Entropy and distance measures of Pythagorean fuzzy soft sets and their applications

1. Introduction

Many complicated problems in economics, engineering, management science, medical science, etc. involve uncertain data. These problems, which one comes face to face in our day-to-day life, cannot be solved using classical mathematical methods due to a large number of uncertainties exists. To complete the process, the most important problem is to choose the appropriate decision about the task and to find the optimal results [1, 2]. Decision making is a crucial task for all those professions where experts apply their knowledge in a given zone to take appropriate decision. However, due to various constraints in day-to-day life, decision makers may unable to give their judgments in exactly crisp form. Thus, to handle it, they prefer to give their preferences under the uncertain and imprecise in nature by using the theory of fuzzy set (FS) [3]. In this theory, the measurement of each element is done with the help of the membership degree. But with the growing complexity, there always exists a degree of hesitancy between the preferences of the decision making and hence the analysis conducted under such circumstances is not ideal. To address it, an important extension of FS named as intuitionistic fuzzy set (IFS) introduced by Atanassov [4] by incorporated the degree of non-membership ν into the analysis along with the membership degree μ in such a way that μ + ν ≤ 1. Since it appearance, many researchers are paying more attentions towards it and presents different algorithms for solving the decision-making problems by using aggregation operators (AOs) or information measures.

For example, in the field of AOs, Xu [5] presented the weighted averaging AOs for different intuitionistic fuzzy numbers (IFNs). Garg [6, 7] gave the improved AOs by Einstein operators laws by degree of hesitancy into the analysis for IFNs. Huang Huang [8] and Garg [9] gave the AOs based on the Hamachar operational laws for IFNs. Chen and Chang [10] presented a geometric AOs by using the transformation technique to solve the multiple-attribute group decision-making (MAGDM) problems.

The above considered work is based on the restriction on the decision makers that they have given their preferences only under the environment where the constraint μ + ν ≤ 1 satisfy. However, if an expert gives their preferences towards the object as 0.7 as membership and 0.5 as non-membership then clearly seen that 0.7 + 0.5nleq1 and thus the above IFSs theories cannot handle such problems. To overcome it, Yager [11] introduced the concept of Pythagorean Fuzzy (PF) sets (PFSs), which is a generalization of IFSs, by relaxing the condition μ + ν ≤ 1 to μ² + ν² ≤ 1. Thus the PFS handle much more information than IFS. After its existence, several studies are investigated on it by using AOs, other information measures and graphs which are summarized in [12–19].

In 1999, Molodtsov introduced soft sets [20] to overcome the lack of parametrization tool when managing vagueness. A soft set is a parameterized family of sets which has been extended into different hybrid structures such as fuzzy soft sets [21], intuitionistic fuzzy soft sets [22] and Pythagorean fuzzy soft sets [23]. Since the Pythagorean fuzzy set is highly capable of dealing with vagueness or uncertainty, the parameterized family of Pythagorean fuzzy sets, that is Pythagorean fuzzy soft set also can perform well. In the case of PFSSs, corresponding to each parameter, a Pythagorean fuzzy set is obtained, and Pythagorean fuzzy sets can manage several situations in real life where the intuitionistic fuzzy sets fail to explain. Suppose there is a case in which someone expresses his degree of satisfaction to particular criteria as 0.6 and degree of dissatisfaction is 0.7. Then their sum exceeds one but the square sum does not. So Pythagorean fuzzy sets can handle this. Thus a Pythagorean fuzzy soft set is an effective parameterizing tool as well as a good medium to represent the vagueness which occurs in many real life situations.

Entropy and distance measures are two important concepts in generalized set theory. The entropy quantifies the degree of vagueness and Zadeh [24] introduced fuzzy entropy. The entropy of a system directly proportional to the irregularity. Thus one can identify that which one is more stable if the entropy of each system is given. The axiomatic Definition of entropy is proposed by De Luca and Termini [25]. They introduced some requirements for entropy known as De Luca and Termini axioms. Majumdar and Samanta [26] obtained the entropy of soft sets by extending the De Luca and Termini’s axioms to measure the uncertainty attached to soft sets. The entropy of intuitionistic fuzzy soft set is explained in [27], and it can be seen that entropy plays an important role in decision making problems. Garg and Arora [28] presented an idea of generalized and group generalized intuitionistic fuzzy soft sets. Also, there are studies for solving decision making problems using many hybrid structures of soft sets like fuzzy soft set [21], intuitionistic fuzzy soft set [29–31] etc. The authors [21, 32–34] have put forward some ideas for solving decision making problems in the corresponding hybrid structures.

