Some New Operations on Pythagorean Fuzzy Sets AdakAmal Kumar KumarDheeraj EdalatpanahSeyyed Ahmad Department of Mathematics, Ganesh Dutt College, Begusarai, India. Department of Mathematics, Lalit Narayn Mithila University, Darbhanga, India. Department of Applied Mathematics; Ayandegan Institute of Higher Education, Tonekabon, Iran.

1,∗ 2 3

Uncertainty Uncert. Amal Keywords: 1|Introduction Corresponding Received: 08 September 2023 Revised: 29 December 2023 Accepted: 18 February 2024 Adak, A. K., Kumar Disc. Kumar, D., Author: Intuitionistic Appl. Adak amaladak17 & Discourse Edalatpanah, S. A. (2024). Some new operations on Vol. fuzzy , @ 1, Dheeraj gmail.com sets, No. Pythagorean 1 (2024) and Kumar 11–19. Applications fuzzy , Seyyed sets, Operations Ahmad Edalatpanah on Pythagorean fuzzy sets.

The https://doi.org/10.48313/uda.v1i1.17 Pythagorean fuzzy sets. concept of Pythagorean Uncertainty discourse and applications, 1 Fuzzy Sets (PFSs) was(1), 11-19. initially developed by Yager in 2013, which provides a novel 1 way to model uncertainty and vagueness with high precision and accuracy compared to Pythagorean Zadeh Department [1] introduced of Mathematics, theGanesh idea of Dutt fuzzy College, setBegusarai, which has India; a membership amaladak17@gmail.com. function, µ that assigns to each element Fuzzy 2 Sets (PFSs).The concept was concretely designed to represent uncertainty and vagueness in

of Department the universe of Mathematics, of discourse, www.uda.reapress.com Lalita Narayn number Mithila from University, the unit Darbhanga, interval India; [0, 1] dheerajkumarbgs1990@gmail.com. to indicate the degree of belongingness to

math Department ematical of Applied way and Mathematics; to furnish Ayandegan a formalized Institute of tool Higher forEducation, tacklingTonekabon, imprecision Iran;to s.a.edalatpanah@aihe.ac.ir. real problems. In this the set under consideration. The notion of fuzzy sets generalizes classical sets theory by allowing intermediate situations paper, various between operations the whole in Pythagorean and nothing. Fuzzy In aSets fuzzy are set, discussed. a membership Some theorems function are is defined proved to fordescribe estab- the degree lishing of the membership properties of an Pythagorean element tofuzzy a class. operators The membership with respect value to ranges different from Pythagorean 0 to 1, where fuzzy 0 shows sets. that the element does not belong to a class, 1 means belongs, and other values indicate the degree of membership to a class. For fuzzy sets, the membership function replaced the characteristic function in crisp sets. Since the pioneering work of Zadeh [1], the fuzzy set theory has been used in different disciplines such as management sciences, engineering, mathematics, social sciences, statistics, signal processing, artificial intelligence, automata theory, medical and life sciences.

The concept of fuzzy sets theory seems to be inconclusive because of the exclusion of nonmembership function and the disregard for the possibility of hesitation margin. Atanassov [2] critically studied these short comings and proposed a concept called Intuitionistic Fuzzy Sets (IFSs) . The construct (that is, IFSs) incorporates both membership function, µ and nonmembership function, ν with hesitation margin, π (that is, neither membership nor nonmembership functions), such that µ + ν ≤ 1 and µ + ν + π = 1. The notion of IFSs provides a flexible framework to elaborate uncertainty and vagueness. There are lot of research work done in area of IFSs in [3-10].

There are many situations, where µ + ν ≥ 1 which violate the conditions IFSs. This limitation in IFS naturally led to a construct a new idea, called Pythagorean fuzzy sets (PFSs). Pythagorean fuzzy sets (PFSs) proposed by Yager [11-13], a new tool to deal with vagueness considering the membership grade (µ) and non-membership grade (ν) satisfying the conditions 0 ≤ µ ≤ 1; 0 ≤ ν ≤ 1, and also, it follows that µ + ν + π ≤ 1, where π is the hesitant index of Pythagorean fuzzy sets. PFS is more capable than IFS to model the vagueness in the practical problem.

