(3, 2)-Fuzzy Sets and Their Applications to Topology and Optimal Choices

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.


Introduction
e concept of fuzzy sets was proposed by Zadeh [1]. e 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. e integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [2][3][4]. e 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][7][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. e concept of fuzzy topological spaces was introduced by Chang [11]. He studied the topological concepts like continuity and compactness via fuzzy topological spaces. en, 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. e 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.

Theorem 1.
e 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 r 1 , r 2 ∈ [0, 1], we have en, we get Hence, the space of (3, 2)-fuzzy membership grades is larger than the space of intuitionistic membership grades and Pythagorean membership grades. is development can be evidently recognized in Figure 1.

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. en, 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.
Remark 3. We showed that every fuzzy set D on a set X is a (3, 2)-fuzzy set having the form D � 〈r, α D (r), 1 − α D (r)〉: r ∈ X}. en, every fuzzy topological space (X, τ 1 ) in the sense of Chang is obviously a (3, 2)-fuzzy topological space in the form τ � D: α D ∈ τ 1 whenever we identify a fuzzy set in X whose membership function is α D with its counterpart D � 〈r, α D (r), 1 − α D (r)〉: r ∈ X . Similarly, one can note that every intuitionistic fuzzy topology (Pythagorean fuzzy topology) is (3, 2)-fuzzy topology. e following examples explain this note.
en, τ is fuzzy topology on X, and hence it is (3, 2)fuzzy topology.
On the other hand, from the facts int(D 1 ) ⊂ D 1 and (4) can be proved similar to (3). □ Theorem 6. Let (X, τ) be a (3, 2)-fuzzy topological space and D be (3, 2)-FS in X. en, the following properties hold: Proof. We only prove (1); the other parts can be proved similarly.
x ∈ X and suppose that the family of open (3, 2)-fuzzy sets contained in D is indexed by the family

(3, 2)-Fuzzy Continuous Maps
Definition 7. Let f: X ⟶ Y be a map and A and B be (3, 2)-fuzzy subsets of X and Y, respectively. e functions of membership and non-membership of the image of A, denoted by f [A], are, respectively, calculated by e functions of membership and non-membership of preimage of B, denoted by f − 1 [B], are, respectively, calculated by In contrast, f − 1 (y) � ϕ leads to the fact that It is easy to prove the case of f − 1 [B].

Proof
(1) Consider v ∈ X and let B be a (3, 2)-fuzzy subset of Y. en, Similarly, one can have (2) For any w ∈ Y such that f − 1 (w) ≠ ϕ and for any (3, 2)-fuzzy subset A of X, we can write Now from (18) e proof is easy when us, On the other hand, □ e proof of the following result is easy, and hence it is omitted.
Theorem 8. Let X and Y be two non-empty sets and f: X ⟶ Y be a map. en, the following statements are true: Proof. e proof is easy. (

(4) For each (3, 2)-FS A of X and for each nbd V of f[A], f − 1 [V] is a nbd of A.
Proof.
(1)⇒ (2) is means that 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.
Proof. Let us define a class of (3, 2)-fuzzy subsets τ 1 of X by We prove that τ 1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous.
(1) We can write for any x ∈ X that Similarly, we immediately have In a similar manner, it is easy to see that 1 Similarly, it is not difficult to see that . Hence, we get D 1 ∩ D 2 ∈ τ 1 . (3) Assume that D i i∈I is an arbitrary subfamily of τ 1 .
en, for any i ∈ I, there exists B i ∈ τ 1 such that On the other hand, it is easy to see that From eorem 11, the (3, 2)-fuzzy continuity of f is trivial. Now, we prove that τ 1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. Let τ 2 ⊆τ 1 be a (3, 2)-fuzzy topology over X such that f is (3, 2)fuzzy continuous.  [16,17].
for y ∈ X. In this case, x is called the support of p x (r 1 ,r 2 ) . A (3, 2)-fuzzy point p x (r 1 ,r 2 ) is said to belong to a (3, 2)-fuzzy set D � 〈x, α D (x), β D (x)〉 of X denoted by p x (r 1 ,r 2 ) ∈ D if r 1 ≤ α D (x) and r 2 ≥ β D (x). Two (3, 2)-fuzzy points are said to be distinct if their supports are distinct.
e proof is straightforward from Definition 11. □ Here is an example which shows that the converse of above proposition is not true in general.

