An IFS A in X is defined by A = { ⟨ x , μ A ( x ) , ν A ( x ) ⟩ : x ∈ X } , where μ A , ν A : X → [ 0 , 1 ] are functions with the condition μ A + ν A ⩽ 1 that are called the membership function and the non-membership function, respectively.

A PFS A in X is defined by A = { ⟨ x , μ A ( x ) , ν A ( x ) ⟩ : x ∈ X } , where μ A , ν A : X → [ 0 , 1 ] are functions with the condition μ A 2 + ν A 2 ⩽ 1 that are called themembership function and the non-membership function, respectively. Let μ α , ν α ∈ [ 0 , 1 ] such that μ α 2 + ν α 2 ⩽ 1. Then the pair α = ⟨ μ α , ν α ⟩ is called a Pythagorean fuzzy value (PFV).

A t-norm is a function T : [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , 1 ] that satisfies the following conditions: (T1)

T ( x , 1 ) = x for all x ∈ [ 0 , 1 ] (border condition),

A t-conorm is a function S : [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , 1 ] that satisfies the following conditions: (S1)

S ( x , 0 ) = x for all x ∈ [ 0 , 1 ] (border condition),

A strictly decreasing function g : [ 0 , 1 ] → [ 0 , ∞ ] with g ( 1 ) = 0 is called the additive generator of a t-norm T if we have T ( x , y ) = g − 1 ( g ( x ) + g ( y ) ) for all ( x , y ) ∈ [ 0 , 1 ] × [ 0 , 1 ].

A fuzzy complement is a function N : [ 0 , 1 ] → [ 0 , 1 ] satisfying the following conditions: (N1)

N ( 0 ) = 1 and N ( 1 ) = 0 (boundary conditions),

Let T be a t-norm and let S be a t-conorm on [ 0 , 1 ]. If T ( x , y ) = N ( S ( N ( x ) , N ( y ) ) ) and S ( x , y ) = N ( T ( N ( x ) , N ( y ) ) ), then T and S are called dual with respect to a fuzzy complement N.

Let T be a t-norm on [ 0 , 1 ]. Then the dual t-conorm S with respect to the Pythagorean fuzzy complement N is S ( x , y ) = 1 − T 2 ( 1 − x 2 , 1 − y 2 ) .

Let r ∈ [ 0 , 1 ]. A circular C-IFS A r in X is defined by A r = { ⟨ x , μ A ( x ) , ν A ( x ) ; r ⟩ : x ∈ X } , where μ A , ν A : X → [ 0 , 1 ] are functions such that μ A + ν A ⩽ 1 . r is the radius of the circle around the point ( μ A ( x ) , ν A ( x ) ) on the plane. This circle represents the membership degree and non-membership degree of x ∈ X.

Since each IFS A has the form A = A 0 = { ⟨ x , μ A ( x ) , ν A ( x ) ; 0 ⟩ : x ∈ X } , any IFS can be considered as a C-IFS. Hence, the notion of C-IFS is a generalization of the notion of IFS.

Let r ∈ [ 0 , 1 ]. A C-PFS A r in X is defined by A r = { ⟨ x , μ A ( x ) , ν A ( x ) ; r ⟩ : x ∈ X } , where μ A , ν A : X → [ 0 , 1 ] are functions such that μ A 2 + ν A 2 ⩽ 1 . r is the radius of the circle around the point ( μ A ( x ) , ν A ( x ) ) on the plane. This circle represents the membership degree and non-membership degree of x ∈ X.

Let X = { x 1 , x 2 , x 3 }. An example of a C-PFS on X can be given by A 0.2 = { ⟨ x 1 , 0.3 , 0.8 ; 0.2 ⟩ , ⟨ x 2 , 0.1 , 0.9 ; 0.2 ⟩ , ⟨ x 3 , 0.5 , 0.6 ; 0.2 ⟩ } .

Let μ α , ν α ∈ [ 0 , 1 ], such that μ α 2 + ν α 2 ⩽ 1 and r α ∈ [ 0 , 1 ]. Then the triple α = ⟨ μ α , ν α ; r α ⟩ is called a C-PFV.

Since each PFS A has the form A = A 0 = { ⟨ x , μ A ( x ) , ν A ( x ) ; 0 ⟩ : x ∈ X } , any PFS is also a C-PFS but the converse is not true in general. Consider the C-PFS A 0.2 given in Example 1. Since 0.3 + 0.8 = 1.1 > 1, it is not a C-IFS.