In this paper, we delve into the realm of Pythagorean fuzzy sets and explore various operations that can be performed within this framework. These operations play a pivotal role in manipulating Pythagorean fuzzy sets, facilitating effective decision-making and inference processes in diverse applications such as expert systems, pattern recognition, decision analysis, and more. Through theoretical analysis and illustrative examples, we aim to elucidate the underlying principles and significance of these operations, showcasing their applicability and effectiveness in handling uncertainty and vagueness inherent in real-world decision scenarios.

The rest of the paper organized as follows. In Section 2, the preliminaries and some definitions are given and present some operations of Pythagorean fuzzy sets. In Section 3, introduce the new operations of Pythagorean fuzzy sets and discussed some important results Pythagorean fuzzy sets. At the end, a conclusion is made in Section 4.

 Preliminary Concepts

This part provides a concise overview of the key concepts and outcomes that are essential for understanding the subsequent sections. In this text, we discussed fundamental concepts fuzzy sets, Intuitionistic fuzzy sets and Pythagorean fuzzy set that are used in the rest of the paper.

Definition 1. The fuzzy set A is defined as the collection of pairs A = (ξ,αA(ξ)), where ξbelongs to X, a universal set. Here, α A(ξ ) represents the membership function of ξin A, which assigns a real number between 0 and 1 to each element in X.

Definition 2. An Intuitionistic fuzzy set (IFS), denoted by A, is an entity that exists in a nonempty set X and is is defined as the collection of pairs A = (ξ , α A(ξ ), β A(ξ )), where ξbelongs to X. The degree of membership function α A(ξ ), maps elements from X to the interval [0, 1].

The non-membership function β A(ξ ) maps the set X to the interval [0, 1]. They satisfy the condition 0 ≤α A(ξ )) + β A(ξ ) ≤1 for every ξ∈ X. An Intuitionistic fuzzy set A is represented symbolically as A = (α A, β A).

The degree of indeterminacy hA(ξ ) = p1 −α A(ξ ) −β A(ξ ).

In practice, the condition 0 ≤ρ (ξ ) + σ (ξ ) ≤1 may not be true for any reason. For example 0.5 + 0.7 = 1.2 > 1, but 0.52 + 0.72 < 1, or 0.6 + 0.6 = 1.2 > 1, but 0.62 + 0.62 < 1. To address this issue, Yager [21, 22] pro-posed the provide the notion of the Pythagorean fuzzy set in 2013.

Proof: 13 from definition, we have P1 ∩ P2 = {⟨x, min ρP (x), ρP (x), max σP (x), σP (x)⟩ : x ∈ E} Definition 3. A Pythagorean fuzzy set, Pˆ in a finite universe of discourse X is given by and P1@P2 = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} Let P1 and P2 be two Pythagorean fuzzy sets, then the following operations and relations can be defined as Now, P ˆ = {⟨ξ,ρ (ξ), σ (ξ)⟩|ξ ∈ X}, P ⊆ P iff (ρ (x) ≤ ρ (x)) and (σ (x) ≥ σ (x P ˆ )) (forP ˆall x ∈ E)

Some Comparison Operation spaces for intuitionistic of Pythagorean fuzzy and pythagorean Fuzzy Sets membership aea|Uncert. Disc. Appl. 1(1) (2024) 11-19 grades.

