Abstract
The purpose of this paper is to define the concept of (3, 2)-fuzzy sets and discuss their relationship with other kinds of fuzzy sets. We describe some of the basic set operations on (3, 2)-fuzzy sets. (3, 2)-Fuzzy sets can deal with more uncertain situations than Pythagorean and intuitionistic fuzzy sets because of their larger range of describing the membership grades. Furthermore, we familiarize the notion of (3, 2)-fuzzy topological space and discuss the master properties of (3, 2)-fuzzy continuous maps. Then, we introduce the concept of (3, 2)-fuzzy points and study some types of separation axioms in (3, 2)-fuzzy topological space. Moreover, we establish the idea of relation in (3, 2)-fuzzy set and present some properties. Ultimately, on the basis of academic performance, the decision-making approach of student placement is presented via the proposed (3, 2)-fuzzy relation to ascertain the suitability of colleges to applicants.
1. Introduction
The concept of fuzzy sets was proposed by Zadeh [1]. The theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. After the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. The integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [2–4].
The idea of intuitionistic fuzzy sets suggested by Atanassov [5] is one of the extensions of fuzzy sets with better applicability. Applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multicriteria decision making [6–8]. Yager [9] offered a new fuzzy set called a Pythagorean fuzzy set, which is the generalization of intuitionistic fuzzy sets. Fermatean fuzzy sets were introduced by Senapati and Yager [10], and they also defined basic operations over the Fermatean fuzzy sets.
The concept of fuzzy topological spaces was introduced by Chang [11]. He studied the topological concepts like continuity and compactness via fuzzy topological spaces. Then, Lowen [12] presented a new type of fuzzy topological spaces. Çoker [13] subsequently initiated a study of intuitionistic fuzzy topological spaces. Recently, Olgun et al. [14] presented the concept of Pythagorean fuzzy topological spaces and Ibrahim [15] defined the concept of Fermatean fuzzy topological spaces.
The main purpose of this paper is to introduce the concept of (3, 2)-fuzzy sets and compare them with the other types of fuzzy sets. We introduce the set of operations for the (3, 2)-fuzzy sets and explore their main features. Following the idea of Chang, we define a topological structure via (3, 2)-fuzzy sets as an extension of fuzzy topological space, intuitionistic fuzzy topological space, and Pythagorean fuzzy topological space. We discuss the main topological concepts in (3, 2)-fuzzy topological spaces such as continuity and compactness. In addition, the concept of relation to (3, 2)-fuzzy sets is investigated. Finally, an improved version of max-min-max composite relation for (3, 2)-fuzzy sets is proposed.
2. (3, 2)-Fuzzy Sets
In this section, we initiate the notion of (3, 2)-fuzzy sets and study their relationship with other kinds of fuzzy sets. Then, we furnish some operations to (3, 2)-fuzzy sets.
Definition 1. Let be a universal set. Then, the (3, 2)-fuzzy set (briefly, (3, 2)-FS) is defined by the following:where is the degree of membership and is the degree of non-membership of to , with the conditionThe degree of indeterminacy of to is defined byIt is clear that , and whenever . In the interest of simplicity, we shall mention the symbol for the (3, 2)-FS .
Definition 2. Let be a universal set. Then, the intuitionistic fuzzy set (IFS) [5] (resp. Pythagorean fuzzy set (PFS) [9] and Fermatean fuzzy set (FFS) [10]) is defined by the following:with the condition (resp. , ), where is the degree of membership and is the degree of non-membership of every to .
To illustrate the importance of (3, 2)-FS to extend the grades of membership and non-membership degrees, assume that and for . We obtain and which means that neither follows the condition of IFS nor follows the condition of PFS. On the other hand, which means we can apply the (3, 2)-FS to control it. That is, is a (3, 2)-FS.
Theorem 1. The set of (3, 2)-fuzzy membership grades is larger than the set of intuitionistic membership grades and Pythagorean membership grades.
Proof. It is well known that for any two numbers , we haveThen, we getHence, the space of (3, 2)-fuzzy membership grades is larger than the space of intuitionistic membership grades and Pythagorean membership grades. This development can be evidently recognized in Figure 1.
