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.
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 stands for a “tall” characteristic, then 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. Section 2 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, denote the sets of parameters, and denote the initial universal sets.
Definition 1 (see [4]). A map of into the power set of a nonempty set is called a soft set over , where is an initial universal set and is a set of parameters.
A soft set is symbolized by an ordered pair and it is expressed as a set of ordered pairs:
Definition 2 (see [17]). A bipolar soft set is a triple over a nonempty set with a set of parameters , where and are two crisp maps such that for each .
A bipolar soft set is expressed as a set of ordered triples:A class of all bipolar soft sets defined over with all sets of parameters which are subsets of is symbolized by .
Henceforth, we consider that any bipolar soft set is defined on , unless otherwise specified.
Example 1. Let be the universe containing set six cars and be a set of parameters, where , and stand for “expensive,” “in good repair,” “red colored,” and “made in Japan,” respectively.
Let and be two maps given as follows:Now, we can describe this system using a bipolar soft set as follows:
Definition 3 (see [17]). A bipolar soft set is said to be(i)A relative null bipolar soft set if equals the empty set and equals the universal set for each . It is symbolized by .(ii)A relative absolute bipolar soft set if equals the universal set and equals the empty set for each . It is symbolized by .
Definition 4 (see [17]). The intersection of two bipolar soft sets and is a bipolar soft set such that and the two mappings and are given byIt is symbolized by .
Definition 5 (see [17]). The union of two bipolar soft sets and is a bipolar soft set , where and the two mappings and are given byIt is symbolized by .
Definition 6 (see [17]). A bipolar soft set is said to be a subset of a bipolar soft set , denoted by , provided that(i).(ii)For all , we have and .The bipolar soft sets and are said to be soft equal if and .
Definition 7 (see [17]). The relative complement of a bipolar soft set is a bipolar soft set , where and are defined as follows:
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. Let be a bipolar soft set and . We say that(i), reading as positively partially belongs to , if for some .(ii), reading as negatively partially belongs to , if for some .(iii), reading as partially belongs to , if and for some and .(iv), reading as positively totally belongs to , if for all .(v), reading as negatively totally belongs to , if for all .
Definition 9. Let be a bipolar soft set and . We say that(i), reading as does not positively partially belong to , if for some .(ii), reading as does not negatively partially belong to , if for some .(iii), reading as does not partially belong to , if and for some and .(iv), reading as does not positively totally belong to , if for all .(v), reading as does not negatively totally belong to , if for all .(vi), reading as does not totally belong to , if and for all and .To well understand the results initiated in this work, we give the following example and remark.
Example 2. Let be a set of parameters and be bipolar soft set over defined as follows:Then, it follows from Definition 7 that . We note the following:That is, the intersection of a bipolar soft set and its relative complement need not be the null bipolar soft set.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 Example 2, it can be seen that
Proposition 1. For two bipolar soft sets and and , we have the following results:(i).(ii).(iii).
Proof. (i) for some for some .(ii) and for some and and for some and .(iii) for all for all .The proofs of the following results follow from Definition 8.
Proposition 2. For two bipolar soft sets and and , we have the following results:(i) and .(ii).(iii).
Proposition 3. For two bipolar soft sets and and , we have the following results:(i) and .(ii) and .(iii).(iv).(v).To show that the converse of (ii) and (iii) of Proposition 2 and (ii) to (v) of Proposition 3 fails, we give the following example.
Example 3. In Example 2, we note the following:(i) and , but and do not hold true.(ii), and , but , , and do not hold true.
Proposition 4. Let and be bipolar soft sets such that . Then, we have the following results:(i)if (resp., ), then (resp., ).(ii)if , then .(iii)if (resp., ), then (resp., ).(iv)if , then .
Proof. The proof is straightforward.
Proposition 5. For two bipolar soft sets and and , we have the following results:(i) or .(ii) and .(iii) or .(iv) or .(v) and .(vi) and .(vii) or .(viii) or .(ix) and .(x) and .
Proof. We will just prove (i), (iv), (v), (vi), and (ix).
Since and are subsets of , then the necessary parts of (i), (iv), and (vi) hold; since are subsets of and , then the sufficient parts of (v) and (ix) hold.
