# Bipolar Soft Sets: Relations between Them and Ordinary Points and Their Applications

Authors: Tareq M. Al-shami
Journal: Complexity
 (2021)

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## Abstract

Bipolar soft set is formulated by two soft sets; one of them provides us the positive information and the other provides us the negative information. The philosophy of bipolarity is that human judgment is based on two sides, positive and negative, and we choose the one which is stronger. In this paper, we introduce novel belong and nonbelong relations between a bipolar soft set and an ordinary point. These relations are considered as one of the unique characteristics of bipolar soft sets which are somewhat expression of the degrees of membership and nonmembership of an element. We discuss essential properties and derive the sufficient conditions of some equivalence of these relations. We also define the concept of soft mappings between two classes of bipolar soft sets and study the behaviors of an ordinary point under these soft mappings with respect to all relations introduced herein. Then, we apply bipolar soft sets to build an optimal choice application. We give an algorithm of this application and show the method for implementing this algorithm by an illustrative example. In conclusion, it can be noted that the relations defined herein give another viewpoint to explore the concepts of bipolar soft topology, in particular, soft separation axioms and soft covers.


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## Body

## 1. Introduction

Many problems in engineering, artificial intelligence, economy, environmental science, social science, etc. involve data that contain ambiguity/vagueness. Therefore, traditional methods which were based on the exact case may not be convenient for solving or modeling them.From this point, the need of new theories that help to surpass these types of instabilities arose. With the passage of time, engineers and mathematicians found alternative approaches to solve the problems that contain ambiguity/vagueness such as probability theory, fuzzy set [1], intuitionistic fuzzy set [2], and rough set [3].However, all these tools require the prespecification of some parameters to start with, for example, an equivalence relation in rough set theory and density function in probability theory. According to the fuzzy set theory, the difficulties in many problems appear in two sides: the first one is how we can determine a membership function for each particular case, and the second difficulty is the extremely individual characteristic of a membership function. That is, everyone understands the meaning of the membership function equal to 0.85 in his own manner.To cope with these difficulties, Molodtsov [4] proposed a new approach, namely, soft sets. Simply, soft set is defined as a map of a set of parameters into the power set of the universe of discourse. He demonstrated the efficiency of soft sets in handling complicated problems compared with the probability theory and fuzzy sets theory. Then, many researchers have studied the properties, operations, and applications of soft set theory (see, for example, [5–13]). The authors of [14, 15] explained the relationship between soft set, rough set, and fuzzy set.In recent years, a number of authors have extensively explored some extensions of soft set. These studies go in two directions: The first one is given by generalizing the structure of soft set. This leads to defining double framed soft set [16], bipolar soft set [17], binary soft set [18], and N-soft set [19]. The second one is introduced by combining soft set (or its updated forms) with rough set or fuzzy set or both. This leads to defining fuzzy soft set [12], fuzzy bipolar soft set [20], bipolar fuzzy soft set [21], soft rough set [14], bipolar soft rough set [22], and modified rough bipolar soft set [23].In [24], the authors pointed out that human decisions rely on two types of information having positive and negative flavor. In this case, if we determine a set of parameters which gives us positive data, we also need to know an associated set of oppositely meaning parameters called the “not set of parameters.” For instance, if a parameter a stands for a “tall” characteristic, then nota stands for a characteristic of “not tall.” It should be noted that a characteristic of “not tall” does not mean “short.” That is, the members who are tall and not tall need not be the universal set. Due to the importance of providing positive and negative aspects of data at a time, Shabir and Naz [17] formulated the idea of bipolar soft sets and discussed its application to decision-making problems. After this study, bipolar soft sets are gaining momentum among researchers. In 2015, Karaaslan and Karataş [25] redefined a bipolar soft set using a bijective map between a set of parameters and its negative. They also provided a decision-making manner using bipolar soft set with the aid of an example. In [26], the authors revealed some algebraic structures of bipolar soft sets. It was followed by Karaaslan et al. [27] who studied a group structure on bipolar soft sets.In 2017, Shabir and Bakhtawar [28] first introduced the concept of bipolar soft topological spaces over a crisp set along with an investigation into bipolar soft compactness and connectedness. Then, Öztürk [29] further discussed the concepts of interior and closure operators, basis, and subspace in the bipolar soft topological spaces. Fadel and Hassan [30] presented the concepts of bipolar soft separation axioms and established fundamental properties. Recently, Fadel and Dzul-Kifli [31] have generalized the concept of bipolar soft topological spaces given in [28] by redefining it on a bipolar soft set. They have presented its main notions and described properties along with some illustrative examples.In 2018, the authors of [32] came up with the idea of partial belong and total nonbelong relations between an ordinary point and soft set which somewhat indicate the degree of membership and nonmembership of an element. In fact, these relations widely open the door to the studying and redefining of many soft topological notions and the obtaining of many fruitful properties. Some applications of these two relations in the domains of soft separation axioms and decision-making problems were introduced in [33–35]. The authors of [36–38] applied these relations to study separation axioms on soft topological ordered spaces and supra soft topological ordered spaces.The rest of this paper is organized as follows. Section2 involves some operations and properties of bipolar soft sets. In Section 3, we define five sorts of belong relations between bipolar soft set and ordinary point called positively partial belong, negatively partial belong, partial belong, positively total belong, and negatively total belong relations; we also define six sorts of nonbelong relations between a bipolar soft set and an ordinary point called positively partial nonbelong, negatively partial nonbelong, partial nonbelong, positively total nonbelong, negatively total nonbelong, and total nonbelong relations. Then, we ascertained their behaviors with respect to the operations of soft union and intersection. Later, we define the concept of soft mappings between two classes of weak bipolar soft sets and discuss the relationship between an ordinary point and its image and preimage with respect to the different types of belong and nonbelong relations. In Section 4, we make use of bipolar soft sets to construct an application of optimum choice and present its algorithm. We give a practical example to illustrate how this algorithm can be applied. In the end, we outline the main obtained results and suggest some future work in Section 5.