Lemma 1. Let be a universal set and be (3, 2)-FS. If , then .
Proof. Presume that is (3, 2)-FS and for ; then,
Example 1. Let be (3, 2)-FS and such that and . Then, .
Definition 3. Let be a positive real number . If and are two (3, 2)-FSs, then their operations are defined as follows:(1).(2).(3).(4).(5).
Remark 1. We will use supremum “sup” instead of maximum “max” and infimum “inf” instead of minimum “min” if the union and the intersection are infinite.
Example 2. Assume that and are both (3, 2)-FSs. Then,(1).(2).(3).(4), for .(5), for .
Theorem 2. Let and be two (3, 2)-FSs; then, the following properties hold:(1).(2).(3).(4).
Proof. From Definition 3, we can obtain(1).(2)The proof is similar to (1).(3).(4)The proof is similar to (3).
Theorem 3. Let , and be three (3, 2)-FSs and ; then,(1).(2).(3).(4).
Proof. For the three (3, 2)-FSs , and and , according to Definition 3, we can obtain(1)(2)The proof is similar to (1).(3)(4)The proof is similar to (3).
In the following result, we claim that is (3, 2)-FS for any (3, 2)-FS .
Theorem 4. Let and be two (3, 2)-FSs such that and are (3, 2)-FSs. Then,(1).(2).
Proof. For the two (3, 2)-FSs and , according to Definition 3, we can obtain(1)(2)The proof is similar to (1).
Definition 4. Let and be two (3, 2)-FSs; then,(1) if and only if and .(2) if and only if and .(3) or if .
Example 3. (1)If and for , then .(2)If and for , then and .
3. Topology with respect to (3, 2)-Fuzzy Sets
In this section, we formulate the concept of (3, 2)-fuzzy topology on the family of (3, 2)-fuzzy sets whose complements are (3, 2)-fuzzy sets and scrutinize main properties. Then, we define (3, 2)-fuzzy continuous maps and give some characterizations. Finally, we establish two types of (3, 2)-fuzzy separation axioms and reveal the relationships between them.
3.1. (3, 2)-Fuzzy Topology
Definition 5. Let be a family of (3, 2)-fuzzy subsets of a non-empty set . If(1) where and ,(2), for any ,(3), for any ,then is called a (3, 2)-fuzzy topology on and is a (3, 2)-fuzzy topological space. We call an open (3, 2)-FS if it is a member of and call its complement a closed (3, 2)-FS.
Remark 2. We call the indiscreet (3, 2)-fuzzy topology on . If contains all (3, 2)-fuzzy subsets, then we call the discrete (3, 2)-fuzzy topology on .
Example 4. Let be the family of (3, 2)-fuzzy subsets of , whereHence, is (3, 2)-fuzzy topology on .
Remark 3. We showed that every fuzzy set on a set is a (3, 2)-fuzzy set having the form . Then, every fuzzy topological space in the sense of Chang is obviously a (3, 2)-fuzzy topological space in the form whenever we identify a fuzzy set in whose membership function is with its counterpart . Similarly, one can note that every intuitionistic fuzzy topology (Pythagorean fuzzy topology) is (3, 2)-fuzzy topology. The following examples explain this note.
Example 5. Consider as family of fuzzy subsets of , whereThen, is fuzzy topology on , and hence it is (3, 2)-fuzzy topology.
Example 6. Let be the family of (3, 2)-fuzzy subsets on whereHence, is (3, 2)-fuzzy topology. On the other hand, is neither intuitionistic fuzzy topology nor Pythagorean fuzzy topology.
Definition 6. Let be a (3, 2)-fuzzy topological space and be a (3, 2)-FS in . Then, the (3, 2)-fuzzy interior and (3, 2)-fuzzy closure of are, respectively, defined by(1) is a closed (3, 2)-FS in and .(2) is an open (3, 2)-FS in and .
Remark 4. Let be a (3, 2)-fuzzy topological space and be any (3, 2)-FS in . Then,(1) is an open (3, 2)-FS.(2) is a closed (3, 2)-FS.(3) and .
