New similarity measure and distance measure for Pythagorean fuzzy set

Pythagorean fuzzy set (PFS), disposing the indeterminacy portrayed by membership and non-membership, is a more viable and effective means to seize indeterminacy. Due to the defects of existing Pythagorean fuzzy similarity measures or distance measures (cannot obey third or fourth axiom; have no power to differentiate positive and negative difference; have no power to deal the division by zero problem), the major key of this paper is to explore the novel Pythagorean fuzzy distance measure and similarity measure. Meanwhile, some interesting properties of distance measure and similarity measure are proved. Some counterintuitive examples are presented to state their availability of similarity measure among PFSs.

Similarity measure (distance measure) is a significant means for measuring the uncertain information. The fuzzy similarity measure (distance measure) is a measure that depicts the closeness (difference) among fuzzy sets. Zhang [33] proposed the Pythagorean fuzzy similarity measures for dealing the multi-attribute decision-making problems. Peng et al. [23] proposed the many new distance measures and similarity measures for dealing the issues of pattern recognition, medical diagnosis and clustering analysis, and discussed their transformation relations. Wei and Wei [32] presented some Pythagorean fuzzy cosine function for dealing with the decision-making problems. However, some existing similarity measures/distance measures cannot the obey third or fourth axiom, and also have no power to differentiate positive difference and negative difference or deal with the division by the zero problem. Due to the above counterintuitive phenomena [32][33][34][36][37][38][39][40][41][42][43][44][45][46] of the existing similarity measures of PFSs, they may be hard for DMs to choose convincible or optimal alternatives. As a consequence, the goal of this paper is to deal with the above issue by proposing a novel similarity measure and distance measure for Pythagorean fuzzy set, which can be without counterintuitive phenomena.
For counting the distance measure and similarity measure of the two PFSs, we introduce a novel method to build the distance measure and similarity measure which rely on four parameters, i.e., a, b, t and p, where p is the L p norm and a, b, t identify the level of vagueness. Meanwhile, their relation with the similarity measures for PFSs are discussed in detail.
The rest of the presented paper is listed in the following. In Sect. 2, the fundamental notions of PFSs and IFSs are shortly retrospected, which will be employed in the analysis in each section. In Sect. 3, some new distance measures and similarity measures are proposed and proved. In Sect. 4, some counterintuitive examples are given to show the effectiveness of Pythagorean fuzzy similarity measure. The paper is concluded in Sect. 5.

Preliminaries
In this section, we briefly review the fundamental concepts related to IFS and PFS.
Definition 1 [1] Let X be a universe of discourse. An IFS I in X is given by where μ I : X → [0, 1] denotes the degree of membership and ν I : X → [0, 1] denotes the degree of nonmembership of the element x ∈ X to the set I , respectively, with the . For convenience, Xu and Yager [35] called (μ I (x), ν I (x)) an intuitionistc fuzzy number (IFN) denoted by i = (μ I , ν I ).
Definition 2 [2] Let X be a universe of discourse. A PFS P in X is given by where μ P : X → [0, 1] denotes the degree of membership and ν P : X → [0, 1] denotes the degree of nonmembership of the element x ∈ X to the set P, respectively, with the condition that 0 For convenience, Zhang and Xu [3] called (μ P (x), ν P (x)) a Pythagorean fuzzy number (PFN) denoted by p = (μ P , ν P ).
For any two PFNs p 1 , Definition 5 [3,4] If M, N ∈ PFSs, then the operations can be defined as follows: .

Distance measure and similarity measure of PFSs
Theorem 1 Let M and N be two PFSs in X where X = {x 1 , x 2 , · · · , x n }, then D(M, N ) is the distance measure between two PFSs M and N in X .
where p is the L p norm, and t, a and b denote the level of uncertainty with the condition a After solving, we can obtain (D5) According to the formula of the distance measure, we have Consequently, However, in most real environment, the diverse sets may possess diverse weights. Therefore, the weight w i (i = 1, 2, . . . , n) of the alternative x i ∈ X should be taken into consideration. We present a weighted distance measure D w (M, N ) between PFSs in the following.
where N ) is the distance measure between two PFSs M and N in X .
Proof (D1) If we obtain the product of the inequality defined above with w i , then we can easily have Furthermore, we can write the following inequality: It is easy to know that Hence, by Eq.
Theorem 4 Let M and N be two PFSs, then we have Proof We only prove the (1), and (2)-(2) can be proved in a homologous way.
(1) According to Definition 5 and Eq. (5), and for D(M, M N ) with ∀x i ∈ X , we can have Consequently, we can obtain D 1 (M, M ⊗ N ) = D 1 (N , M ⊕ N ).

Theorem 5 Let M and N be two PFSs, and x
Proof We only prove (1), and (2)-(2) can be proved in a homologous way.

Apply the similarity measure between PFSs to pattern recognition
For stating the advantage of the explored similarity measure S, a comparison among the initiated similarity measure with the current similarity measures is established. Some existing similarity measures are presented in Table 1.
(1) It is easily seen that the third axiom of similarity measure (S3) is not satisfied by S ) "Bold" represents unreasonable results. "N/A" represents the division by zero problem" (2) Some similarity measures [34,36,39,40,43,44] have no power to differentiate positive and negative difference. ). In such case, it is expected that the degree of similarity between M and N is bigger than or equal to the degree of similarity between M and N 1, since they are ranked as N 1 N M by means of score function shown in Definition 3. However, the degree of similarity between M and N 1 is bigger than the degree of similarity between M and N when S L , S HY 1 , S HY 2 , S HY 3 , S H K , S LC , S L X , S L S1 , S L S2 , S C , S L S3 , S M , S W , S Z , S P2 , S P3 are used, which does not seem to be reasonable. On the other hand, our proposed similarity measures S(M, N ) = 0.9625 and S(M, N 1) = 0.9438. Therefore, the developed similarity measure is the same as score function. The presented similarity measure S and the existing similarity measures (S CC and S B A ) are the similarity measures that have no such counter-intuitive issues as stated in Table  2. To continue digging the defects of the existing similarity measures (S CC and S B A ), we give the following tables for further discussion.

Conclusion
The main contributions can be illustrated and reviewed in the following.
(1) The formulae of Pythagorean fuzzy similarity measures and distance measures are proposed, and their properties are proved. Meanwhile, the diverse desirable relations between the developed similarity measures and distance measures have also been elicited.
In future, we will employ some similarity measures in other domains, such as medical diagnosis and machine learning. Besides, as this paper is just an applied research focusing on the similarity measures of PFSs, we will attempt to design some softwares to preferably realize the initiated information measure in daily life. Meanwhile, we also will take them into diverse fuzzy environment [47][48][49][50][51][52][53][54].