## 2. Preliminaries

In the following, we recall some definitions related to bipolar soft sets.Through this paper,A,B,C,D,E,M,N denote the sets of parameters, and X,Y denote the initial universal sets.Definition 1 (see [4]).
A mapf of A into the power set of a nonempty set X is called a soft set over X, where X is an initial universal set and A is a set of parameters.
A soft set is symbolized by an ordered pairf,A and it is expressed as a set of ordered pairs:(1)f,A=a,fa:a∈Aandfa∈2X.Definition 2 (see [17]).
A bipolar soft set is a triplef,g,A over a nonempty set X with a set of parameters A, where f:A⟶2X and g:¬A⟶2X are two crisp maps such that fa∩g¬a=∅ for each a∈A.
A bipolar soft set is expressed as a set of ordered triples:(2)f,g,A=a,fa,g¬a:a∈Aandfa,g¬a∈2X.
A class of all bipolar soft sets defined overX with all sets of parameters which are subsets of A is symbolized by CXA.
Henceforth, we consider that any bipolar soft set is defined onX≠∅, unless otherwise specified.Example 1.
LetX=x1,x2,…,x6 be the universe containing set six cars and A=a1,a2,a3,a4 be a set of parameters, where a1,a2,a3, and a4 stand for “expensive,” “in good repair,” “red colored,” and “made in Japan,” respectively.
Letf:A⟶2X and g:¬A⟶2X be two maps given as follows:(3)fa1=x3,x4,g¬a1=x1,fa2=x3,x5,x6,g¬a2=∅,fa3=x1,x2,x4,g¬a3=x3,x5,x6,fa4=x3,x6,g¬a4=x1,x2,x4,x5.
Now, we can describe this system using a bipolar soft set as follows:(4)f,g,A=a1,x3,x4,x3,x1,a2,x3,x5,x6,∅,a3,x1,x2,x4,x3,x5,x6,a4,x3,x6,x1,x2,x4,x5.Definition 3 (see [17]).
A bipolar soft setf,g,A is said to be(i)
A relative null bipolar soft set iffa equals the empty set and g¬a equals the universal set for each a∈A. It is symbolized by Φ,X,A.(ii)
A relative absolute bipolar soft set iffa equals the universal set and g¬a equals the empty set for each a∈A. It is symbolized by X,Φ,A.Definition 4 (see [17]).
The intersection of two bipolar soft setsf1,f2,A and g1,g2,B is a bipolar soft set h1,h2,C such that C=A∩B≠∅ and the two mappings h1:C⟶2X and h2:¬C⟶2X are given by(5)h1c=f1c∩g1c,h2¬c=f2¬c∪g2¬c.
It is symbolized byf1,f2,A∩˜g1,g2,B.Definition 5 (see [17]).
The union of two bipolar soft setsf1,f2,A and g1,g2,B is a bipolar soft set h1,h2,C, where C=A∪B and the two mappings h1:C⟶2X and h2:¬C⟶2X are given by(6)h1c=f1c:c∈A\B,g1c:c∈B\A,f1c∪g1c:c∈A∩B,h2¬c=f2¬c:¬c∈¬A\¬B,g2¬c:¬c∈¬B\¬A,f2¬c∩g2¬c:¬c∈¬A∩¬B.
It is symbolized byf1,g1,A∪˜f2,g2,B.Definition 6 (see [17]).
A bipolar soft setf,g,A is said to be a subset of a bipolar soft set h,l,B, denoted by f,g,A⊆˜h,l,B, provided that(i)
A⊆B.(ii)
For alla∈A, we have fa⊆ha and l¬a⊆g¬a.
The bipolar soft setsf,g,A and h,l,B are said to be soft equal if f,g,A⊆˜h,l,B and h,l,B⊆˜f,g,A.Definition 7 (see [17]).
The relative complement of a bipolar soft setf,g,A is a bipolar soft set f,g,A′=f′,g′,A, where f′:A⟶2X and g′:¬A⟶2X are defined as follows:(7)f′a=g¬a,g′¬a=fa,for alla∈Aand¬a∈¬A.