1 2 P1 P2 P1 P2

P1 = P2 iff (ρP (x P) 1 = ∩ P ρP 2)@ (xP )) 3 and = (σ {⟨ Px, (x min ) ={σ ρP (x)) ), ρ(for P (xall )},x max ∈ E {) σP (x), σP (x)}⟩ : x ∈ E}, where ρ Pˆ(ξ ) : X → [0, 1] indicates the grade to which the element ξ∈ X and σ Pˆ(ξ ) : X → [0, 1] represents P1 ∩ P2 = {⟨x. min(ρP (x), ρP (x)), max(σP (x), σPP (x))⟩ : x ∈ E} 1 2 @{⟨x, ρ1 (x), σ 2(x)⟩ : x ∈ E} the grade to which the element ξ∈ X is not P3 a member P3 of the set Pˆ, with the condition that P1 ∪ P2 = {⟨x. max(ρP (x), ρP (x)), min(σP (x), σP (x))⟩ : x ∈ E} LetρP (x) < ρP ( 2x) and σP 2 (x) > σP (x) p1 + P2 = {⟨x, ρP (x) + ρP (x) − ρP (x)ρ0 P ≤ (x, (ρ), ˆσ (ξ P))(x+ ).σ (P σ ˆ (( xξ))) ⟩ : x ≤∈1,E}

1 2 1 2 P 1 2 P

= {⟨x, ρP (x), σP (x)⟩ : x ∈ E}@{⟨x, ρP (x), σP (x)⟩ : x ∈ E} P1.P2 = {⟨x, ρP (x)ρP (x), σP (x) + σP (x)1− σP (x 2 ).σP (x)⟩ : x ∈ E} 3 3 for P @ all P ξ = ∈{⟨ Xx, . ρ (x) + ρ (x)/2= , σ {⟨ (x x, ) ρ + σ(x)(x + )/ ρ2⟩ ( :x x)/ ∈2,E σ} (x) + σ (x)/2⟩ : x ∈ E} (1)

1 2 P1 P2 P1 P1 P2 P3 P2 P3

The Theorem degree 1. of indeterminacy Let P1 = (ρ P1,hP σ P1), ˆ(ξ ) = P2 p1 =− (ρ (ρ P2, Pˆ(σ ξ ))2 P2)− and (σ PP3 ˆ(ξ ))2. = (ρ P3, σ P3) be three Pythagorean fuzzy sets.

Then (P1 ∩P2)@P3 = (P1@P3) ∩(P2@P3).

Again, (P1@P3) ∩ (P2@P3) = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} ∩{⟨x, ρP (x) + ρP (x)/2, σP (x) + ρP (x)/2⟩ : x ∈ E} = {⟨x, min{ρP (x) + ρP (x) + ρP (x)/2}, max{(σP (x) + σP (x))/2; (σP (x) + σP (x))/2} : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, σP (x) + ρP (x)/2⟩ : x ∈ E} (2) From equation (1) and (2), we get (P1 ∩ P2)@P3 = (P1@P3) ∩ (P2@P3).

Theorem 2. Let P1 = (ρP , σP ), P2 = (ρP , σP ) and P3 = (ρP , σP ) be three Pythagorean fuzzy sets, then P1@(P2 ∩ P3) = (P1@P2) ∩ (P1@P3)

Proof: We know that, P1 ∩ P2 = {⟨x, min ρP (x), ρP (x), max σP (x), σP (x)⟩ : x ∈ E} P1@P2 = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} P1@(P2 ∩ P3) = {⟨x, ρP (x), σP (x)⟩ : x ∈ E} @{⟨x, min{ρP (x), ρP (x)}, max{σP (x), σP (x)}⟩ : x ∈ E} LetρP (x) < ρP (x) and σP (x) < σP (x) = {⟨x, ρP (x), σP (x)⟩ : x ∈ E}@{⟨x, ρP (x), σP (x)⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} (3) Also, (P1@P2) ∩ (P1@P3) = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} ∩{⟨x, ρP (x) + ρP (x)/2, σP (x) + ρP (x)/2⟩ : x ∈ E} = {⟨x, min{ρP (x) + ρP (x) + ρP (x)/2}, max{(σP (x) + σP (x))/2; (σP (x) + σP (x))/2}⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, σP (x) + ρP (x)/2⟩ : x ∈ E} (4) From equation (3) and (4), conveys P1@(P2 ∩ P3) = (P1@P2) ∩ (P1@P3).