To prove the sufficient part of (i), let . Then, for some . Therefore, or for some , and hence or .
To prove the necessary part of (v), let and . Then, for all , we have and . Therefore, for all . Hence, .
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 Proposition 5 fails.
Example 4. Let be a set of parameters and be bipolar soft sets over defined as follows:Then, , and
We note the following:(i) and , but does not hold true.(ii) and , but and do not hold true.(iii) , but or does not hold true.(iv) and , but does not hold true.(v) , but and do not hold true.One can prove the following result similarly.
Proposition 6. For two bipolar soft sets and over and , we have the following results:(i) and .(ii) and .(iii) and .(iv) and .(v) or .(vi) or .(vii) or .(viii) or .(ix) or .(x) and .(xi); ; and .
Definition 10. A bipolar soft set is said to be stable if there are two disjoint subsets of such that for each and and for each .
is said to be positive stable if ; and is said to be negative stable if .
It is clear that positive (negative) stable bipolar soft set is stable; however, the converse is not true.
Proposition 7. Let be a stable bipolar soft set. Then,(i).(ii).(iii).(iv).(v).
Proof. Since is a stable bipolar soft set, then there are two disjoint subsets of such that for each and for each . Therefore, the following properties hold:(i) for some if and only if for each .(ii) for some if and only if for each .(iii) for some and for some if and only if for each and for each .(iv) for some if and only if for each .(v) for some if and only if for each .Hence, the desired results are proved.
Corollary 1. Let be a positive stable bipolar soft set. Then,(i).(ii).
Corollary 2. Let be a negative stable bipolar soft set. Then,(i).(ii).
Definition 11. The Cartesian product of two bipolar soft sets and , denoted by , is defined as for each and for each .
Proposition 8. We have the next four results:(i) if and only if and .(ii) if and only if and .(iii) if and only if and .(iv) if and only if and .(v) if and only if and .
Proof. We will just prove (iii). The other cases can be proved similarly.(iii) and for some and and for some and ; and for some and and for some and ; and for some and and .
Definition 12. A weak bipolar soft set is a triple over a nonempty set with a set of parameters , where and are two crisp maps.
A weak bipolar soft set is expressed as a set of ordered triples:A class of all weak bipolar soft sets defined over with all sets of parameters which are subsets of is symbolized by .
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 mapping of into is a pair of crisp mappings and such that is defined as follows: the image of a weak bipolar soft set in is a weak bipolar soft set in such that and the two maps and are given byThe following example shows that the image of a bipolar soft set need not be a bipolar soft set.
Example 5. Consider that and are two sets of parameters and are the universal sets. We define two crisp maps and as follows:Let and be a bipolar soft set over with defined as follows:Then, is a soft mapping such that the image of is in such that and the two maps and are given byIt is clear that is not a bipolar soft set because .
Proposition 9. Let be a soft mapping such that and are injective maps. Then, the image of a bipolar soft set is a bipolar soft set.
Proof. Let be a bipolar soft set in . Then, its image is in such that and the two maps and are given bySince is injective, is empty or a singleton set. This with the injectiveness of leads to the fact that and are empty or singleton sets such that for each . Hence, is a bipolar soft set.
Definition 14. Let be a soft mapping. Then a crisp map is defined as follows: , where is a crisp map given in Definition 13.
It is clear that .
Definition 15. A soft map is said to be injective (resp., surjective, bijective) if and are injective (resp., surjective, bijective).
Definition 16. Let be a soft mapping. Then, the preimage of a bipolar soft set in is a bipolar soft set in such that and the two maps and are given by
Proposition 10. Let be a soft mapping. Then, the preimage of a bipolar soft set is a bipolar soft set.
Proof. Let be a bipolar soft set in . Then, in such that and the two maps and are given byLet .
Then, . Hence, is a bipolar soft set.
Proposition 11. Let be a soft mapping and let and be two bipolar soft sets in . Then, we have the following results:(i). The equality holds if and are surjective.(ii). The equality holds if and are surjective.(iii)If , then .(iv).(v).The equality holds if and are injective.
Proof. To prove (i), let , where and , and for each . Then, for each , and for each . Therefore, . Since , and , then .
If and are surjective, then and . Hence, .
Following similar argument above, one can prove (ii).