## 3. Belong and Nonbelong Relations between Bipolar Soft Sets and Ordinary Points

In this section, we initiate five types of memberships and six types of nonmemberships between bipolar soft set and ordinary point and ascertain the relationships between them. We investigate their main properties in terms of soft union and intersection operators, the product of bipolar soft sets and soft mappings.Definition 8.
Letf,g,A be a bipolar soft set and x∈X. We say that(i)
x⋐ppf,g,A, reading as x positively partially belongs to f,g,A, if x∈fa for some a∈A.(ii)
x⋐npf,g,A, reading as x negatively partially belongs to f,g,A, if x∈g¬a for some ¬a∈¬A.(iii)
x⋐pf,g,A, reading as x partially belongs to f,g,A, if x∈fa and x∈g¬a′ for some a∈A and ¬a′∈¬A.(iv)
x∈ptf,g,A, reading as x positively totally belongs to f,g,A, if x∈fa for all a∈A.(v)
x∈ntf,g,A, reading as x negatively totally belongs to f,g,A, if x∈g¬a for all ¬a∈¬A.Definition 9.
Letf,g,A be a bipolar soft set and x∈X. We say that(i)
x⋐ppf,g,A, reading as x does not positively partially belong to f,g,A, if x∉fa for some a∈A.(ii)
x⋐npf,g,A, reading as x does not negatively partially belong to f,g,A, if x∉g¬a for some ¬a∈¬A.(iii)
x⋐pf,g,A, reading as x does not partially belong to f,g,A, if x∉fa and x∉g¬a′ for some a∈A and ¬a′∈¬A.(iv)
x∉ptf,g,A, reading as x does not positively totally belong to f,g,A, if x∉fa for all a∈A.(v)
x∉ntf,g,A, reading as x does not negatively totally belong to f,g,A, if x∉g¬a for all ¬a∈¬A.(vi)
x∉tf,g,A, reading as x does not totally belong to f,g,A, if x∉fa and x∉g¬a for all a∈A and ¬a∈¬A.
To well understand the results initiated in this work, we give the following example and remark.Example 2.
LetA=a1,a2,a3 be a set of parameters and f,g,A be bipolar soft set over X=x1,x2,…,x10 defined as follows:(8)f,g,A=a1,x1,x2,x4,x10,a2,∅,x4,x10,a3,x2,x3,x4,x5,x10.
Then, it follows from Definition7 that f′,g′,A=g,f,A. We note the following:(9)f,g,A∩˜f′,g′,A=a1,∅,x1,x2,x4,x10,a2,∅,x4,x10,a3,∅,x2,x3,x4,x5,x10≠Φ,X,A.
That is, the intersection of a bipolar soft set and its relative complement need not be the null bipolar soft set.(10)f,g,A∪˜f′,g′,A=a1,x1,x2,x4,x10,∅,a2,x4,x10,∅,a3,x2,x3,x4,x5,x10,∅≠X,Φ,A.
That is, the union of a bipolar soft set and its relative complement need not be the absolute bipolar soft set.Remark 1.
The possibility of existence and nonexistence of an element in the same place is well-known in the quantum physics, that is, corresponding of a thing and its opposite. This matter also occurs here with respect to positively partial belong and positively partial nonbelong relations; negatively partial belong and negatively partial nonbelong relations; and partial belong and partial nonbelong relations. For instance, in Example2, it can be seen that(11)x1⋐ppf,g,A,x1⋐ppf,g,A,x2⋐npf,g,A,x2⋐npf,g,A,x2⋐pf,g,A,x2⋐pf,g,A.Proposition 1.
For two bipolar soft setsf,g,A and h,l,A and x∈X, we have the following results:(i)
x⋐ppf,g,A⟺x⋐npf′,g′,A.(ii)
x⋐pf,g,A⟺x⋐pf′,g′,A.(iii)
x∈ptf,g,A⟺x∈ntf′,g′,A.Proof.
(i)
x⋐ppf,g,A⇔x∈fa for some a∈A⇔x∈g′¬a=fa for some ¬a∈¬A⇔x⋐npf′,g′,A.(ii)
x⋐pf,g,A⇔x∈fa and x∈g¬a for some a∈A and ¬a∈¬A⇔x∈f′a=g¬a and x∈g′¬a=fa for some a∈A and ¬a∈¬A⇔x⋐pf′,g′,A.(iii)
x∈ptf,g,A⇔x∈fa for all a∈A⇔x∈g′¬a=fa for all ¬a∈¬A⇔x∈ntf′,g′,A.
The proofs of the following results follow from Definition8.Proposition 2.
For two bipolar soft setsf,g,A and h,l,A and x∈X, we have the following results:(i)
x⋐pf,g,A⇔x⋐ppf,g,A and x⋐npf,g,A.(ii)
x∈ptf,g,A⇔x⋐ppf,g,A.(iii)
x∈ntf,g,A⇔x⋐npf,g,A.Proposition 3.
For two bipolar soft setsf,g,A and h,l,A and x∈X, we have the following results:(i)
x⋐pf,g,A⇔x⋐ppf,g,A and x⋐npf,g,A.(ii)
x∉pf,g,A⇔x∉ppf,g,A and x∉npf,g,A.(iii)
x∉ptf,g,A⇔x⋐ppf,g,A.(iv)
x∉ntf,g,A⇔x⋐npf,g,A.(v)
x∉tf,g,A⇔x⋐pf,g,A.