Theorem 3. Let P1 = (ρP , σP ), P2 = (ρP , σP ) and P3 = (ρP , σP ) be three Pythagorean fuzzy sets. Then (P1 ∪ P2)@P3 = (P1@P3) ∪ (P2@P3)

Proof: For three PFSs A, B and C, from definition (P1 ∪ P2)@P3 = {⟨x, max{ρP (x), ρP (x)}, min{σP (x), σP (x)⟩ : x ∈ E}@{⟨x, ρC(x), σCx)⟩ : x ∈ E} LetρA < ρB, and σA(x) < σB(x) = {⟨x, , ρB(x), σA(x)⟩ : x ∈ E}@{⟨x, ρC(x), σC(x)⟩ : x ∈ E} = {⟨x, ρB(x) + ρC(x)/2, σA(x) + σC(x)/2⟩ : x ∈ E} (5) Also, (P1@P3) ∪ (P2@P3) = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} ∪{⟨x, ρP (x) + ρP (x)/2.σP (x) + σP (x)/2⟩ : x ∈ E} = {⟨x, max{ρP (x) + ρP (x)/2, ρP (x) + ρP (x)/2}, min{σP (x) + σP (x)/2, σP (x) + σP (x)/2}⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} (6) From equation (5) and (6) proposes (P1 ∪ P2)@P3 = (P1@P3) ∪ (P2@P3).

Proof: For three PFS A, B and C, From definition (P1@P2).P3 = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E}, {⟨x, ρP (x), σP (x)⟩ : x ∈ E} = {⟨x, ρP (x).[ρP (x) + ρP (x)]/2, (9) [σP (x) + σP (x)]/2 + σC − [σP (x) + σP (x)]/2.σP (x)⟩ : x ∈ E}

1 2 1 2 3

Also, P1.P3@P2.P3 = {⟨x, ρP (x).ρP (x), σP (x) + σP (x) − σP (x).σP (x)⟩ : x ∈ E} @{⟨x, ρP (x).ρP (x), σP (x) + σP (x) − σP (x).σP (x)⟩ : x ∈ E} = {⟨x, [ρP (x).ρP (x) + ρP (x).ρP (x)]/2, σP (x) + σP (x) − σP (x).σP (x)σP (x) + σP (x) − σP (x).σP (x)/2⟩ : x ∈ E} = {⟨x, [ρP (x) + ρP (x)]/2.ρP (x), {σP (x) + σP (x) + σP (x) − σP (x)[σP (x) + σP (x)]}/2⟩ : x ∈ E} = {⟨x, [ρP (x) + ρP (x)]/2}.ρP (x), (10) [σP (x) + σP (x)]/2 + σP (x) − σP (x)[σP (x) + σP (x)]/2⟩ : x ∈ E} From equation (9) and (10) presents (P1@P2).P3 = P1.P3@P2.P3.

Proof: 15 For three PFS A, B and C, From definition Theorem Theorem 5. 6. 4. Let Let P1 = (ρ P1, σ P1), P2 = (ρ P2, σ P2) and P3aea = (ρ P3,|Uncert. Disc. Appl. 1(1) (2024) 11-19 σ P3) be three Pythagorean fuzzy sets. sets, Then then Then(P1 (P1@P2).P3 P1@(P2 ∩P2).P3 ∪P P3) 1 = = @( P1.P3@P2.P3 (P1.P3) = P2(P1@P2) ∪ P3)∩(P2.P3) =∪ @ (P1@P3) {⟨x, ρP (x), σP (x)⟩ : x ∈ E} = {⟨X,max{ρP (x), ρP (x)}, min{σP (x), σP (x)⟩ : x ∈ E} LetρP < ρP , and σP (x) < σP (x) @{⟨x, ρP (x), σP (x)⟩ : x ∈ E} = @{⟨x, , ρP (x), σP (x)⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, (7) σP (x) + σP (x)/2⟩ : x ∈ E} Also, (P1@P2) ∪ (P1@P3) = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} ∪{⟨x, ρP (x) + ρP (x)/2.σP (x) + σP (x)/2⟩ : x ∈ E} = {⟨x, max{ρP (x) + ρP (x)/2, ρP (x) + ρP (x)/2}, min{σP (x) + σP (x)/2, σP (x) + σP (x)/2}⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, (8) σP (x) + σP (x)/2⟩ : x ∈ E}

From Eq. (7) and Eq.(8) yields P1@(P2 ∪ P3) = (P1@P2) ∪ (P1@P3).

