^{+}},$

${\underline{S}}_{P^{+}} (U) = ⋃_{e \in A} f (e) = {\bar{S}}_{P^{+}} (U),$ ${\underline{U}}_{φ^{+}} = U = {\bar{U}}_{φ^{+}},$

${\underline{S}}_{P^{-}} (\emptyset) = ⋃_{\neg e \in ⌉ A} g (\neg e) = {\bar{S}}_{P^{-}} (\emptyset),$ ${\underline{\emptyset}}_{ψ^{-}} = \emptyset = {\bar{\emptyset}}_{ψ^{-}},$

${\underline{S}}_{P^{-}} (U) = \emptyset = {\bar{S}}_{P^{-}} (U),$ ${\underline{U}}_{ψ^{-}} = U = {\bar{U}}_{ψ^{-}},$

${\underline{S}}_{P^{+}} (X \cap Y) \subseteq {\underline{S}}_{P^{+}} (X) \cap {\underline{S}}_{P^{+}} (Y),$ ${\underline{(X \cap Y)}}_{φ^{+}} = {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}},$

${\bar{S}}_{P^{+}} (X \cap Y) \subseteq {\bar{S}}_{P^{+}} (X) \cap {\bar{S}}_{P^{+}} (Y),$ ${\bar{(X \cap Y)}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}},$

${\underline{S}}_{P^{+}} (X \cup Y) \supseteq {\underline{S}}_{P^{+}} (X) \cup {\underline{S}}_{P^{+}} (Y),$ ${\underline{(X \cup Y)}}_{φ^{+}} \supseteq {\underline{X}}_{φ^{+}} \cup {\underline{Y}}_{φ^{+}},$

${\bar{S}}_{P^{+}} (X \cup Y) = {\bar{S}}_{P^{+}} (X) \cup {\bar{S}}_{P^{+}} (Y),$ ${\bar{(X \cup Y)}}_{φ^{+}} = {\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}},$

${\underline{S}}_{P^{-}} (X \cap Y) = {\underline{S}}_{P^{-}} (X) \cup {\underline{S}}_{P^{-}} (Y),$ ${\underline{(X \cap Y)}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}},$

${\bar{S}}_{P^{-}} (X \cap Y) \supseteq {\bar{S}}_{P^{-}} (X) \cup {\bar{S}}_{P^{-}} (Y),$ ${\bar{(X \cap Y)}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}},$

${\underline{S}}_{P^{-}} (X \cup Y) \subseteq {\underline{S}}_{P^{-}} (X) \cap {\underline{S}}_{P^{-}} (Y),$ ${\underline{(X \cup Y)}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}},$

${\bar{S}}_{P^{-}} (X \cup Y) = {\bar{S}}_{P^{-}} (X) \cap {\bar{S}}_{P^{-}} (Y),$ ${\bar{(X \cup Y)}}_{ψ^{-}} \supseteq {\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}} .$