To show that the converse of (ii) and (iii) of Proposition2 and (ii) to (v) of Proposition 3 fails, we give the following example.Example 3.
In Example2, we note the following:(i)
x1⋐ppf,g,A and x5⋐npf,g,A, but x1∈ptf,g,A and x5∈ntf,g,A do not hold true.(ii)
x2⋐pf,g,A, x2⋐ppf,g,A and x2⋐npf,g,A, but x2∉tf,g,A, x2∉ptf,g,A, and x2∉ntf,g,A do not hold true.Proposition 4.
Letf,g,A and h,l,A be bipolar soft sets such that f,g,A⊆˜h,l,A. Then, we have the following results:(i)
ifx⋐ppf,g,A (resp., x∈ptf,g,A), then x⋐pph,l,A (resp., x∈pth,l,A).(ii)
ifx∈nth,l,A, then x∈ntf,g,A.(iii)
ifx⋐pph,l,A (resp., x∉pth,l,A), then x⋐ppf,g,A (resp., x∉ptf,g,A).(iv)
ifx∉ntf,g,A, then x∉nth,l,A.Proof.
The proof is straightforward.Proposition 5.
For two bipolar soft setsf,g,A and h,l,A and x∈X, we have the following results:(i)
x⋐ppf,g,A or x⋐pph,l,A⇔x⋐ppf,g,A∪˜h,l,A.(ii)
x⋐npf,g,A∪˜h,l,A⇒x⋐npf,g,A and x⋐nph,l,A.(iii)
x⋐pf,g,A∪˜h,l,A⇒x⋐pf,g,A or x⋐ph,l,A.(iv)
x∈ptf,g,A or x∈pth,l,A⇒x∈ptf,g,A∪˜h,l,A.(v)
x∈ntf,g,A and x∈nth,l,A⇒x∈ntf,g,A∪˜h,l,A.(vi)
x⋐ppf,g,A∩˜h,l,A⇒x⋐ppf,g,A and x⋐pph,l,A.(vii)
x⋐npf,g,A∩˜h,l,A⇒x⋐npf,g,A or x⋐nph,l,A.(viii)
x⋐pf,g,A∩˜h,l,A⇒x⋐pf,g,A or x⋐ph,l,A.(ix)
x∈ptf,g,A and x∈pth,l,A⇒x∈ptf,g,A∩˜h,l,A.(x)
x∈ntf,g,A and x∈nth,l,A⇒x∈ntf,g,A∩˜h,l,A.Proof.
We will just prove (i), (iv), (v), (vi), and (ix).
Sincef,g,A and h,l,A are subsets of f,g,A∪˜h,l,A, then the necessary parts of (i), (iv), and (vi) hold; since f,g,A∩˜h,l,A are subsets of f,g,A and h,l,A, then the sufficient parts of (v) and (ix) hold.
To prove the sufficient part of (i), letx⋐ppf,g,A∪˜h,l,A. Then, x∈fa∪ha for some a∈A. Therefore, x∈fa or ha for some a∈A, and hence x⋐ppf,g,A or x⋐pph,l,A.
To prove the necessary part of (v), letx∈ntf,g,A and x∈nth,l,A. Then, for all ¬a∈¬A, we have x∈g¬a and x∈l¬a. Therefore, x∈g¬a∩l¬a for all ¬a∈¬A. Hence, x∈ntf,g,A∪˜h,l,A.
Similarly, one can prove the necessary part of (ix).
We provide the next example to clarify that the converse of results (ii), (iii), (vi), and (viii) to (x) of Proposition5 fails.Example 4.
LetA=a1,a2 be a set of parameters and f,g,A,h,l,A be bipolar soft sets over X=x1,x2,x3,x4,x5,x6,x7 defined as follows:(12)f,g,A=a1,x1,x3,x5,x7,a2,x4,x6,x3,x5,h,l,A=a1,x1,x4,x3,x4,x5,a2,x2,x1,x3,x5,x7.
Then,f,g,A∪˜h,l,A=a1,x1,x3,x4,x5,a2,x2,x4,x6,x3,x5, and f,g,A∩˜h,l,A=a1,x1,x3,x4,x5,x7,a2,∅,x1,x3,x5,x7
We note the following:(i)
x7⋐npf,g,A and x7⋐nph,l,Ax7⋐nph,l,A, but x7⋐npf,g,A∪˜h,l,A does not hold true.(ii)
x7⋐pf,g,A and x7⋐ph,l,Ax7⋐ph,l,A, but x7⋐pf,g,A∪˜h,l,A and x7⋐pf,g,A∩˜h,l,A do not hold true.(iii)
x4⋐ptf,g,A∪˜h,l,Ax4⋐ptf,g,A∪˜h,l,A, but x4⋐ptf,g,A or x4⋐pth,l,A does not hold true.(iv)
x4⋐ppf,g,A and x4⋐pph,l,Ax4⋐pph,l,A, but x4⋐ppf,g,A∩˜h,l,A does not hold true.(v)
x7⋐ntf,g,A∩˜h,l,Ax7⋐ntf,g,A∩˜h,l,A, but x7⋐ntf,g,A and x4⋐pth,l,A do not hold true.
One can prove the following result similarly.Proposition 6.
For two bipolar soft setsf,g,A and h,l,A over X and x∈X, we have the following results:(i)
x⋐ppf,g,A∪˜h,l,A⇒x⋐ppf,g,A and x⋐pph,l,A.(ii)
x⋐npf,g,A∪˜h,l,A⇔x⋐npf,g,A and x⋐nph,l,A.(iii)
x⋐pf,g,A∪˜h,l,A⇒x⋐pf,g,A and x⋐ph,l,A.(iv)
x∉ptf,g,A∪˜h,l,A⇔x∉ptf,g,A and x∉pth,l,A.(v)
x∉ntf,g,A or x∉nth,l,A⇒x∉ntf,g,A∪˜h,l,A.(vi)
x⋐ppf,g,A or x⋐pph,l,A⇒x⋐ppf,g,A∩˜h,l,A.(vii)
x⋐npf,g,A∩˜h,l,A⇒x⋐npf,g,A or x⋐nph,l,A.(viii)
x⋐pf,g,A∩˜h,l,A⇒x⋐pf,g,A or x⋐ph,l,A.(ix)
x∉ptf,g,A or x∉pth,l,A⇒x∉ptf,g,A∩˜h,l,A.(x)
x∉ntf,g,A∩˜h,l,A⇔x∉ntf,g,A and x∉nth,l,A.(xi)
x∉tf,g,A; x∉th,l,A⇔x∉tf,g,A∩˜h,l,A; and x∉tf,g,A∪˜h,l,A.Definition 10.
A bipolar soft setf,g,A is said to be stable if there are two disjoint subsets U,V of X such that fa=U for each a∈A and g¬a=V and for each ¬a∈¬A.
f,g,A is said to be positive stable if V=∅; and f,g,A is said to be negative stable if U=∅.
It is clear that positive (negative) stable bipolar soft set is stable; however, the converse is not true.Proposition 7.