Let A r = { ⟨ x , μ A ( x ) , ν A ( x ) ; r ⟩ : x ∈ X } and B s = { ⟨ x , μ B ( x ) , ν B ( x ) ; s ⟩ : x ∈ X } be two C-PFSs in X. Some set operations among C-PFSs are defined as follows: a)

A r ⊂ B s if and only if r ⩽ s and μ A ( x ) ⩽ μ B ( x ) and ν A ( x ) ⩾ ν B ( x ) , for any x ∈ X .

Let X = { x 1 , x 2 , x 3 }. Consider the C-PFSs given with A 0.2 = { ⟨ x 1 , 0.3 , 0.8 ; 0.2 ⟩ , ⟨ x 2 , 0.1 , 0.9 ; 0.2 ⟩ , ⟨ x 3 , 0.5 , 0.6 ; 0.2 ⟩ } and B 0.6 = { ⟨ x 1 , 0.7 , 0.5 ; 0.6 ⟩ , ⟨ x 2 , 0.2 , 0.5 ; 0.6 ⟩ , ⟨ x 3 , 0.6 , 0.3 ; 0.6 ⟩ } . Then A 0.2 ⊂ B 0.6 . On the other hand, A 0.2 c = { ⟨ x 1 , 0.8 , 0.3 ; 0.2 ⟩ , ⟨ x 2 , 0.9 , 0.1 ; 0.2 ⟩ , ⟨ x 3 , 0.6 , 0.5 ; 0.2 ⟩ } , A 0.2 ∪ min B 0.6 = { ⟨ x 1 , 0.7 , 0.5 ; 0.2 ⟩ , ⟨ x 2 , 0.2 , 0.5 ; 0.2 ⟩ , ⟨ x 3 , 0.6 , 0.3 ; 0.2 ⟩ } , A 0.2 ∪ max B 0.6 = { ⟨ x 1 , 0.7 , 0.5 ; 0.6 ⟩ , ⟨ x 2 , 0.2 , 0.5 ; 0.6 ⟩ , ⟨ x 3 , 0.6 , 0.3 ; 0.6 ⟩ } , A 0.2 ∩ min B 0.6 = { ⟨ x 1 , 0.3 , 0.8 ; 0.2 ⟩ , ⟨ x 2 , 0.1 , 0.9 ; 0.2 ⟩ , ⟨ x 3 , 0.5 , 0.6 ; 0.2 ⟩ } , and A 0.2 ∩ max B 0.6 = { ⟨ x 1 , 0.3 , 0.8 ; 0.6 ⟩ , ⟨ x 2 , 0.1 , 0.9 ; 0.6 ⟩ , ⟨ x 3 , 0.5 , 0.6 ; 0.6 ⟩ } .

Let A r = { ⟨ x , μ A ( x ) , ν A ( x ) ; r ⟩ : x ∈ X } and B s = { ⟨ x , μ B ( x ) , ν B ( x ) ; s ⟩ : x ∈ X } be two C-PFSs in X. Then we have: 1)

( A r ∪ min B s ) c = A r i c ∩ min B s i c ;

The proof is trivial from Definition 11.  □

Let { ⟨ μ 1 , ν 1 ⟩ , ⟨ μ 2 , ν 2 ⟩ , … , ⟨ μ k , ν k ⟩ } be a collection of PFVs. Then ⟨ μ , ν ; r ⟩ is a C-PFV where μ = ∑ j = 1 k μ j 2 k , ν = ∑ j = 1 k ν j 2 k and r = min { max 1 ⩽ j ⩽ k ( μ − μ j ) 2 + ( ν − ν j ) 2 , 1 } .