Since the Pythagorean fuzzy soft set is a good tool to handle more than that of intuitionistic fuzzy soft sets, obtaining the entropy is also relevant. It is identified that entropy can be applied in decision making problem and thus decision making problems manage some bigger domain. Distance measures can be applied to decision making problems, pattern recognition problems, machine learning problems, etc. The most widely used distance measures are Hamming distance and Euclidean distance. The distance measure is inversely proportional to the measure of similarity. So it is helpful to identify the similarity between sets. A distance measure of Pythagorean fuzzy sets are introduced in [35] and demonstrates with a numerical example that the proposed distance measures are practical and adequate. This Definition for PFSs is extended to define the distance measure of PFSSs. Both the proposed entropy and distance measure may help in explain the real life situations adequately. Based on the motivation of the structure of the soft sets and the advantages of Pythagorean fuzzy sets to handle the uncertain and imprecise information, this paper explore the theory of Pythagorean fuzzy soft sets by defining some new information measures named as entropy and distance measures.

The rest of the manuscript is organized as follows. Section 2 briefly overview the basic definitions related to the soft sets and Pythagorean fuzzy sets. In Section 3, we defined the concept of distance and entropy for PFSSs and studied their desirable relations. Section 4 deals with an algorithm to solve the decision making problems based on the proposed measures and demonstrate with a numerical example. Also, a comparative study is conducted with some of the existing approach to show the feasibility of the approach. Finally, a concrete conclusion is drawn in Section 5.

2. Preliminaries

In this section, some basic concepts useful for further discussions are introduced.

Entropy is an essential tool to measure uncertain information. If the entropy is less the uncertainty is also less thus one can quickly identify that which one is the more stable information. PFSS being a more generalized structure, it is able to represent an information in which other existing structures fail. Thus introducing the measure for entropy is important in the current scenario. In this section, by introducing some definitions and results, the expression for entropy and distance measure for PFSSs are obtained and illustrated with examples.

Definition 3.1. Let (F, A) and (G, B) are two PFSSs over U with parameter set E. Define a relation (F, A) is less than or equal to (G, B), denoted as (F, A) ⪯ (G, B) if, for all x ∈ U and e ∈ E, μ_F(e) (x) ≤ μ_G(e) (x) and ν_F(e) (x) ≤ ν_G(e) (x).

The following definition is about a mapping which maps every Pythagorean fuzzy soft set to a fuzzy soft set. It is also shown that the collection of images of PFSSs with α ∈ [0, 1] and with the relation ⊆ is a totally ordered family of fuzzy soft sets.

Definition 3.2. Let α ∈ [0, 1]. The mapping f _α  : PFSS (U) → FSS (U) is defined as, f _α  ((F, E)) = (F _α , E), for every Pythagorean fuzzy soft set (F, E) with membership value μ_F(e) and nonmembership value ν_F(e) and F _α  (e) = f _α  (F (e)) and,

f α ( F ( e ) ) = { ( x , μ F ( e ) 2 ( x ) + α π F ( e ) 2 ( x ) , 1 - μ F ( e ) 2 ( x ) ) - α π F ( e ) 2 ( x ) : x ∈ U } .(8) Thus the map f _α assign every PFSSs to a fuzzy soft set.

To demonstrate the Definition 3, we provide a numerical example as follows:

Example 3.1. Let α = 0.6 and ( F , E ) = [ a ij ] = [ ( 0.6 , 0.6 ) ( 0.5 , 0.3 ) ( 0.2 , 0.9 ) ( 0.6 , 0.4 ) ] . Then, we can compuite F α ( e 1 ) = f α ( ( u 1 , 0.6 , 0.6 ) , ( u 2 , 0.5 , 0.3 ) ) = { ( u 1 , 0.58 , 0.472 ) , ( u 2 , 0.646 , 0.354 ) } , and F α ( e 2 ) = f α ( ( u 1 , 0.2 , 0.9 ) , ( u 2 , 0.6 , 0.4 ) ) = { ( u 1 , 0.13 , 0.87 ) , ( u 2 , 0.648 , 0.352 ) } Thus fuzzy soft set is obtained and is represented by matrix [ ( 0.528 , 0.472 ) ( 0.646 , 0.354 ) ( 0.13 , 0.87 ) ( 0.648 , 0.352 ) ] .