Letf,g,A be a stable bipolar soft set. Then,(i)
x⋐ppf,g,A⇔x⋐ptf,g,A.(ii)
x⋐npf,g,A⇔x⋐ntf,g,A.(iii)
x⋐pf,g,A⇔x∉tf,g,A.(iv)
x⋐ppf,g,A⇔x⋐ptf,g,A.(v)
x⋐npf,g,A⇔x⋐ntf,g,A.Proof.
Sincef,g,A is a stable bipolar soft set, then there are two disjoint subsets U,V of X such that fa=U for each a∈A and g¬a=V for each ¬a∈¬A. Therefore, the following properties hold:(i)
x∈fa for some a∈A if and only if x∈fa for each a∈A.(ii)
x∈g¬a for some ¬a∈¬A if and only if x∈g¬a for each ¬a∈¬A.(iii)
x∉fa for some a∈A and x∉g¬a for some ¬a∈¬A if and only if x∉fa for each a∈A and x∉g¬a for each ¬a∈¬A.(iv)
x∉fa for some a∈A if and only if x∉fa for each a∈A.(v)
x∉g¬a for some ¬a∈¬A if and only if x∉g¬a for each ¬a∈¬A.
Hence, the desired results are proved.Corollary 1.
Letf,g,A be a positive stable bipolar soft set. Then,(i)
x⋐ppf,g,A⇔x⋐ptf,g,A.(ii)
x⋐ppf,g,A⇔x⋐ptf,g,A⇔x⋐pf,g,A⇔x∉tf,g,A.Corollary 2.
Letf,g,A be a negative stable bipolar soft set. Then,(i)
x⋐npf,g,A⇔x⋐ntf,g,A.(ii)
x⋐npf,g,A⇔x⋐ntf,g,A⇔x⋐pf,g,A⇔x∉tf,g,A.Definition 11.
The Cartesian product of two bipolar soft setsf,g,A and h,l,B, denoted by f×h,g×l,A×B, is defined as f×ha,b=fa×hb for each a,b∈A×B and g×l¬a,¬b=g¬a×l¬b for each ¬a,¬b∈¬A×¬B.Proposition 8.
We have the next four results:(i)
x,y⋐ppf,g,A×h,l,B if and only if x⋐ppf,g,A and y⋐pph,l,B.(ii)
x,y⋐npf,g,A×h,l,B if and only if x⋐npf,g,A and y⋐nph,l,B.(iii)
x,y⋐pf,g,A×h,l,B if and only if x⋐pf,g,A and y⋐ph,l,B.(iv)
x,y⋐ptf,g,A×h,l,B if and only if x⋐ptf,g,A and y⋐pth,l,B.(v)
x,y⋐ntf,g,A×h,l,B if and only if x⋐ntf,g,A and y⋐nth,l,B.Proof.
We will just prove (iii). The other cases can be proved similarly.(iii)
x,y⋐pf,g,A×h,l,B=f×h,g×l,A×B⇔x,y∈f×ha,b=fa×hb and x,y∈g×l¬a′,¬b′=g¬a′×l¬b′ for some a,b∈A×B and ¬a′,¬b′∈¬A×¬B⇔x∈fa and y∈hb for some a∈A and b∈B; x∈g¬a′ and y∈l¬b′ for some ¬a′∈¬A and ¬b′∈¬B⇔x∈fa and x∈g¬a′ for some a∈A and ¬a′∈¬A; y∈hb and y∈l¬b′ for some b∈B and ¬b′∈¬B⇔x⋐pf,g,A and y⋐ph,l,B.Definition 12.
A weak bipolar soft set is a triplef,g,A over a nonempty set X with a set of parameters A, where f:A⟶2X and g:¬A⟶2X are two crisp maps.
A weak bipolar soft set is expressed as a set of ordered triples:(13)f,g,A=a,fa,g¬a:a∈Aandfa,g¬a∈2X.
A class of all weak bipolar soft sets defined overX with all sets of parameters which are subsets of A is symbolized by WCXA.Remark 2.
It is clear that a bipolar soft set is a special case of a weak bipolar soft set, and a weak bipolar soft set is not necessarily a bipolar soft set. We will prove that the image of a bipolar soft set is not necessarily a bipolar soft set; however, the image of a weak bipolar soft set is a weak bipolar soft set. Therefore, we define a soft mapping between two classes of weak bipolar soft sets instead of two classes of bipolar soft sets.Definition 13.
A soft mappingπφ of WCXA into WCYB is a pair π,φ of crisp mappings π:X⟶Y and φ:A⟶B such that πφ is defined as follows: the image of a weak bipolar soft set f1,f2,M in WCXA is a weak bipolar soft set πφf1,f2,M=g1,g2,E in WCYB such that E=φM⊆B and the two maps g1 and g2 are given by(14)g1e=π∪λ∈φ−1e∩Mf1λ,for eache∈E,g2¬e=π∪¬λ∈¬φ−1e∩Mf2¬λ,for each¬e∈¬E.
The following example shows that the image of a bipolar soft set need not be a bipolar soft set.Example 5.
Consider thatA=a1,a2,a3 and B=b1,b2 are two sets of parameters and X=Y=x1,x2,x3 are the universal sets. We define two crisp maps π:X⟶Y and φ:A⟶B as follows:(15)πxi=xi,for eachxi∈X,φa1=b1,φa2=φa3=b2.
LetM=a2,a3⊂A and f1,f2,M be a bipolar soft set over X with M defined as follows:(16)f1a2=x1,x2,f1a3=x1,f2¬a2=x3,f2¬a3=x2.
Then,πφ:WCXA⟶WCYB is a soft mapping such that the image of f1,f2,M is g1,g2,E in WCYB such that E=φM=b2⊆B and the two maps g1 and g2 are given by(17)g1b2=πf1a2∪f1a3=πx1,x2=x1,x2,g2¬b2=πf1¬a2∪f¬a3=πx2,x3=x2,x3.
