Zadeh [1] introduced the idea of fuzzy set which has a membership function, µ that assigns to each element of the universe of discourse, a number from the unit interval [0, 1] to indicate the degree of belongingness to the set under consideration. The notion of fuzzy sets generalizes classical sets theory by allowing intermediate situations between the whole and nothing. In a fuzzy set, a membership function is defined to describe the degree of membership of an element to a class. The membership value ranges from 0 to 1, where 0 shows 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.

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.

Definition 3. A Pythagorean fuzzy set, Pˆ in a finite universe of discourse X is given by P ˆ = {⟨ξ,ρ (ξ), σ (ξ)⟩|ξ ∈ X},

P ˆ P ˆ

where ρ Pˆ(ξ ) : X → [0, 1] indicates the grade to which the element ξ∈ X and σ Pˆ(ξ ) : X → [0, 1] represents the grade to which the element ξ∈ X is not a member of the set Pˆ, with the condition that 0 ≤ (ρ ˆ(ξ)) + (σ ˆ(ξ)) ≤ 1, for all ξ∈ X.

The degree of indeterminacy hPˆ(ξ ) = p1 −(ρ Pˆ(ξ ))2 −(σ Pˆ(ξ ))2.

Fig. Comparison spaces for intuitionistic fuzzy and pythagorean membership grades.

Let P1 and P2 be two Pythagorean fuzzy sets, then the following operations and relations can be defined as P1 ⊆ P2 iff (ρP (x) ≤ ρP (x)) and (σP (x) ≥ σP (x)) (for all x ∈ E) P1 = P2 iff (ρP (x) = ρP (x)) and (σP (x) = σP (x)) (for all x ∈ E) P1 ∩ P2 = {⟨x. min(ρP (x), ρP (x)), max(σP (x), σPP (x))⟩ : x ∈ E} P1 ∪ P2 = {⟨x. max(ρP (x), ρP (x)), min(σP (x), σP (x))⟩ : x ∈ E} p1 + P2 = {⟨x, ρP (x) + ρP (x) − ρP (x)ρP (x, ), σP (x).σP (x)⟩ : x ∈ E} P1.P2 = {⟨x, ρP (x)ρP (x), σP (x) + σP (x) − σP (x).σP (x)⟩ : x ∈ E} P1@P2 = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} Theorem 1. Let P1 = (ρ P1, σ P1), P2 = (ρ P2, σ P2) and P3 = (ρ P3, σ P3) be three Pythagorean fuzzy sets. Then (P1 ∩P2)@P3 = (P1@P3) ∩(P2@P3).

Proof: from definition, we have P1 ∩ P2 = {⟨x, min ρP (x), ρP (x), max σP (x), σP (x)⟩ : x ∈ E} and P1@P2 = {⟨x, ρP (x) + ρP (x)/2, σP (x) + σP (x)/2⟩ : x ∈ E} Now, (P1 ∩ P2)@P3 = {⟨x, min{ρP (x), ρP (x)}, max{σP (x), σP (x)}⟩ : x ∈ E}, @{⟨x, ρP (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} (1) 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).

Theorem 4. Let P1 = (ρ P1, σ P1), P2 = (ρ P2, σ P2) and P3 = (ρ P3, σ P3) be three Pythagorean fuzzy sets. Then P1@(P2 ∪ P3) = (P1@P2) ∪ (P1@P3) Proof: For three PFS A, B and C, From definition P1@(P2 ∪ 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).

Theorem 5. Let P1 = (ρ P1, σ P1), P2 = (ρ P2, σ P2) and P3 = (ρ P3, σ P3) be three Pythagorean fuzzy sets. Then (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}

1 3 1 3 2 3 2 3

= {⟨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.

Theorem 6. Let P1 = (ρ P1, σ P1), P2 = (ρ P2, σ P2) and P3 = (ρ P3, σ P3) be three Pythagorean fuzzy sets, then (P1 ∩P2).P3 = (P1.P3) ∩(P2.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}

1 3 1 3 2 3 2 3

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).

Theorem 7. Let P1 = (ρ P1, σ P1) and P2 = (ρ P2, σ P2) 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 −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.

Theorem 8. Let P1 = (ρ P1, σ P1) and P2 = (ρ P2, σ P2) be two Pythagorean fuzzy sets. Then (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 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).

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.

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.

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.

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.

All authors have provided their consent for the publication of this manuscript.

This article does not involve studies with human participants or animals conducted by any authors.

The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions.

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

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.