Bipolar soft rough sets	Modified rough bipolar soft sets
${\underline{BS}}_{P} (X) ⊑ {\underline{BS}}_{P} (Y) \Leftrightarrow {\underline{S}}_{P^{+}} (X) \subseteq {\underline{S}}_{P^{+}} (Y) and {\underline{S}}_{P^{-}} (X) \supseteq {\underline{S}}_{P^{-}} (Y)$	${\underline{MBS}}_{P} (X) ⊑ {\underline{MBS}}_{P} (Y) \Leftrightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}} and {\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}}$
${\bar{BS}}_{P} (X) ⊑ {\bar{BS}}_{P} (Y) \Leftrightarrow {\bar{S}}_{P^{+}} (X) \subseteq {\bar{S}}_{P^{+}} (Y) and {\bar{S}}_{P^{-}} (X) \supseteq {\bar{S}}_{P^{-}} (Y)$	${\bar{MBS}}_{P} (X) ⊑ {\bar{MBS}}_{P} (Y) \Leftrightarrow {\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}} and {\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}}$
$X \subseteq Y \Rightarrow {\underline{S}}_{P^{+}} (X) \subseteq {\underline{S}}_{P^{+}} (Y),$ ${\bar{S}}_{P^{+}} (X) \subseteq {\bar{S}}_{P^{+}} (Y),$ ${\underline{S}}_{P^{-}} (X) \supseteq {\underline{S}}_{P^{-}} (Y)$ and ${\bar{S}}_{P^{-}} (X) \supseteq {\bar{S}}_{P^{-}} (Y) .$	$X \subseteq Y \Rightarrow {\underline{X}}_{φ^{+}} \subseteq {\underline{Y}}_{φ^{+}},$ ${\bar{X}}_{φ^{+}} \subseteq {\bar{Y}}_{φ^{+}},$ ${\underline{X}}_{ψ^{-}} \subseteq {\underline{Y}}_{ψ^{-}},$ and ${\bar{X}}_{ψ^{-}} \subseteq {\bar{Y}}_{ψ^{-}} .$
${\underline{S}}_{P^{+}} (X) \subseteq {\bar{S}}_{P^{+}} (X)$ ,	${\underline{X}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}},$
${\bar{S}}_{P^{-}} (X) \subseteq {\underline{S}}_{P^{-}} (X)$ ,	${\bar{X}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}},$
${\underline{S}}_{P^{+}} (\emptyset) = \emptyset = {\bar{S}}_{P^{+}} (\emptyset),$	${\underline{\emptyset}}_{φ^{+}} = \emptyset = {\bar{\emptyset}}_{φ^{+}},$
${\underline{S}}_{P^{+}} (U) = ⋃_{e \in A} f (e) = {\bar{S}}_{P^{+}} (U),$	${\underline{U}}_{φ^{+}} = U = {\bar{U}}_{φ^{+}},$
${\underline{S}}_{P^{-}} (\emptyset) = ⋃_{\neg e \in ⌉ A} g (\neg e) = {\bar{S}}_{P^{-}} (\emptyset),$	${\underline{\emptyset}}_{ψ^{-}} = \emptyset = {\bar{\emptyset}}_{ψ^{-}},$
${\underline{S}}_{P^{-}} (U) = \emptyset = {\bar{S}}_{P^{-}} (U),$	${\underline{U}}_{ψ^{-}} = U = {\bar{U}}_{ψ^{-}},$
${\underline{S}}_{P^{+}} (X \cap Y) \subseteq {\underline{S}}_{P^{+}} (X) \cap {\underline{S}}_{P^{+}} (Y),$	${\underline{(X \cap Y)}}_{φ^{+}} = {\underline{X}}_{φ^{+}} \cap {\underline{Y}}_{φ^{+}},$
${\bar{S}}_{P^{+}} (X \cap Y) \subseteq {\bar{S}}_{P^{+}} (X) \cap {\bar{S}}_{P^{+}} (Y),$	${\bar{(X \cap Y)}}_{φ^{+}} \subseteq {\bar{X}}_{φ^{+}} \cap {\bar{Y}}_{φ^{+}},$
${\underline{S}}_{P^{+}} (X \cup Y) \supseteq {\underline{S}}_{P^{+}} (X) \cup {\underline{S}}_{P^{+}} (Y),$	${\underline{(X \cup Y)}}_{φ^{+}} \supseteq {\underline{X}}_{φ^{+}} \cup {\underline{Y}}_{φ^{+}},$
${\bar{S}}_{P^{+}} (X \cup Y) = {\bar{S}}_{P^{+}} (X) \cup {\bar{S}}_{P^{+}} (Y),$	${\bar{(X \cup Y)}}_{φ^{+}} = {\bar{X}}_{φ^{+}} \cup {\bar{Y}}_{φ^{+}},$
${\underline{S}}_{P^{-}} (X \cap Y) = {\underline{S}}_{P^{-}} (X) \cup {\underline{S}}_{P^{-}} (Y),$	${\underline{(X \cap Y)}}_{ψ^{-}} \subseteq {\underline{X}}_{ψ^{-}} \cap {\underline{Y}}_{ψ^{-}},$
${\bar{S}}_{P^{-}} (X \cap Y) \supseteq {\bar{S}}_{P^{-}} (X) \cup {\bar{S}}_{P^{-}} (Y),$	${\bar{(X \cap Y)}}_{ψ^{-}} = {\bar{X}}_{ψ^{-}} \cap {\bar{Y}}_{ψ^{-}},$
${\underline{S}}_{P^{-}} (X \cup Y) \subseteq {\underline{S}}_{P^{-}} (X) \cap {\underline{S}}_{P^{-}} (Y),$	${\underline{(X \cup Y)}}_{ψ^{-}} = {\underline{X}}_{ψ^{-}} \cup {\underline{Y}}_{ψ^{-}},$
${\bar{S}}_{P^{-}} (X \cup Y) = {\bar{S}}_{P^{-}} (X) \cap {\bar{S}}_{P^{-}} (Y),$	${\bar{(X \cup Y)}}_{ψ^{-}} \supseteq {\bar{X}}_{ψ^{-}} \cup {\bar{Y}}_{ψ^{-}} .$