It is clear thatg1,g2,E is not a bipolar soft set because g1b2∩g2¬b2≠∅.Proposition 9.
Letπφ:WCXA⟶WCYB be a soft mapping such that π:X⟶Y and φ:A⟶B are injective maps. Then, the image of a bipolar soft set is a bipolar soft set.Proof.
Letf1,f2,M be a bipolar soft set in WCXA. Then, its image is g1,g2,E in WCYB such that E=φM⊆B and the two maps g1 and g2 are given by(18)g1e=π∪λ∈φ−1e∩Mf1λ,for eache∈E,g2¬e=π∪¬λ∈¬φ−1e∩Mf2¬λ,for each¬e∈¬E.
Sinceφ is injective, φ−1e is empty or a singleton set. This with the injectiveness of π leads to the fact that g1e and g2¬e are empty or singleton sets such that g1e∩g2¬e=∅ for each e∈E. Hence, g1,g2,E is a bipolar soft set.Definition 14.
Letπφ:WCXA⟶WCYB be a soft mapping. Then a crisp map φ′:¬A⟶¬B is defined as follows: φ′¬a=¬φa, where φ:A⟶B is a crisp map given in Definition 13.
It is clear thatφ′−1¬b=¬φ−1b.Definition 15.
A soft mapπφ:WCXA⟶WCYB is said to be injective (resp., surjective, bijective) if πφ:WCXA⟶WCYBaπ and φ are injective (resp., surjective, bijective).Definition 16.
Letπφ:WCXA⟶WCYB be a soft mapping. Then, the preimage of a bipolar soft set h1,h2,N in CYB is a bipolar soft set πφ−1h1,h2,N=l1,l2,D in CXA such that D=φ−1N⊆A and the two maps l1 and l2 are given by(19)l1d=π−1h1φd,for eachd∈D,l2¬d=π−1h2¬φd,for each¬d∈¬D.Proposition 10.
Letπφ:WCXA⟶WCYB be a soft mapping. Then, the preimage of a bipolar soft set is a bipolar soft set.Proof.
Leth1,h2,N be a bipolar soft set in CYB. Then, πφ−1h1,h2,N=l1,l2,D in CXA such that D=φ−1N⊆A and the two maps l1 and l2 are given by(20)l1d=π−1h1φd,for eachd∈D,l2¬d=π−1h2¬φdfor each¬d∈¬D.
Letφd=d′.
Then,l1d∩l2¬d=π−1h1d′∩π−1h2¬d′=π−1h1d′∩h2¬d′=π−1∅=∅. Hence, l1,l2,D is a bipolar soft set.Proposition 11.
Letπφ:WCXA⟶WCYB be a soft mapping and let f1,f2,M and h1,h2,M′ be two bipolar soft sets in WCXA. Then, we have the following results:(i)
ΦφA,XφA,φA⊆˜πφΦA,XA,A. The equality holds if π and φ are surjective.(ii)
XφA,ΦφA,φA⊆˜πφXA,ΦA,A. The equality holds if π and φ are surjective.(iii)
Iff1,f2,M⊆˜h1,h2,M′, then πφf1,f2,M⊆˜πφh1,h2,M′.(iv)
πφf1,f2,M∪˜h1,h2,M′=πφf1,f2,M∪˜πφh1,h2,M′.(v)
πφf1,f2,M∩˜h1,h2,M′⊆˜πφf1,f2,M∩˜πφh1,h2,M′.
The equality holds ifπ and φ are injective.Proof.
To prove (i), letπφΦA,XA,A=πφu1,u2,A=f1,f2,E, where u1a=∅ and E=φA, and u2¬a=X for each ¬a∈¬A. Then, f1e=π∪λ∈φ−1eu1λ=π∅=∅ for each e∈E, and f2¬e=π∪¬λ∈¬φ−1¬eu2¬λ=πX=Z⊆Y for each ¬e∈¬E. Therefore, f1,f2,E=ΦE,ZE,E. Since E=φA, f1e⊆∅ and f2¬e=Z⊆Y, then ΦφA,XφA,φA⊆˜πφΦA,XA,A=ΦE,ZE,E.
Ifπ and φ are surjective, then E=φA=B and πX=Y. Hence, πφΦA,XA,A=ΦB,XB,B.
Following similar argument above, one can prove (ii).
One can prove (iii) easily.(iv)
First, letπφf1,f2,M∪˜h1,h2,M′=πφu1,u2,M∪M′=g1,g2,E, where E=φM∪M′. Now, for each e∈E, we have g1e=π∪λ∈φ−1e∩Eu1λ. Since(21)u1λ=f1λ,λ∈M−M′,h1λ,λ∈M′−M,f1λ∪h1λ,λ∈M∩M′,then(22)π∪λ∈φ−1e∩Eu1λ=π∪f1λλ∈M−M′∩φ−1eh1λλ∈M′−M∩φ−1ef1λ∪h1λλ∈M∩M′∩φ−1e.
Also, for each¬e∈¬E, we have g2¬e=π∪¬λ∈¬φ−1e∩Eu2¬λ. Since(23)u2¬λ=f2¬λ,¬λ∈¬M−¬M′,h2¬λ,¬λ∈¬M′−¬M,f2¬λ∪h2¬λ,¬λ∈¬M∩¬M′,then(24)π∪¬λ∈¬φ−1e∩Eu2¬λ=π∪f2¬λ¬λ∈¬M−M′∩φ−1eh2¬λ¬λ∈¬M′−M∩φ−1ef2¬λ∪h2¬λ¬λ∈¬M∩M′∩φ−1e.
Second, letπφf1,f2,M∪˜πφh1,h2,M′=w1,w2,N, where N=φM∪φM′. Now, for each n∈N, we have(25)w1n=π∪λ∈φ−1n∩Nf1λ∪π∪λ∈φ−1n∩Nh1λ=π∪λ∈φ−1n∩Nf1λ∪∪λ∈φ−1n∩Nh1λ=π∪f1λλ∈M−M′∩φ−1nh1λλ∈M′−M∩φ−1nf1λ∪h1λλ∈M∩M′∩φ−1n.
Also, for each¬n∈¬N, we have(26)w2¬n=π∪¬λ∈¬φ−1n∩Nf2¬λ∪π∪¬λ∈¬φ−1n∩Nh2¬λ=π∪¬λ∈¬φ−1n∩Nf2¬λ∪∪¬λ∈¬φ−1n∩Nh2¬λ=π∪f2¬λ¬λ∈¬M−M′∩φ−1nh2¬λ¬λ∈¬M′−M∩φ−1nf2¬λ∪h2λλ∈¬M∩M′∩φ−1n.
SinceφM∪φM′=φM∪φM′, then E=N. Thus v1e=w1e for each e∈E=N and v2¬e=w2¬e for each ¬e∈¬E=¬N. Hence, we obtain the desired result.
One can prove (v) similarly.
By using a similar technique, one can prove the following result.Proposition 12.
Letπφ:WCXA⟶WCYB be a soft mapping and let g1,g2,N and l1,l2,N′ be two bipolar soft sets in WCYB. Then, we have the following results:(i)