The following Theorem establishes some properties of the map f _α and as a corollary of this Theorem it is obtained that ({p _α } _α∈[0,1], ⊆) is a totally ordered family of fuzzy soft sets.

Theorem 3.1. For any α, β ∈ [0, 1] and p , p ¯ ∈ PFSS ( U ) , the following statements are true.

If α ≤ β then f _α  (p) ⊂ f _β  (p).

Proof. Let p = (F, E), F (e) = {(x, μ_F(e) (x), ν_F(e) (x)) : x ∈ U} ∀ e ∈ U, p ¯ = ( G , E ) , G (e) = {(x, μ_G(e) (x), ν_G(e) (x)) : x ∈ U} ∀ e ∈ U and f _α  (p) = (F _α , E), where f _α  (e) = {(x, μ F ( e ) 2 ( x ) + α π F ( e ) 2 ( x ) , 1 - μ F ( e ) 2 ( x ) - α π F ( e ) 2 ( x ) ) : x ∈ U } ∀ e ∈ U .

Given that α ≤ β. Then, for all x ∈ U and e ∈ E, μ F ( e ) 2 ( x ) + α π F ( e ) 2 ( x ) ≤ μ F ( e ) 2 ( x ) + β π F ( e ) 2 ( x ) . Thus, μ_Fα(e) (x) ≤ μ_Fβ(e) (x) ∀ x ∈ Uande ∈ E. Therefore, f _α  (p) ⊂ f _β  (p) .

Corollary 3.1. For all p = (F, E) ∈ PFSS (U), ({p _α } _α∈[0,1], ⊆) is a totally ordered family of fuzzy soft sets.

Proof. Since {p _α } _α∈[0,1] is a collection of PFSSs with common universe U and parameter set E, it satisfies reflexivity, antisymmetry and transitivity under the relation ⊆. Thus ({p _α } _α∈[0,1], ⊆) is a partial ordered family.

Now ∀p _β , p _γ  ∈ {p _α } _α∈[0,1]. Without loss of generality, assume that β ≤ γ. Then property 1 of Theorem 3 implies that p _β  ⊆ p _γ . Thus any two elements of {p _α } _α∈[0,1] is comparable.

Next, we have defined the axiomatic definition of the Pythagorean fuzzy soft entropy.

Definition 3.3. A real function E : PFSS ( U ) → ℝ + is called a Pythagorean fuzzy soft entropy on PFSS (U), if E has following properties;

E (p) =0 if and only if p ∈ S (U)

From the definition it is clear that entropy is minimum (zero) when the Pythagorean fuzzy soft set degenerates into soft set. The following Theorem discussed the case when the entropy is maximum.

Theorem 3.2. Entropy of a Pythagorean fuzzy soft set p is maximum if and only if p = (F, E) = [a _ij ] _m×n = [0] _m×n i.e., μ_{F(e j )} (u _i ) = ν_{F(e j )} (u _i ) =0∀ e _j  ∈ E, u _i  ∈ U where 0 ≤ i ≤ m and 0 ≤ j ≤ n and p∈ PFSS (U).

Proof. Let p = (F, E) = [0] _m×n. Let p ¯ = ( G , E ) be any Pythagorean fuzzy soft set. Since μ_{G(e j )} (u _i ) ≥0 and ν_{G(e j )} (u _i ) ≥0 for all e _j  ∈ E, u _i  ∈ U, where 0 ≤ i ≤ m and 0 ≤ j ≤ n, by Definition 3, p ⪯ p ¯ . So, by property 4 of Definition 3 E ( p ) ≥ E ( p ¯ ) for all p ¯ . Thus E (p) is maximum.