(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. e system of fuzzy relation equations was first studied by Sanchez [18][19][20][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. e notion of maxmin-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-minmax 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 X and Y be two (crisp) sets. e (3, 2)fuzzy relation R (briefly, (3, 2)-FR) from X to Y is a (3, 2)-FS of X × Y characterized by the degree of membership function α R and degree of non-membership function β R . e (3,2) for all n ∈ Y. (36) for all n ∈ Y. (37) Definition 13. Let Q(X ⟶ Y) and R(Y ⟶ Z) be two (3, 2)-FRs. en, for all (m, r) ∈ X × Z and n ∈ Y, (1) e max-min-max composition R o Q is the (3, 2)fuzzy relation from X to Z defined by (2) e improved composite relation R o Q is the (3, 2)fuzzy relation from X to Z such that Remark 7. e improved composite and max-min-max composite relations for (3, 2)-fuzzy sets are calculated by the following: Example 11. Let D 1 and D 2 be two (3, 2)-fuzzy sets for X � x 1 , x 2 , x 3 , x 4 . Assume that By using Definitions 12 (1) and 13 (1), respectively, we find the max-min-max composite relation with application to D 1 and D 2 as follows: It is obvious that the minimum value of the membership values of the elements (that is, x 1 , x 2 , x 3 , x 4 ) in D 1 and D 2 , respectively, is 0.7, 0.5, 0.6, and 0.8. Also, the maximum value of the non-membership values of the elements (that is, Again, by using Definitions 12 (2) and 13 (2), respectively, we find the improved composite relation with application to D 1 and D 2 as follows:  (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.
Let S � r 1 , . . . , r l be a finite set of subjects related to the colleges, C � b 1 , . . . , b m be a finite set of colleges, and A � t 1 , . . . , t n be a finite set of students. Suppose that we have two (3, 2)-FRs, U(A ⟶ S) and R(S ⟶ C), such that where α U (t, r) denotes the degree to which the student (t) passes the related subject requirement (r). β U (t, r) denotes the degree to which the student (t) does not pass the related subject requirement (r). α R (r, b) denotes the degree to which the related subject requirement (r) determines the college (b). β R (r, b) denotes the degree to which the related subject requirement (r) does not determine the college (b).
T � RoU is the composition of R and U. is describes the state in which the students, t i , with respect to the related subject requirement, r j , fit the colleges, b k . us, ∀t i ∈ A and b k ∈ C, where i, j, and k take values from 1, . . . , n.
e values of α RoU (t i , b k ) and β RoU (t i , b k ) 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.

Discussion
e 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. (3,2)-FS

IFS PFS FFS
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. e next example elaborates that Fermatean fuzzy topological space need not be a (3, 2)-fuzzy topological space.
Example 12. Let X � x 1 , x 2 . Consider the following family of Fermatean fuzzy subsets τ � 1 X , 0 X , D 1 , D 2 , where Computational Intelligence and Neuroscience Table 4: Greatest value given by T � α T (t i , b k ) − β T (t i , b k ) · π T (t i , b k ). Step 1. e (3, 2)-fuzzy relation U(A ⟶ S) and the (3, 2)-fuzzy relation R(S ⟶ C) are given as in Tables 2 and 3, respectively. ese data in (3, 2)-F values are assumably obtained after students finished from preparatory school.
Step 2. Compute the composition R o U as in Table 1. Table 4.
Step 4. We present the decision making from Table 4. e greatest value of relation between students and colleges is taken for decisions.
ALGORITHM 1: Determination of the optimal college for students.

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. e 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. e 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.  Computational Intelligence and Neuroscience 13