πφ−1ΦB,XB,B=ΦA,XA,A.(ii)
πφ−1XB,ΦB,B=XA,ΦA,A.(iii)
Ifg1,g2,N⊆˜l1,l2,N′, then πφ−1g1,g2,N⊆˜πφ−1l1,l2,N′.(iv)
πφ−1g1,g2,N∪˜l1,l2,N′=πφ−1g1,g2,N∪˜πφ−1l1,l2,N′.(v)
πφ−1g1,g2,N∩˜l1,l2,N′=πφ−1g1,g2,N∩˜πφ−1l1,l2,N′.Proposition 13.
Letπφ:WCXA⟶WCYB be a soft mapping and let f1,f2,M be a bipolar soft set in WCXA. Then, we have the following results:(i)
Ifx⋐ppf1,f2,M, then πx⋐ppπφf1,f2,M.(ii)
Ifx⋐npf1,f2,M, then πx⋐npπφf1,f2,M.(iii)
Ifx⋐pf1,f2,M, then πx⋐pπφf1,f2,M.(iv)
Ifx∈ptf1,f2,M, then πx∈ptπφf1,f2,M.(v)
Ifx∈ntf1,f2,M, then πx∈ntπφf1,f2,M.(vi)
Ifx⋐ppf1,f2,M and φ is injective, then πx⋐ppπφf1,f2,M.(vii)
Ifx⋐npf1,f2,M and φ is injective, then πx⋐npπφf1,f2,M.(viii)
Ifx⋐pf1,f2,M and φ is injective, then πx⋐pπφf1,f2,M.(ix)
Ifx∉ptf1,f2,M, then πx∉ptπφf1,f2,M.(x)
Ifx∉ntf1,f2,M, then πx∉ntπφf1,f2,M.(xi)
Ifx∉tf1,f2,M, then πx∉tπφf1,f2,M.Proof.
We only prove (i), (v), (vi), and (xi). The other cases can be proved similarly.
To prove (i), letx⋐ppf1,f2,M and πφf1,f2,M=g1,g2,φM. Then, there exists a parameter a∈M⊆A such that x∈fa. Therefore, there is a parameter b∈φM⊆B such that a∈φ−1b. Obviously, a∈φ−1b∩M, so that it follows from Definition 13 that πx∈g1b=π∪λ∈φ−1b∩Mf1λ. Thus, πx⋐ppg1,g2,φM=πφf1,f2,M, as required.
To prove (v), letx⋐ntf1,f2,M and πφf1,f2,M=g1,g2,φM. Then, x∈f2¬a for each ¬a∈¬M⊆¬A. Therefore, for each parameter ¬b∈φ′¬M=¬φM⊆¬B, there is ¬a∈φ′−1¬b=¬φ−1b. Obviously, ¬a∈¬φ−1b∩¬M, so that it follows from Definition 13 that for each ¬b∈¬φM we have πx∈g2¬b=π∪¬λ∈¬φ−1b∩Mf2¬λ. Thus, πx⋐ntg1,g2,φM=πφf1,f2,M, as required.
To prove (vi), letx⋐ppf,g,M and πφf1,f2,M=g1,g2,φM. Then, there exists a parameter a∈M⊆A such that x∉f1a. Therefore, there is a parameter b∈φM⊆B such that a∈φ−1b. Since φ is injective, a=φ−1b. This means that a=φ−1b∩M. Therefore, πx∈g1b=π∪λ∈φ−1b∩Mf1λ=πf1a. Therefore, πx⋐ppg1,g2,φM=πφf1,f2,M, as required.
To prove (xi), letx∉tf1,f2,M and πφf1,f2,M=g1,g2,φM. Then, x∉f1a for all a∈M⊆A and x∉f2¬a for all ¬a∈¬M⊆¬A. Therefore, for each parameter b∈φM⊆B, there is a∈M such that a∈φ−1b, and for each parameter ¬b∈¬φM⊆¬B, there is ¬a∈¬M such that ¬a∈¬φ−1b. Thus, we obtain πx∉g1b=π∪λ∈φ−1b∩Mf1λ=πf1a for each b∈φM and πx∉g2¬b=π∪¬λ∈¬φ−1b∩Mf2¬λ=πf2¬a for each ¬b∈¬φM. Hence, πx∉tg1,g2,φM=πφf1,f2,M, as required.Proposition 14.
Letπφ:WCXA⟶WCYB be a soft mapping and let h1,h2,N be a bipolar soft set in WCYB. If φ is surjective, then we have the following results:(i)
Ify⋐pph1,h2,N, then x⋐ppπφ−1h1,h2,N for each x∈π−1y.(ii)
Ify⋐nph1,h2,N, then x⋐npπφ−1h1,h2,N for each x∈π−1y.(iii)
Ify⋐ph1,h2,N, then x⋐pπφ−1h1,h2,N for each x∈π−1y.(iv)
Ify∈pth1,h2,N, then x∈ptπφ−1h1,h2,N for each x∈π−1y.(v)
Ify∈nth1,h2,N, then x∈ntπφ−1h1,h2,N for each x∈π−1y.(vi)
Ify⋐pph1,h2,N such that π is injective, then π−1y⋐ppπφ−1h1,h2,N.(vii)
Ify⋐nph1,h2,N such that π is injective, then π−1y⋐npπφ−1h1,h2,N.(viii)
Ify⋐ph1,h2,N such that π is injective, then π−1y⋐pπφ−1h1,h2,N.(ix)
Ify∉pth1,h2,N such that π is injective, then π−1y∉ptπφ−1h1,h2,N.(x)
Ify∉nth1,h2,N such that π is injective, then π−1y∉ntπφ−1h1,h2,N.(xi)
Ify∉th1,h2,N such that π is injective, then π−1y∉tπφ−1h1,h2,N.Proof.
We only prove (i) and (x). The other cases can be proved similarly.
To prove (i), lety⋐pph1,h2,N and πφ−1h1,h2,N=l1,l2,φ−1N. Then, there exists a parameter b∈N⊆B such that y∈hb. Since φ is surjective, there is a parameter a∈φ−1N⊆A such that φa=b. It follows from Definition 16 that l1a=π−1h1φa=π−1h1b. Now, for each x∈π−1y, we obtain x⋐ppl1,l2,φ−1N=πφ−1h1,h2,N, as required.
To prove (x), lety∉nth1,h2,N and πφ−1h1,h2,N=l1,l2,φ−1N. Then, y∉h2¬b for all ¬b∈¬N⊆¬B. Since φ is surjective, there exists a parameter ¬a∈¬φ−1N⊆¬A such that ¬φa=¬b. It follows from Definition 16 that l2¬a=π−1h2¬φa=π−1h2¬b. Since π is injective, π−1y∉ntl1,l2,φ−1N=πφ−1h1,h2,N, as required.