Conversely, let E (p) be maximum. Assume that p = (F, E) ≠ [0] _m×n, then there exist e _j  ∈ E and x _i  ∈ U such that μ_{F(e j )} (u _i ) ≠0 or ν_{F(e j )} (u _i ) ≠0 . Now construct the Pythagorean fuzzy soft set as p ¯ = ( G , E ) with μ_{G(e j )} (u _i ) = μ_{F(e j )} (u _i )/2 and ν_{G(e j )} (u _i ) = ν_{F(e j )} (u _i )/2 for all e _j  ∈ E and x _i  ∈ U. Then by Definition 3, p ≤ p ¯ . This implies that E ( p ¯ ) ≥ E ( p ) which contradicts the hypothesis E (p) is maximum. Thus p = [0] _m×n.

In order to obtain an expression for entropy, a function Φ _D is constructed as, Φ _D  : D → [0, 1] where the domain D is given by {(x, y) ∈ [0, 1] × [0, 1] : x² + y² ≤ 1} and Φ _D satisfies following conditions.

Φ _D  (x, y) =1 if and only if (x, y) = (0, 1) or (1, 0).

Theorem 3.3. Let E: PFSS ( U ) → ℝ + and p = (F, E) = [a _ij ] _m×n ∈ PFSS (U). If E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - ( Φ D ( μ F ( e j ) ( u i ) , ν_{F(e j )} (u _i )))) where Φ _D satisfies the conditions (1) - (4) given above then E is a PFSS entropy.

Proof. E (p) =0 if and only if ∑ j = 1 n ∑ i = 1 m ( 1 - ( Φ D ( μ F ( e j ) ( u i ) , ν_{F(e j )} (u _i )))) =0 if and only if Φ _D  (μ_{F(e j )} (u _i ), ν_{F(e j )} (u _i )) =1 ∀ e _j  ∈ E and u _i  ∈ U if and only if μ_{F(e j )} (u _i ) =1, ν_{F(e j )} (u _i ) =0or μ_{F(e j )} (u _i ) =0, ν_{F(e j )} (u _i ) =1 if and only if p is a soft set. Thus E satisfies property 1 of Definition 3.3.

E (p) = mn if and only if ∑ j = 1 n ∑ i = 1 m 1 - ( ( Φ D ( μ F ( e j ) ( u i ) , ν_{F(e j )} (u _i )))) = mn if and only if Φ _D  (μ_{F(e j )} (u _i ), ν_{F(e j )} (u _i )) =0 ∀e _j  ∈ E and u _i  ∈ U if and only if μ_{F(e j )} (u _i ) =0 = ν_{F(e j )} (u _i ) ∀e _j  ∈ E and u _i  ∈ U. Therefore, E satisfies property 2 of Definition 3.3.

Since p = (F, E)  ^c  = (F ^c , ¬ E) where F ^c  (e) = {(u _i , ν_{F(e j )} (u _i ), μ_{F(e j )} (u _i )) : u _i  ∈ U} ∀ ¬ e _j  ∈ ¬ E, then, E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - ( Φ D ( μ F ( e j ) ( u i ) , ν_{F(e j )} ( u i ) ) ) ) = ∑ j = 1 n ∑ i = 1 m ( 1 - ( Φ D ( ν F ( e j ) ( u i ) , μ F ( e j ) ( u i ) ) ) ) = E ( p c ) . Thus E satisfies property 3 of Definition 3.

Let p ¯ = ( G , E ) = [ b ij ] m × n . If p ≤ p ¯ then, μ_{F(e j )} (u _i ) ≤ μ_{G(e j )} (u _i ) & ν_{F(e j )} (u _i ) ≤ ν_{G(e j )} (u _i ) which implies Φ _D  (μ_{F(e j )} (u _i ) , ν_{F(e j )} (u _i )) ≤ Φ _D  (μ_{G(e j )} (u _i ) , ν_{G(e j )} (u _i )).

Hence E ( p ) ≥ E ( p ¯ ) . Therefore E satisfies property 4 of Definition 3.3. Thus E is a PFSS entropy. ◻

Example 3.2. E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - ( μ F ( e ) 3 + ν F ( e ) 3 ) ) .