We have μ 2 + ν 2 = ( ∑ j = 1 k μ j 2 k ′ , ) 2 + ( ∑ j = 1 k ν j 2 k ) 2 = ∑ j = 1 k μ , j 2 k i + ∑ j = 1 k ν j 2 k i = ∑ j = 1 k ( μ , j 2 + ν , j 2 ) k ⩽ ∑ j = 1 k 1 k = 1 . Furthermore, it is clear that 0 ⩽ r ⩽ 1. Therefore, A r = ⟨ μ , ν ; r ⟩ is a C-PFV.  □

We present the following collections of PFVs: { ⟨ 0.3 , 0.8 ⟩ , ⟨ 0.4 , 0.6 ⟩ , ⟨ 0.5 , 0.7 ⟩ , ⟨ 0.4 , 0.8 ⟩ } , { ⟨ 0.2 , 0.3 ⟩ , ⟨ 0.1 , 0.4 ⟩ , ⟨ 0.2 , 0.5 ⟩ , ⟨ 0.1 , 0.6 ⟩ } , and { ⟨ 0.9 , 0.2 ⟩ , ⟨ 0.8 , 0.3 ⟩ , ⟨ 0.8 , 0.2 ⟩ , ⟨ 0.7 , 0.5 ⟩ } . By applying Proposition 1, we derive the corresponding C-PFVs, respectively: ⟨ 0.41 , 0.73 ; 0.13 ⟩ , ⟨ 0.16 , 0.46 ; 0.17 ⟩ and ⟨ 0.8 , 0.32 ; 0.2 ⟩ . We visualize this transformations in Fig. 5.

Let α = ⟨ μ α , ν α ; r α ⟩ and β = ⟨ μ β , ν β ; r β ⟩ be two C-PFVs. Some algebraic operations among C-PFVs are defined as follows: a)

α ⊕ min β = ⟨ μ α 2 + μ β 2 − μ α 2 μ β 2 , ν 1 ν 2 ; min ( r α , r β ) ⟩,

Let α = ⟨ μ α , ν α ; r α ⟩ and β = ⟨ μ β , ν β ; r β ⟩ be two C-PFVs. Assume that T , S are dual t-norm and t-conorm with respect to the Pythagorean fuzzy complement N ( a ) = 1 − a 2 , respectively, and Q is a t-norm or a t-conorm. General algebraic operations among C-PFVs are defined as follows: a)

α ⊕ Q β = ⟨ S ( μ α , μ β ) , T ( ν α , ν β ) ; Q ( r α , r β ) ⟩,

Let α and β be two C-PFVs. Assume that T , S are dual t-norm and t-conorm with respect to Pythagorean fuzzy complement N ( a ) = 1 − a 2 , respectively, and Q is a t-norm or a t-conorm. Then α ⊕ Q β and α ⊗ Q β are also C-PFVs.

We know that the dual t-conorm S with respect to the Pythagorean fuzzy complement N is S ( x , y ) = 1 − T 2 ( 1 − x 2 , 1 − y 2 ) from Remark 1. Since μ α ⩽ 1 − ν α 2 and T is increasing, it is obtained that T 2 ( μ α , μ β ) + S 2 ( ν α , ν β ) = T 2 ( μ α , μ β ) + ( 1 − T 2 ( 1 − ν α 2 , 1 − ν β 2 ) ) 2 = T 2 ( μ α , μ β ) + 1 − T 2 ( 1 − ν α 2 , 1 − ν β 2 ) ⩽ T 2 ( 1 − ν α 2 , 1 − ν β 2 ) + 1 − T 2 ( 1 − ν α 2 , 1 − ν β 2 ) = 1 . Moreover, since the domain of Q is the unit closed interval we conclude that α ⊗ Q β is a C-PFV. Similarly, it can be shown that α ⊕ Q β is also a C-PFV.  □

Let α = ⟨ μ α , ν α ; r α ⟩ and β = ⟨ μ β , ν β ; r β ⟩ be two C-PFVs and let λ > 0. Assume that g : [ 0 , 1 ] → [ 0 , ∞ ] is the additive generator of a continuous Archimedean t-norm and h ( t ) = g ( 1 − t 2 ) and q : [ 0 , 1 ] → [ 0 , ∞ ] is the additive generator of a continuous Archimedean t-norm or t-conorm. Some algebraic operations among C-PFVs are defined as follows: a)

α ⊕ q β = ⟨ h − 1 ( h ( μ α ) + h ( μ β ) ) , g − 1 ( g ( ν α ) + g ( ν β ) ) ; q − 1 ( q ( r α ) + q ( r β ) ) ⟩,

Let α = ⟨ μ α , ν α ; r α ⟩ and β = ⟨ μ β , ν β ; r β ⟩ be two C-PFVs and let λ > 0. Assume that g : [ 0 , 1 ] → [ 0 , ∞ ] is the additive generator of a continuous Archimedean t-norm and h ( t ) = g ( 1 − t 2 ) and q : [ 0 , 1 ] → [ 0 , ∞ ] is the additive generator of a continuous Archimedean t-norm or t-conorm. Then α ⊕ q β, α ⊗ q β, λ q α and α λ q are C-PFVs.