Example 7. Consider the (3, 2)-fuzzy topological space in Example 4. If , then and .
Theorem 5. Let be a (3, 2)-fuzzy topological space and be (3, 2)-FSs in . Then, the following properties hold:(1) and .(2)If , then and .(3) is an open (3, 2)-FS if and only if .(4) is a closed (3, 2)-FS if and only if .
Proof. (1) and (2) are obvious.
(3) and (4) follow from Definition 6.
Corollary 1. Let be a (3, 2)-fuzzy topological space and be (3, 2)-FSs in . Then, the following properties hold:(1).(2).(3).(4).
Proof. (1) and (2) follows from (1) of the above theorem.
(3): since and , we obtain . On the other hand, from the facts and , we have and ; we see that , and hence .
(4) can be proved similar to (3).
Theorem 6. Let be a (3, 2)-fuzzy topological space and be (3, 2)-FS in . Then, the following properties hold:(1).(2).(3).(4).
Proof. We only prove (1); the other parts can be proved similarly.
Let and suppose that the family of open (3, 2)-fuzzy sets contained in is indexed by the family . Then, . Therefore, . Now, such that , for each . This implies that is the family of all closed (3, 2)-fuzzy sets containing . That is, . Hence, .
3.2. (3, 2)-Fuzzy Continuous Maps
Definition 7. Let be a map and and be (3, 2)-fuzzy subsets of and , respectively. The functions of membership and non-membership of the image of , denoted by , are, respectively, calculated byThe functions of membership and non-membership of preimage of , denoted by , are, respectively, calculated by
Remark 5. To show that and are (3, 2)-fuzzy subsets, consider . If is non-empty, then we obtainIn contrast, leads to the fact that .
It is easy to prove the case of .
Theorem 7. Let be a map s.t. and are (3, 2)-fuzzy subsets of and , respectively. Then, we have(1).(2).(3)If , then where and are (3, 2)-fuzzy subsets of .(4)If , then where and are (3, 2)-fuzzy subsets of .(5).(6).
Proof. (1)Consider and let be a (3, 2)-fuzzy subset of . Then, Similarly, one can have . Therefore, , as required.(2)For any such that and for any (3, 2)-fuzzy subset of , we can write Now from (18), we have The proof is easy when . Following a similar technique, we obtain , which means that .(3)Assume that . Then, for each , . Also, . Hence, we obtain the desired result.(4)Assume that and . The proof is easy when . So, presume that . Then, Thus, follows. Similarly, we have .(5)For any s.t. , we find that On the other hand, we have when . Similarly, we have .(6)For any , we have Similarly, we have .
The proof of the following result is easy, and hence it is omitted.
Theorem 8. Let and be two non-empty sets and be a map. Then, the following statements are true:(1) for any (3, 2)-fuzzy subset of .(2) for any (3, 2)-fuzzy subset of .(3) for any two (3, 2)-fuzzy subsets and of .(4) for any (3, 2)-fuzzy subset of .
Definition 8. In a (3, 2)-fuzzy topological space, consider that and are two (3, 2)-fuzzy subsets. We call a neighborhood of , briefly nbd, if there exists an open (3, 2)-fuzzy subset such that .
Theorem 9. A (3, 2)-fuzzy subset is open iff it contains a nbd of its each subset.
Proof. The proof is easy.
Definition 9. A map is said to be (3, 2)-fuzzy continuous if for any (3, 2)-fuzzy subset of and for any nbd of there is a nbd of s.t. .
Theorem 10. The following statements are equivalent for a map :(1) is (3, 2)-fuzzy continuous.(2)For each (3, 2)-FS of and each nbd of , there is a nbd of s.t. for each , we obtain .(3)For each (3, 2)-FS of and for each nbd of , there is a nbd of s.t. .(4)For each (3, 2)-FS of and for each nbd of , is a nbd of .