Proof: Let ρP (x) < ρP (x) and σP (x) > σP (x), then

(P1 ∩ P2).P3 = {⟨x, ρP (x), σP (x)⟩ : x ∈ E}.{⟨x, ρP (x), σP (x)⟩ : x ∈ E} = {⟨x, ρP (x).ρP (x).σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} (11) Also (P1.P3) ∩ (P2.P3) = {⟨x, ρP (x), σP (x)⟩ : x ∈ E}.{⟨x, ρP (x), σP (x)⟩ : x ∈ E} ∩{⟨x, ρP (x), σP (x)⟩ : x ∈ E}.{⟨: x, ρP (x), σP (x)⟩ : x ∈ E} = {⟨x, ρP (x)ρP (x), σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} ∩{⟨x, ρP (x)ρP (x), σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} = {⟨x, min{ρP (x)ρP (x).ρP (x)ρP (x)} max{σP (x)σP (x) − σP (x)σP (x), σP (x) + σP (x) − σP (x)σP (x)}⟩ : x ∈ E} SinceρP (x) < ρP (x), σP (x) > σP (x) = {⟨x, ρP (x)ρP (x), σP (x) + σP (x) − σP (x)σP (x) : x ∈ E} (12) From equation (11) and (12) gives Therefore, (P1 ∩ P2).P3 = (P1.P3) ∩ (P2.P3).

Proof: P1 and P2 be two PFSs, then (P1 ∩ P2) + (P1 ∪ P2) = {⟨x, min{ρP (x).ρP (x)}, max{σP (x).σP (x)}⟩ : x ∈ E} +{⟨x, max{ρP (x).ρP (x)}, min{σP (x).σP (x)}⟩ : x ∈ E} LetρP (x) < ρP (x) and σP (x) < σP −2(x) = {⟨x, ρP (x).σP (x)⟩ : x ∈ E} + {⟨x, ρP (x).σP (x)⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x) − ρP (x)ρP (x), σP (x)σP (x)⟩ : x ∈ E} = P1 + P2 by definition of” + ”

As a result, (P1 ∩ P2) + (P1 ∪ P2) = P1 + P2.

Proof: A and B be two PFSs, then (P1 ∩ P2).(P1 ∪ P2) = {⟨x, min{ρP (x), ρP (x)}, max{σP (x), σP (x)}⟩ : x ∈ E}.

{⟨x, max{ρP (x), ρP (x)}, min{σP (x), σP (x)}⟩ : x ∈ E} = {⟨x, ρP (x), σP (x)⟩ : x ∈ E}.{⟨x, ρP (x), σP (x)⟩ : x ∈ E} = {⟨x, ρP (x)ρP (x), σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} = {⟨x, ρP (x)ρP (x), σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} = P1.P2 by definition of”.”

Consequently, (P1 ∩ P2).(P1 ∪ P2) = P1.P2.

Theorem 9. Let P1 = (ρP , σP ) and P2 = (ρP , σP ) be two Pythagorean fuzzy sets. Then (P1 + P2)@(P1.P2) = P1@P2

Theorem 8. 7. Let LetP1 P1== (ρ ( P1, ρ P1, σ P1) σ P1) and and P2 P2 = (ρ = P2, (ρ σ P2, P2)σ be P2) two bePythagorean two Pythagorean fuzzy sets. fuzzyThen sets.(P1 Then ∩P2). (P1 (P1 ∩P2)+(P1 ∪ P2) ∪ = P2) P1.P2 = P1 + P2 Proof: P1 and P2 be two PFSs, then (P1 + P2)@(P1.P2) = {⟨x, ρP (x) + ρP (x) − ρP (x).ρP (x), σP (x)σP (x)⟩ : x ∈ E} @{⟨x, ρP (x).ρP (x).σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} = {⟨, {ρP (x) + ρP (x) − ρP (x)ρP (x) + ρP (x)ρP (x)}/2, {σP (x).σP (x) + σP (x) + σP (x) − σP (x)σP (x)}/2⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} = P1@P2 by definition.