To verify that above expression is an entropy of PFSSs it is enough to verify that μ F ( e ) 3 + ν F ( e ) 3 satisfy the four conditions of Φ _D defined above. Φ D ( x , y ) = μ F ( e ) 3 + ν F ( e ) 3 is a function from D = {(μ_F(e),) ∈ [0, 1] × [0, 1] : x² + y² ≤ 1} to [0, 1]. Also, μ F ( e ) 3 + ν F ( e ) 3 = 1 if and only if μ_F(e) = 1, ν_F(e) = 0 or μ_F(e) = 0, ν_F(e) = 1 in the domain D. All other conditions are trivial.

Definition 3.4. Let φ, φ′ : [0, 1] → [0, 1] such that if x² + y² ≤ 1 then φ (x²) + φ′ (y²) ≤1 with x, y ∈ [0, 1]. Define E_φ,φ′ function of the Pythagorean fuzzy soft set p = (F, E) = [a _ij ] _m×n to ℝ + as, E φ , φ ′ ( p ) = mn - ∑ j = 1 n ∑ i = 1 m φ ( μ F ( e j ) ( u i ) + φ ′ ( ν F ( e j ) ( u i ) )(9)

Obviously 0 ≤ E_φ,φ′ (p) ≤ mn and ∀p = [a _ij ] _m×n belonging to PFSS (U).

Theorem 3.4. If φ : [0, 1] → [0, 1] satisfies,

φ is increasing,

Then φ (x) + φ (y) satisfies the conditions 1 - 4 of the Φ _D function defined previously.

Proof. Consider Φ _D  (x, y) = φ (x) + φ (y) then the property 3 of this Theorem is same as the condition 1 of Φ _D . Now, Φ _D  (x, y) = φ (x) + φ (y) =0 if and only if φ (x) =0 = φ (x) from property 2, x = y = 0. Thus second condition of Φ _D is obtained. Also, φ (x) + φ (y) = φ (y) + φ (x) thus Φ _D  (x, y) = Φ _D  (y, x). Since φ is increasing, condition 4 of Φ _D is obtained. Hence φ (x) + φ (y) satisfies the conditions 1 - 4 of the Φ _D function.

The following Theorem characterizes the entropies of PFSSs in a general way.

Theorem 3.5. Let E : PFSS ( U ) → ℝ + , φ : [0, 1] → [0, 1] and p = (F, E) = [a _ij ] _m×n ∈ PFSS (U). E is a Pythagorean fuzzy soft entropy and a E_φ,φ-function if and only if E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ ( ν F ( e j ) ( u i ) ) ) where φ satisfies the conditions 1 - 3 of Theorem 3.

Proof. Consider Φ : [0, 1] × [0, 1] → [0, 1] such that Φ (x, y) = φ (x) + φ (y) and take the following set {(x, y) ∈ [0, 1] × [0, 1] : x² + y² ≤ 1}. Restrict the function Φ _D  : D → [0, 1], which is simply the notation.

If E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ (ν_{F(e j )} (u _i ))) where φ satisfies the conditions 1 - 3 of Theorem 3. Then E (p) is a Pythagorean fuzzy soft entropy by Theorem 3. Now it is enough to prove that E is an E_φ,φ function.

Let α, β ∈ [0, 1] and α² + β² ≤ 1 to prove φ (α²) +φ (β²) ≤1, construct the following Pythagorean fuzzy soft set:

[ a ij ] m × n = [ ( α 2 , β 2 ) ( 1 , 0 ) . . . ( 1 , 0 ) ( α 2 , β 2 ) ( 1 , 0 ) . . . ( 1 , 0 ) . . . . . . . . . ( α 2 , β 2 ) ( 1 , 0 ) . . . ( 1 , 0 ) ](10) Thus, E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ ( ν F ( e j ) ( u i ) ) ) = mn - m ( φ ( α 2 ) + φ ( β 2 ) ) - m ( n - 1 ) ( φ ( 1 ) + φ ( 0 ) ) By property 3 of Theorem 3, φ (0) + φ (1) =1. Then, E (p) = mn - m (φ (α²) + φ (β²)) - m (n - 1) = m (1 - (φ (α²) + φ (β²))). Since E is an entropy, E (p) ≥0. Therefore m (1 - (φ (α²) + φ (β²))) ≥0, which implies that φ (α²) + φ (β²) ≤1. Therefore E is an entropy and E_φ,φ function.