Proof. : let be a (3, 2)-fuzzy continuous map. Consider as a (3, 2)-FS of and as a nbd of . Then, there is a nbd of s.t. . If , we obtain . : assume as a (3, 2)-FS of and as a nbd of . According to (2), there is a nbd of s.t. for each , we find . Therefore, . Since is chosen arbitrarily, . : presume is a (3, 2)-FS of and is a nbd of . According to (3), there is a nbd of s.t. . Since is a nbd of , there is an open (3, 2)-FS of s.t. . On the other hand, we obtain because . This means that is a nbd of . : suppose that is a (3, 2)-FS of and is a nbd of . By hypothesis, is a nbd of . So, there is an open (3, 2)-FS of s.t. which means . Moreover, is an open (3, 2)-FS, so it is a nbd of . Hence, we obtain the proof that is (3, 2)-fuzzy continuous.
Theorem 11. A map is (3, 2)-fuzzy continuous iff is an open (3, 2)-FS of for each open (3, 2)-FS of .
Proof. Necessity: presume as a (3, 2)-fuzzy continuous map. Consider an open (3, 2)-FS of s.t. . This directly gives that . It follows from Theorem 9 that there is a nbd of satisfying . Now, is (3, 2)-fuzzy continuous, so by (4) of Theorem 10, we obtain that is a nbd of . Also, it follows from (3) of Theorem 7 that . So, is a nbd of . Since is an arbitrary subset of , then by Theorem 9, the (3, 2)-FS is open.
3.2.1. Sufficiency
Presume is a (3, 2)-FS of and is a nbd of . Then, contains a (3, 2)-FS of s.t. . By hypothesis, is an open (3, 2)-FS. Also, we have . Thus, is a nbd of which demonstrates that is (3, 2)-fuzzy continuous.
We build the following two examples such that the first one provides a (3, 2)-fuzzy continuous map, whereas the second one presents a fuzzy map that is not (3, 2)-fuzzy continuous.
Example 8. Consider with the (3, 2)-fuzzy topology and with the (3, 2)-fuzzy topology , whereLet be defined as follows:Since , and are open (3, 2)-fuzzy subsets of , thenare open (3, 2)-fuzzy subsets of . Thus, is (3, 2)-fuzzy continuous.
Example 9. Consider with the (3, 2)-fuzzy topology and with the (3, 2)-fuzzy topology , where .
Let be defined as follows:Since is an open (3, 2)-fuzzy subset of , but is not an open (3, 2)-fuzzy subset of , is not (3, 2)-fuzzy continuous.
Theorem 12. The following are equivalent to each other:(1) is (3, 2)-fuzzy continuous.(2)For each closed (3, 2)-fuzzy subset of we have that is a closed (3, 2)-fuzzy subset of .(3) for each (3, 2)-fuzzy set in .(4) for each (3, 2)-fuzzy set in .
Proof. They can be easily proved using Theorems 6, 7, and 11.
Theorem 13. Let be a (3, 2)-fuzzy topological space and be a map. Then, there is a coarsest (3, 2)-fuzzy topology over such that is (3, 2)-fuzzy continuous.
Proof. Let us define a class of (3, 2)-fuzzy subsets of byWe prove that is the coarsest (3, 2)-fuzzy topology over such that is (3, 2)-fuzzy continuous.(1)We can write for any that Similarly, we immediately have for any which implies . Now, as , we have . In a similar manner, it is easy to see that .(2)Assume that . Then, for , there exists such that which implies and . Thus, we obtain for any that Similarly, it is not difficult to see that . Hence, we get .(3)Assume that is an arbitrary subfamily of . Then, for any , there exists such that which implies and . Therefore, one can get for any thatOn the other hand, it is easy to see that . Thus, we have .
From Theorem 11, the (3, 2)-fuzzy continuity of is trivial. Now, we prove that is the coarsest (3, 2)-fuzzy topology over such that is (3, 2)-fuzzy continuous. Let be a (3, 2)-fuzzy topology over such that is (3, 2)-fuzzy continuous. If , then there is such that . Since is (3, 2)-fuzzy continuous with respect to , we have . Hence, , as required.
3.3. (3, 2)-Fuzzy Separation Axioms
Separation axioms are one of the most important and popular notions in topological studies. They have been studied and applied to model some real-life issues in soft setting as explained in [16, 17].