Table 7

Tabular representation of (f, g: A)

(a)
f	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
e ₁	0	0	0	0	1	0
e ₂	1	1	0	0	0	0
e ₃	0	0	0	0	0	0
e ₄	1	1	1	0	0	1
(b)
g	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
¬e₁	1	1	0	0	0	0
¬e₂	0	0	1	1	1	0
¬e₃	1	0	1	0	0	0
¬e₄	0	0	0	1	0	0

So measure of accuracy and measure of roughness for modified rough bipolar soft set (MRBS-set) with respect to X are respectively given as: $\begin{matrix} A (X) & = (A^{X^{+}}, A^{X^{-}}) = (\frac{| {\underline{X}}_{φ^{+}} |}{| {\bar{X}}_{φ^{+}} |}, \frac{| {\bar{X}}_{ψ^{-}} |}{| {\underline{X}}_{ψ^{-}} |}) \\ = (1 / 3, 1 / 3) = (0.333, 0.333) \end{matrix}$ and $\begin{matrix} R (X) & = (1, 1) - A (X) = (1, 1) - (0.333, 0.333) \\ = (0.666, 0.666) . \end{matrix}$ Clearly, $A (X) > A (X)$ . Hence, the measure of accuracy of modified rough bipolar soft sets can be greater than the measure of accuracy of bipolar soft rough sets in some cases.

7 Conclusion

In this article, we have discussed in detail the fundamentals of modified rough bipolar soft sets (MRBS-sets), by defining φ - lower positive, φ - upper positive, ψ - lower negative and ψ - upper negative MBSR - approximations of MRBS-sets in the modified rough bipolar soft approximation space. The notion is further explored by studying its structural properties. Based upon these properties, the accuracy measure and roughness measure for MRBS-sets are also provided. We finally gave an application to show the method can be successfully applied to some problems including uncertainty in the real world. In future, based upon the defined notions and operations in this article, researchers may study on algebraic structures of MRBS-sets. Modeling of supported physical phenomenon is our next goal. Another prospective direction is to study the topological properties and similarity measures of MRBS-sets in order to explore for a solid foundation of the research work and development of working methodologies. Also notion of modified rough bipolar soft sets may be extended to modified rough bipolar fuzzy soft sets and effective decision making methods may be developed.

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Modified rough bipolar soft sets

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 2 Tabular representation of (f, g: A) (a) f u 1 u 2 u 3 u 4 u 5 u 6 e 1 1 0 0 0 0 1 e 2 0 0 1 0 0 0 e 3 0 0 0 0 0 0 e 4 1 1 0 0 1 0 (b) g u 1 u 2 u 3 u 4 u 5 u 6 ¬e1 0 0 1 0 0 0 ¬e2 0 0 0 0 1 0 ¬e3 0 1 1 0 0 1 ¬e4 0 0 1 0 0 0

Table 4 Tabular representation of (f, g: A) (a) f u 1 u 2 u 3 u 4 u 5 u 6 e 1 1 1 1 1 1 1 e 2 1 0 1 1 0 1 e 3 1 0 0 1 0 1 e 4 0 1 0 0 1 0 (b) g u 1 u 2 u 3 u 4 u 5 u 6 ¬e1 0 0 0 0 0 0 ¬e2 0 1 0 0 1 0 ¬e3 0 0 1 0 1 0 ¬e4 1 0 0 1 0 0

References

Table 2
Tabular representation of (f, g: A)

(a)

f u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

e ₁ 1 0 0 0 0 1

e ₂ 0 0 1 0 0 0

e ₃ 0 0 0 0 0 0

e ₄ 1 1 0 0 1 0

(b)

g u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

¬e₁ 0 0 1 0 0 0

¬e₂ 0 0 0 0 1 0

¬e₃ 0 1 1 0 0 1

¬e₄ 0 0 1 0 0 0

Table 4
Tabular representation of (f, g: A)

(a)

f u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

e ₁ 1 1 1 1 1 1

e ₂ 1 0 1 1 0 1

e ₃ 1 0 0 1 0 1

e ₄ 0 1 0 0 1 0

(b)

g u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

¬e₁ 0 0 0 0 0 0

¬e₂ 0 1 0 0 1 0

¬e₃ 0 0 1 0 1 0

¬e₄ 1 0 0 1 0 0