Conversely let E be an entropy and E_φ,φ function, then E has the form, E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ ( ν F ( e j ) ( u i ) ) ) . It is enough to prove that φ satisfies conditions 1 - 4 of Theorem 3.

(1) Let α ≤ β, α, β ∈ [0, 1] construct the following PFSSs: p = ( F , E ) = [ a ij ] m × n = [ ( α , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ( α , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) . . . . . . . . . ( α , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ] and, p ¯ = ( G , E ) = [ b ij ] m × n = [ ( β , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ( β , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) . . . . . . . . . ( β , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ]

Thus, E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ ( ν F ( e j ) ( u i ) ) ) = mn - m ( φ ( α ) + φ ( 0 ) ) - m ( n - 1 ) 2 φ ( 0 ) , and E ( p ¯ ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ ( ν F ( e j ) ( u i ) ) ) = mn - m ( φ ( β ) + φ ( 0 ) ) - m ( n - 1 ) 2 φ ( 0 ) . Since α ≤ β, p ⪯ p ¯ so, E ( p ) ≥ E ( p ¯ ) . Hence, mn - m ( φ ( α ) + φ ( 0 ) ) - m ( n - 1 ) 2 φ ( 0 ) ≥ mn - m ( φ ( β ) + φ ( 0 ) ) - m ( n - 1 ) 2 φ ( 0 ) implies φ (α) ≤ φ (β). Thus φ is increasing.

(2) To prove φ (α) =0 if and only if α = 0;

If α = 0, construct the following Pythagorean fuzzy soft set: p = ( F , E ) = [ a ij ] m × n = [ ( 0 , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) . . . . . . . . . ( 0 , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ]

Thus,

E (p) = mn - mn (φ (0)) + φ (0)) then φ (0) =0 i.e, φ (α) =0.

If φ (α) =0 construct the following Pythagorean fuzzy soft set: p = ( F , E ) = [ a ij ] m × n = [ ( α , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ( α , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) . . . . . . . . . ( α , 0 ) ( 0 , 0 ) . . . ( 0 , 0 ) ]

Thus, E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ ( ν F ( e j ) ( u i ) ) ) = mn - m ( φ ( α ) + φ ( 0 ) ) - m ( n - 1 ) ( φ ( 0 ) + φ ( 0 ) ) . By previous part, φ (0) =0 and given that φ (α) =0 So E (p) = mn. Thus property 2 of Definition 3 implies α = 0.

(3) To prove φ (α) + φ (β) =1 if and only if (α, β) = (0, 1) or (1, 0);

If (α, β) = (1, 0) or (0, 1) take the following Pythagorean fuzzy soft set: p = ( F , E ) = [ a ij ] m × n = [ ( α , β ) ( α , β ) . . . ( α , β ) ( α , β ) ( α , β ) . . . ( α , β ) . . . . . . . . . ( α , β ) ( α , β ) . . . ( α , β ) ] .

Then p ∈ S (U). By property 1 of Definition 3, E (p) =0.

Thus E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - φ ( μ F ( e j ) ( u i ) ) + φ (ν_{F(e j )} (u _i ))) =0. i.e, φ (α) + φ (β) =1.

If φ (α) + φ (β) =1 construct the following Pythagorean fuzzy soft set: p = [ a ij ] m × n = [ ( α , β ) ( 1 , 0 ) . . . ( 1 , 0 ) ( α , β ) ( 1 , 0 ) . . . ( 1 , 0 ) . . . . . . . . . ( α , β ) ( 1 , 0 ) . . . ( 1 , 0 ) ]

Thus, E ( p ) = ∑ j = 1 n ∑ i = 1 m (1 - φ (μ_{F(e j )} (u _i )) + φ (ν_{F(e j )} (u _i ))) = mn - m (φ (α) + φ (β)) - m (n - 1) (φ (1) + φ (0)). Given φ (α) + φ (β) =1 and by previous result φ (0) + φ (1) =1. Thus E (p) =0. By property 1 of Definition 3 p ∈ S (U) i.e, (α, β) = (1, 0) or (0, 1).