## 4. Application of Bipolar Soft Sets

In this section, we apply the idea of bipolar soft sets to initiate an application of optimal choices. We provide an example to demonstrate how we make optimal choices. Then, we construct an algorithm of this method.Example 6.
Suppose that a car sales company has a set of carsX with a set of parameters E. Let X=ci:i=1,2,…,20 be a set of twenty cars and A=ai:i=1,2,…,11 be a set of eleven parameters, where aii=1,2,3,4,5,6,7,8,9,10,11 stand for “expensive,” “cheap,” “modern,” “sport,” “red color,” “white color,” “Japanese industry,” “German industry,” “in good repair,” “in bad repair,” and “low fuel consumption,” respectively.
It should be noted that¬a1 does not mean “cheap” and ¬a2 does not mean “expensive.”
Now, suppose that a car sales company classifies these cars with respect to the set of parameters using a notion of a bipolar soft setf,g,A as follows:(27)fa1=ci:i=4,7,8,…,12,g¬a1=ci:i=14,15,…,18,fa2=ci:i=5,10,14,15,18,g¬a2=ci:i=7,8,9,11,12,19,fa3=ci:i=1,2,3,6,8,10,19,20,g¬a3=ci:i=9,13,16,17,fa4=ci:i=4,9,15,16,17,g¬a4=ci:i=8,10,11,12,13,20,fa5=ci:i=1,2,…,7,g¬a5=ci:i=8,9,…,20,fa6=ci:i=1,4,8,12,13,…,16,20,g¬a6=ci:i=2,…,7,9,10,11,17,18,19,fa7=ci:i=1,2,…,7,14,16,g¬a7=ci:i=8,9,…,13,15,17,18,19,20,fa8=ci:i=4,8,10,13,15,18,19,20,g¬a8=ci:i=1,2,3,5,6,7,9,11,12,14,16,17,fa9=ci:i=1,4,5,8,15,16,…,20,g¬a9=ci:i=9,11,12,14,fa10=ci:i=9,11,12,g¬a10=ci:i=1,5,8,15,16,…,20,fa11=ci:i=1,3,5,8,11,…,17,g¬a11=ci:i=4,6,7,18.
Now, suppose that Mr. Redhwan wants to choose a car with respect to a set of parametersE=ai:i=2,3,6,8,9,11⊆A. To help him, we will construct two tables, one of them with respect to a map f:E⟶2X (see Table 1) and the other with respect to a map g:¬E⟶2X (see Table 2). Then we determine the value of ck,fai and ck,g¬ai by the following two roles:(28)ck,fai=1,ck∈fai,0,ck∉fai,ck,g¬ai=−1,ck∈g¬ai,0,ck∉g¬ai.
Sincefai∩g¬ai=∅ for each ai∈E, we can combine Tables 1 and 2 in Table 3. Note that the value of ck,fai,g¬ai is given by the following two roles:(29)ck,fai,g¬ai=1,ck∈fai,−1,ck∈g¬ai,0,ck∉fai∪g¬ai.
One can note from Table3 that car no. 15 is the optimal car for Mr. Redhwan. Cars no. 8 and no. 20 are the second optimal cars for Mr. Redhwan.
In the following, we present an algorithm for determining the wining students.Step 1. Determine a set of parameters A and the universal set X.Step 2. Define a map f which associates each parameter of A with its corresponding subset of 2X.Step 3. Define a map g which associates each parameter of ¬A with its corresponding subset of 2X.Step 4. Determine the favorite set of parameters E⊆A of Mr. X.Step 5. Construct a bipolar soft set f,g,E.Step 6. Initiate a table which represents a bipolar soft set f,g,E (see Table 3).Step 7. Input the value of ck,fai,g¬ai as given by the following two roles:(30)ck,fai,g¬ai=1,ck∈fai,−1,ck∈g¬ai,0,ck∉fai∪g¬ai.Step 8. Count the values of each arrow by the rule Sumk=∑i=1mck,fai,g¬ai.Step 9. Find the decision, denoted by d, for which S=maxSumk:k=1,2,…,s, where s=X.Step 10. Then, d is the optimal choice car. If d has more than one value, then any one of them could be chosen by Mr. Redhwan satisfying his option.
On the other hand, some of the parameters are of less significance than the other ones so they must be graded with lesser priority. For this reason, we suggest weightswi∈0,1 on the parameters according to the desire of customers.
In this case, we modify the previous algorithm to be convenient for weighted selection.Step 1. Determine a set of parameters A and the universal set X.Step 2. Define a map f which associates each parameter of A with its corresponding subset of 2X.Step 3. Define a map g which associates each parameter of ¬A with its corresponding subset of 2X.Step 4. Determine the favorite set of parameters E⊆A of Mr. X.Step 5. Determine the weight wa of each parameter of the favorite set of parameters.Step 6. Construct a bipolar soft set f,g,E.Step 7. Initiate a table which represents a bipolar soft set f,g,E (see Table 3).Step 8. Input the value of ck,fai,g¬ai as given by the following two roles:(31)ck,fai,g¬ai=wai,ck∈fai,−wai,ck∈g¬ai,0,ck∉fai∪g¬ai.Step 9. Count the values of each arrow by the rule Sumk=∑i=1mck,fai,g¬ai. See Table 4.Step 10. Find the decision, denoted by d, for which S=maxSumk:k=1,2,…,s, where s=X.Step 11. Then, d is the optimal choice car. If d has more than one value, then any one of them could be chosen by Mr. Redhwan satisfying his option.
With respect to our example, suppose that the weights10%,40%,10%,10%,20%, and 10% are, respectively, corresponding to a2,a3,a6,a8,a9, and a11. Then, we update Table 3 to be as shown in Table 4.
Now, one can note from Table4 that cars no. 8 and no. 20 are the optimal cars for Mr. Redhwan. Therefore, any one of them could be chosen by Mr. Redhwan satisfying his option.Table 1
The positive componentsf of bipolar soft sets f,g,E of each car.
Carsfaia2a3a6a8a9a11c1011011c2010000c3010001c4101110c5100011c6010000c7000000c8011111c9000000c10110100c11000001c12001001c13001101c14101001c15101111c16001011c17000011c18100110c19010110c20011110Table 2
The negative componentsg of bipolar soft sets f,g,E of each car.
Carsg¬ai¬a2¬a3¬a6¬a8¬a9¬a11c1000−100c200−1−100c300−1−100c400000−1c500−1−100c600−1−10−1c7−10−1−10−1c8−100000c9−1−1−1−1−10c1000−1000c11−10−1−1−10c12−100−1−10c130−10000c14000−1−10c15000000c160−10−100c170−1−1−100c1800−100−1c19−10−1000c20000000Table 3
Values of all components of the favorite bipolar soft setsf,g,E.
Carsfa,g¬aa2,¬a2a3,¬a3a6,¬a6a8,¬a8a9,¬a9a11,¬a11Sumc1011−1113c201−1−100−1c301−1−1010c410111−13c510−1−1111c601−1−10−1−2c7−10−1−10−1−4c8−1111114c9−1−1−1−1−10−5c1011−11002c11−10−1−1−11−3c12−101−1−11−1c130−111012c14101−1−111c151011115c160−11−1111c170−1−1−111−1c1810−111−11c19−11−11101c200111104Table 4
Weight values of all components of the favorite bipolar soft setsf,g,E.
Carsf,g,waia2,¬a2,10%a3,¬a3,40%a6,¬a6,10%a8,¬a8,10%a9,¬a9,20%a11,¬a11,10%Sumc100.40.1−0.10.20.10.7c200.4−0.1−0.100−0.2c300.4−0.1−0.100.10.3c40.100.10.10.2−0.10.4c50.10−0.1−0.10.20.10.2c600.4−0.1−0.10−0.10.1c7−0.10−0.1−0.10−0.1−0.4c8−0.10.40.10.10.20.10.8c9−0.1−0.4−0.1−0.1−0.20−0.9c100.10.4−0.10.1000.5c11−0.10−0.1−0.1−0.20.1−0.4c12−0.100.1−0.1−0.20.1−0.2c130−0.40.10.100.1−0.1c140.100.1−0.1−0.20.1−0.2c150.100.10.10.20.10.6c160−0.40.1−0.10.20.1−0.1c170−0.4−0.1−0.10.20.1−0.3c180.10−0.10.10.2−0.10.2c19−0.10.4−0.10.10.200.5c2000.40.10.10.200.8


## 5. Conclusion

This study has introduced five types of belong relations and six types of nonbelong relations between a bipolar soft set and an ordinary point. These relations can be considered as a primary indicator of membership and nonmembership degree of an element. Then, the concept of soft mappings between two classes of bipolar soft sets has been defined, and the sufficient conditions to preserve these relations between an ordinary point and its image and preimage have been studied. Finally, we have applied the idea of bipolar soft sets to present an application of choosing the best products according to the favorite set of parameters. We have given an algorithm of the application and provided explanatory example.In the upcoming work, we shall exploit these relations to initiate different types of soft separation axioms and compact spaces on bipolar soft topological spaces. In addition, we shall try to model some natural phenomena using bipolar soft sets.



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*Source: 6621854-2021-01-29.xml*