Definition 10. Let and be a fixed element in . Suppose that and are two fixed real numbers such that . Then, a (3, 2)-fuzzy point is defined to be a (3, 2)-fuzzy set of as follows.for . In this case, is called the support of . A (3, 2)-fuzzy point is said to belong to a (3, 2)-fuzzy set of denoted by if and . Two (3, 2)-fuzzy points are said to be distinct if their supports are distinct.
Remark 6. Let and be two (3, 2)-fuzzy sets of . Then, if and only if implies for any (3, 2)-fuzzy point in .
Definition 11. Let , , and . A (3, 2)-fuzzy topological space is said to be(1) if for each pair of distinct (3, 2)-fuzzy points in , there exist two open (3, 2)-fuzzy sets and such that(2) if for each pair of distinct (3, 2)-fuzzy points in , there exist two open (3, 2)-fuzzy sets and such that
Proposition 1. Let be a (3, 2)-fuzzy topological space. If is , then is .
Proof. The proof is straightforward from Definition 11.
Here is an example which shows that the converse of above proposition is not true in general.
Example 10. Consider with the (3, 2)-fuzzy topology , where . Then, is but not because there does not exist an open (3, 2)-fuzzy set such that .
Theorem 14. Let be a (3, 2)-fuzzy topological space, , and . If is , then for each pair of distinct (3, 2)-fuzzy points , of , .
Proof. Let be and , be any two distinct (3, 2)-fuzzy points of . Then, there exist two open (3, 2)-fuzzy sets and such thatLet exist. Then, is a closed (3, 2)-fuzzy set which does not contain but contains . Since is the smallest closed (3, 2)-fuzzy set containing , then , and therefore . Consequently, .
Theorem 15. Let be a (3, 2)-fuzzy topological space. If is closed (3, 2)-fuzzy set for every , then, is .
Proof. Suppose is a closed (3, 2)-fuzzy set for every . Let , be any two distinct (3, 2)-fuzzy points of ; then, implies that and are two open (3, 2)-fuzzy sets such thatThus, is .
4. (3, 2)-Fuzzy Relations
A relation is a mathematical description of a situation where certain elements of sets are related to one another in some way. The system of fuzzy relation equations was first studied by Sanchez [18–21], who used it in medical research. Biswas [22] defined the method of intuitionistic medical diagnosis which involves intuitionistic fuzzy relations. Kumar et al. [23] used the applications of intuitionistic fuzzy set theory in diagnosis of various types of diseases. The notion of max-min-max composite relation for Pythagorean fuzzy sets was studied by Ejegwa [24], and the approach was improved and applied to medical diagnosis.
In this section, we introduce the notions of max-min-max composite relation and improved composite relation for (3, 2)-FSs. Moreover, we provide a numerical example to elaborate on how we can apply the composite relations to obtain the optimal choices.
Definition 12. Let and be two (crisp) sets. The (3, 2)-fuzzy relation (briefly, (3, 2)-FR) from to is a (3, 2)-FS of characterized by the degree of membership function and degree of non-membership function . The (3, 2)-FR from to will be denoted by . If is a (3, 2)-FS of , then(1)The max-min-max composition of the (3, 2)-FR with is a (3, 2)-FS of denoted by C = R o D and is defined by(2)The improved composite relation of with is a (3, 2)-FS of denoted by C = R o D, such that
Definition 13. Let and be two (3, 2)-FRs. Then, for all and ,(1)The max-min-max composition R o Q is the (3, 2)-fuzzy relation from to defined by(2)The improved composite relation R o Q is the (3, 2)-fuzzy relation from to such that
Remark 7. The improved composite and max-min-max composite relations for (3, 2)-fuzzy sets are calculated by the following:
Example 11. Let and be two (3, 2)-fuzzy sets for . Assume thatBy using Definitions 12 (1) and 13 (1), respectively, we find the max-min-max composite relation with application to and as follows:It is obvious that the minimum value of the membership values of the elements (that is, ) in and , respectively, is , and 0.8. Also, the maximum value of the non-membership values of the elements (that is, ) in and , respectively, is , and 0.69. From Remark 7, we can getAgain, by using Definitions 12 (2) and 13 (2), respectively, we find the improved composite relation with application to and as follows:From Remark 7, we can getHence, from (43) and (45), we obtain that the improved composite relation produces better relation with greater relational value when compared to max-min-max composite relation.