Thus, (P1 + P2)@(P1.P2) = P1@P2.

Theorem 10. Let P1 = (ρP , σP ) and P2 = (ρP , σP ) be two Pythagorean fuzzy sets. Then (P1∩P2)@(P1∪P2) = P1@P2

Proof: P1 and P2 be two PFSs, then (P1 ∩ P2)@(P1 ∪ P2) = {⟨x, min{ρP (x), ρP (x)}, max{σP (x), σP (x)}⟩ : x ∈ E} @{⟨x, max{ρP (x), ρP (x)}, min{σP (x), σP (x)}⟩ : x ∈ E} LetρP (x) < ρP (x) and σP (x) < σP (x) = {⟨X, ρP (x), σP (x)⟩/x ∈ E}@{⟨x, ρP (x).σP (x)⟩ : x ∈ E} = {⟨x, [ρP (x) + ρP (X)]/2, [σP (X) + σP (x)]/2⟩ : x ∈ E} = P1@P2 by definition Therefore, (P1 ∩ P2)@(P1 ∪ P2) = P1@P2.

Theorem 11. Let P1 = (ρP , σP ) and P2 = (ρP , σP ) be two Pythagorean fuzzy sets. Then (P1∩P2)@(P1∪P2) = (P1 + P2)@(P1.P2)

Proof: Let P1 and P2 be two PFSs, then (P1 ∩ P2)@(P1 ∪ P2) = {⟨x, min{ρP (x), ρP (x)}, max{σP (x), σP (x)}⟩ : x ∈ E} @{⟨x, max{ρP (x), ρP (x)}, min{σP (x), σP (x)}⟩ : x ∈ E} LetρP (x) < ρP (x) and σP (x) < σP (x) = {⟨X, ρP (x), σP (x)⟩/x ∈ E}@{⟨x, ρP (x).σP (x)⟩ : x ∈ E} = {⟨x, [ρP (x) + ρP (X)]/2, [σP (X) + σP (x)]/2⟩ : x ∈ E} = P1@P2 by definition (13)

(P1 + P2) @ (P1.P2) = {⟨x, ρP (x) + ρP (x) − ρP (x).ρP (x), σP (x)σP (x)⟩ : x ∈ E} @{⟨x, ρP (x).ρP (x).σP (x) + σP (x) − σP (x)σP (x)⟩ : x ∈ E} = {⟨, {ρP (x) + ρP (x) − ρP (x)ρP (x) + ρP (x)ρP (x)}/2, {σP (x).σP (x) + σP (x) + σP (x) − σP (x)σP (x)}/2⟩ : x ∈ E} = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} = P1@P2 by definition. (14) Hence, (P1 ∩ P2)@(P1 ∪ P2) = (P1 + P2)@(P1.P2).

 Conclusion

This paper provide a comprehensive understanding of various operations on Pythagorean fuzzy sets, offering insights into their theoretical foundations, computational aspects, and practical implications. By elucidating these operations, we seek to contribute to the advancement of decision support systems and intelligent methodologies capable of handling uncertainty in a more nuanced and effective manner.

aea|Uncert. Disc. Appl. 1(1) (2024) 11-19 In future, we will discuss the potential applications of these operations in fields such as engineering, finance, medicine, and artificial intelligence, highlighting the practical utility and versatility of Pythagorean fuzzy sets in addressing complex decision-making problems.

 Acknowledgments

The authors are very grateful and would like to express their sincere thanks to the anonymous referees and Editor for their valuable comments to improve the presentation of the paper.

 Authors' Contributions

A. K. A.: Research Design, Conceptualization, and Validation. D. K.: Data Gathering, Computing, and Edit-ing. S. A. E.: Methodology, Visualization and Formal Analysis. The authors have read and agreed to the pub-lished version of the manuscript.

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All data supporting the reported findings in this research paper are provided within the manuscript.

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The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.

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