Remark 3.1. Let p = (F, E) = [a _ij ] _m×n ∈ PFSS (U) then entropy of p is given by, E ( p ) = ∑ j = 1 n ∑ i = 1 m ( 1 - ( μ F ( e j ) q ( u i ) ) + ( ν F ( e j ) q ( u i ) ) ) , q = 3 , 4 , 5 , ⋯(11)

The following part of this section includes definition of distance measure of PFSSs. According to Li and Zeng [35], distance measure of PFSSs are characterized by four parameters μ _p , ν _p , r _p and k _p , which was introduced by Yager [11]. The Hamming and Euclidean distance measures are obtained through generalizing the corresponding measures of distances for Pythagorean fuzzy sets.

Axiomatic Definition of distance measure between PFSSs is given below.

Definition 3.5. Let p = (F, A) and p ¯ = ( G , A ) be two PFSSs over U. Let d be a mapping given by d : PFSS ( U ) × PFSS ( U ) → ℝ + ∪ { 0 } and d ( p , p ¯ ) satisfies the following axioms.

d ( p , p ¯ ) ≥ 0 .

Definition 3.6. Let U = {u₁, u₂, . . . u _n }, E = {e₁, e₂, . . . e _m } & p₁ = (F, E), p₂ = (G, E) be two PFSSs over U. Then normalized Hamming distance between p₁ and p₂ is defined as follows;

d H ( p 1 , p 2 ) = 1 4 mn ∑ j = 1 m ∑ i = 1 n ( | μ F ( e j ) ( x i ) - μ G ( e j ) ( x i ) | + | ν F ( e j ) ( x i ) - ν G ( e j ) ( x i ) | + | r F ( e j ) ( x i ) - r G ( e j ) ( x i ) | + | k F ( e j ) ( x i ) - k G ( e j ) ( x i ) | ) .(12) The normalized Euclidean distance between p₁ and p₂ is defined as follows;

d E ( p 1 , p 2 ) = [ 1 4 mn ∑ j = 1 m ∑ i = 1 n ( ( μ F ( e j ) ( x i ) - μ G ( e j ) ( x i ) ) 2 + ( ν F ( e j ) ( x i ) - ν G ( e j ) ( x i ) ) 2 + ( r F ( e j ) ( x i ) - r G ( e j ) ( x i ) ) 2 + ( k F ( e j ) ( x i ) - k G ( e j ) ( x i ) ) 2 ) ] 1 2 .(13)

Remark 3.2. It is easy to verify that given Definitions satisfy four conditions given in Definition 3.5.

To illustrate the above Definition 3.5, we provide a numerical example as follows:

Example 3.3. Let p 1 = [ ( 0.9 , 0.2 ) ( 0.4 , 0.6 ) ( 0.2 , 0.8 ) ( 0.3 , 0.6 ) ( 0.4 , 0.8 ) ( 0.6 , 0.9 ) ( 0.1 , 0.8 ) ( 0.5 , 0.8 ) ( 0.6 , 0.3 ) ] and p 2 = [ ( 0.2 , 0.7 ) ( 0.2 , 0.9 ) ( 0.5 , 0.3 ) ( 0.9 , 0.2 ) ( 0.7 , 0.3 ) ( 0.6 , 0.2 ) ( 0.6 , 0.5 ) ( 0.8 , 0.2 ) ( 0.4 , 0.8 ) ]

The matrix representation of PFSSs p₁ = (r_{F(e j )} (u _i ), k_{F(e j )} (u _i )) and p₂ = (r_{G(e j )} (u _i ), k_{G(e j )} (u _i )) are given below, p 1 = [ ( 0.922 , 0.861 ) ( 0.721 , 0.374 ) ( 0.825 , 0.156 ) ( 0.671 , 0.295 ) ( 0.894 , 0.705 ) ( 1.08 , 0.374 ) ( 0.806 , 0.079 ) ( 0.943 , 0.356 ) ( 0.671 , 0.705 ) ] and, p 2 = [ ( 0.729 , 0.177 ) ( 0.922 , 0.861 ) ( 0.583 , 0.656 ) ( 0.922 , 0.861 ) ( 0.762 , 0.742 ) ( 0.632 , 0.795 ) ( 0.781 , 0.558 ) ( 0.825 , 0.844 ) ( 0.894 , 0.295 ) ]