5. Application of (3, 2)-Fuzzy Sets
We localize the idea of (3, 2)-FR as follows.
Let be a finite set of subjects related to the colleges, be a finite set of colleges, and be a finite set of students. Suppose that we have two (3, 2)-FRs, and , such thatwhere denotes the degree to which the student (t) passes the related subject requirement (r). denotes the degree to which the student (t) does not pass the related subject requirement (r). denotes the degree to which the related subject requirement (r) determines the college (b). denotes the degree to which the related subject requirement (r) does not determine the college (b).
is the composition of R and . This describes the state in which the students, , with respect to the related subject requirement, , fit the colleges, . Thus, and , where , , and take values from .
The values of and of the composition T = R o U are as follows (Table 1).
If the value of T is given by the following:then the student placement can be achieved.
5.1. Application Example
By using a hypothetical case with quasi-real data, we apply this method. Let be the set of students for the colleges; = {English Lang., Mathematics, Biology, Physics, Chemistry, Computer Sci.} be the set of related subject requirement to the set of colleges; and = {College of Engineering (E), College of Medicine (M), College of Agricultural Engineering Sciences (AE), College of Sport Sciences (Sp), College of Science (S)} be the set of colleges the students are vying for (Algorithm 1).
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From Table 4 and based on suitability of the students to the list of colleges, this decision making is made:(1) and are suitable to study at College of Agricultural Engineering Sciences.(2) is suitable to study at College of Agricultural Engineering Sciences, College of Sport Sciences, and College of Science.(3) is suitable to study at College of Medicine.(4) is suitable to study at College of Agricultural Engineering Sciences and College of Science.
6. Discussion
The main idea of this work is to introduce a new type of fuzzy set called (3, 2)-FS. We illustrated that this type produces membership grades larger than intuitionistic and Pythagorean fuzzy sets which are already defined in the literature. However, Fermatean fuzzy sets give a larger space of membership grades than (3, 2)-FS. Figure 2 illustrates the relationships between these types of fuzzy sets.
We summarize the relationships in terms of the space of membership and non-membership grades in the following figure.
Regarding topological structure, we illustrated that every fuzzy topology in the sense of Chang (intuitionistic fuzzy topology and Pythagorean fuzzy topology) is a (3, 2)-fuzzy topology. In contrast, every (3, 2)-fuzzy topological space is a Fermatean fuzzy topological space because every (3, 2)-fuzzy subset of a set can be considered as a Fermatean fuzzy subset. The next example elaborates that Fermatean fuzzy topological space need not be a (3, 2)-fuzzy topological space.
Example 12. Let . Consider the following family of Fermatean fuzzy subsets , whereObserve that is a Fermatean fuzzy topological space, but is not a (3, 2)-fuzzy topological space.
7. Conclusions
In this paper, we have introduced a new generalized intuitionistic fuzzy set called (3, 2)-fuzzy sets and studied their relationship with intuitionistic fuzzy, Pythagorean fuzzy, and Fermatean fuzzy sets. In addition, some operators on (3, 2)-fuzzy sets are defined and their relationships have been proved. The notions of (3, 2)-fuzzy topology, (3, 2)-fuzzy neighborhood, and (3, 2)-fuzzy continuous mapping were studied. Furthermore, we introduced the concept of (3, 2)-fuzzy points and studied separation axioms in (3, 2)-fuzzy topological space. We also introduced the concept of relation to (3, 2)-fuzzy sets, called (3, 2)-FR. Moreover, based on academic performance, the application of (3, 2)-FSs was explored on student placement using the proposed composition relation.
In future work, more applications of (3, 2)-fuzzy sets may be studied; also, (3, 2)-fuzzy soft sets may be studied. In addition, we will try to introduce the compactness and connectedness in (3, 2)-fuzzy topological spaces. The motivation and objectives of this extended work are given step by step in this paper.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.