One can prove (iii) easily.(iv)First, let , where . Now, for each , we have . SincethenAlso, for each , we have . SincethenSecond, let , where . Now, for each , we haveAlso, for each , we haveSince , then . Thus for each and for each . Hence, we obtain the desired result.
One can prove (v) similarly.
By using a similar technique, one can prove the following result.
Proposition 12. Let be a soft mapping and let and be two bipolar soft sets in . Then, we have the following results:(i).(ii).(iii)If , then .(iv).(v).
Proposition 13. Let be a soft mapping and let be a bipolar soft set in . Then, we have the following results:(i)If , then .(ii)If , then .(iii)If , then .(iv)If , then .(v)If , then .(vi)If and is injective, then .(vii)If and is injective, then .(viii)If and is injective, then .(ix)If , then .(x)If , then .(xi)If , then .
Proof. We only prove (i), (v), (vi), and (xi). The other cases can be proved similarly.
To prove (i), let and . Then, there exists a parameter such that . Therefore, there is a parameter such that . Obviously, , so that it follows from Definition 13 that . Thus, , as required.
To prove (v), let and . Then, for each . Therefore, for each parameter , there is . Obviously, , so that it follows from Definition 13 that for each we have . Thus, , as required.
To prove (vi), let and . Then, there exists a parameter such that . Therefore, there is a parameter such that . Since is injective, . This means that . Therefore, . Therefore, , as required.
To prove (xi), let and . Then, for all and for all . Therefore, for each parameter , there is such that , and for each parameter , there is such that . Thus, we obtain for each and for each . Hence, , as required.
Proposition 14. Let be a soft mapping and let be a bipolar soft set in . If is surjective, then we have the following results:(i)If , then for each .(ii)If , then for each .(iii)If , then for each .(iv)If , then for each .(v)If , then for each .(vi)If such that is injective, then .(vii)If such that is injective, then .(viii)If such that is injective, then .(ix)If such that is injective, then .(x)If such that is injective, then .(xi)If such that is injective, then .
Proof. We only prove (i) and (x). The other cases can be proved similarly.
To prove (i), let and . Then, there exists a parameter such that . Since is surjective, there is a parameter such that . It follows from Definition 16 that . Now, for each , we obtain , as required.
To prove (x), let and . Then, for all . Since is surjective, there exists a parameter such that . It follows from Definition 16 that . Since is injective, , 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 cars with a set of parameters . Let be a set of twenty cars and be a set of eleven parameters, where 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 does not mean “cheap” and 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 set as follows:Now, suppose that Mr. Redhwan wants to choose a car with respect to a set of parameters . To help him, we will construct two tables, one of them with respect to a map (see Table 1) and the other with respect to a map (see Table 2). Then we determine the value of and by the following two roles:Since for each , we can combine Tables 1 and 2 in Table 3. Note that the value of is given by the following two roles:One can note from Table 3 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 and the universal set . Step 2. Define a map which associates each parameter of with its corresponding subset of . Step 3. Define a map which associates each parameter of with its corresponding subset of . Step 4. Determine the favorite set of parameters of Mr. X. Step 5. Construct a bipolar soft set . Step 6. Initiate a table which represents a bipolar soft set (see Table 3). Step 7. Input the value of as given by the following two roles: Step 8. Count the values of each arrow by the rule . Step 9. Find the decision, denoted by , for which , where . Step 10. Then, is the optimal choice car. If 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 weights 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 and the universal set . Step 2. Define a map which associates each parameter of with its corresponding subset of . Step 3. Define a map which associates each parameter of with its corresponding subset of . Step 4. Determine the favorite set of parameters of Mr. X. Step 5. Determine the weight of each parameter of the favorite set of parameters. Step 6. Construct a bipolar soft set . Step 7. Initiate a table which represents a bipolar soft set (see Table 3). Step 8. Input the value of as given by the following two roles: Step 9. Count the values of each arrow by the rule . See Table 4. Step 10. Find the decision, denoted by , for which , where . Step 11. Then, is the optimal choice car. If 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 weights , and are, respectively, corresponding to , and . Then, we update Table 3 to be as shown in Table 4.
Now, one can note from Table 4 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.
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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that he has no conflicts of interest.