It is clear from Proposition 2 that α ⊕ q β and α ⊗ q β are C-PFVs. We know that h − 1 ( t ) = 1 − [ g − 1 ( t ) ] 2 and g ( t ) = h ( 1 − t 2 ). Since μ α ⩽ 1 − ν α 2 and h, h − 1 are increasing, we have 0 ⩽ [ h − 1 ( λ h ( μ α ) ) ] 2 + [ g − 1 ( λ g ( ν α ) ) ] 2 ⩽ [ h − 1 ( λ h ( 1 − ν α 2 ) ) ] 2 + [ g − 1 ( λ g ( ν α ) ) ] 2 = 1 − [ g − 1 ( λ h ( 1 − ν α 2 ) ] 2 + [ g − 1 ( λ g ( ν α ) ) ] 2 = 1 − [ g − 1 ( λ g ( ν α ) ) ] 2 + [ g − 1 ( λ g ( ν α ) ) ] 2 = 1 . Moreover, since the range of q − 1 is the unit closed interval, we conclude that λ q α is a C-PFV. In a similar way, it can be shown that α λ q is also a C-PFV.  □

Let g , h , q , p : [ 0 , 1 ] → [ 0 , ∞ ] defined by g ( t ) = − log t 2 , h ( t ) = − log ( 1 − t 2 ), q ( t ) = − log t 2 and p ( t ) = − log ( 1 − t 2 ) and λ > 0. Then we obtain the algebraic operators a)

α ⊕ q β = ⟨ μ α 2 + μ β 2 − μ α 2 μ β 2 , ν α ν β ; r α r β ⟩,

Let α = ⟨ μ α , ν α ; r α ⟩, β = ⟨ μ β , ν β ; r β ⟩ and θ = ⟨ μ θ , ν θ ; r θ ⟩ be C-PFVs and let λ , γ > 0. Assume that g : [ 0 , 1 ] → [ 0 , ∞ ] is the additive generator of a continuous Archimedean t-norm and h ( t ) = g ( 1 − t 2 ) and q : [ 0 , 1 ] → [ 0 , ∞ ] is the additive generator of a continuous Archimedean t-norm or t-conorm. We have i)

α ⊕ q β = β ⊕ q α,

(i) and (ii) are trivial. iii)

We obtain ( α ⊕ q β ) ⊕ q θ = ⟨ h − 1 ( h ( μ α ) + h ( μ β ) ) , g − 1 ( g ( ν α ) + g ( ν β ) ) ; q − 1 ( q ( r α ) + q ( r β ) ) ⟩ ⊕ q ⟨ μ θ , ν θ ; r θ ⟩ = ⟨ h − 1 ( h ( h − 1 ( h ( μ α ) + h ( μ β ) ) + h ( μ θ ) ) , g − 1 ( g ( g − 1 ( g ( ν α ) + g ( ν β ) ) + g ( ν θ ) ) ; q − 1 ( q ( q − 1 ( q ( r α ) + q ( r β ) ) + q ( r θ ) ) ⟩ = ⟨ h − 1 ( h ( μ α ) + h ( μ β ) + h ( μ θ ) ) , g − 1 ( g ( ν α ) + g ( ν β ) + g ( ν θ ) ) ; q − 1 ( q ( r α ) + q ( r β ) + q ( r θ ) ) ⟩ = ⟨ h − 1 ( h ( μ α ) + h ( h − 1 ( h ( μ β ) + h ( μ θ ) ) ) ) , g − 1 ( g ( ν α ) + g ( g − 1 ( g ( ν β ) + g ( ν θ ) ) ) ) ; q − 1 ( q ( r α ) + q ( q − 1 ( q ( r β ) + q ( r θ ) ) ) ) ⟩ = ⟨ μ α , ν α ; r α ⟩ ⊕ q ⟨ h − 1 ( h ( μ β ) + h ( μ θ ) ) , g − 1 ( g ( ν β ) + g ( ν θ ) ) ; q − 1 ( q ( r β ) + q ( r θ ) ) ⟩ = α ⊕ q ( β ⊕ q θ ) .