Communications on Applied Nonlinear Analysis 

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𝑴-open Maps and its Applications in Pythagorean Fuzzy Topological 

Spaces 

 

B. Vijayalakshmi 𝟏,  M. Ramalakshmi 𝟐 , A. Vadivel 𝟑,  G. Saravanakumar  𝟒 

 1Department of Mathematics, Government Arts College, Chidambaram, Tamil Nadu-608 102; Mathematics Section 

(FEAT), Annamalai University, Annamalai Nagar - 608 002, TamilNadu. 

 2Department of Mathematics, Sri Meenakshi Government Arts College for Women (A), Madurai- 625 002 

 3 Arignar Anna Government Arts College, Namakkal - 637 002, India. 

 3Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India. 

 4Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology (Deemed 

to be University), Avadi, Chennai-600062, India 

1 mathvijaya2006au@gmail.com,2 ramalakshmikrishnan23@gmail.com, 3 avmaths@gmail.com,4 

saravananguru2612@gmail.com 

 

Article History: 

Received: 11-09-2024 

Revised: 16-11-2024 

Accepted: 26-11-2024 

Abstract:  

In this paper, we introduce and investigate Pythagorean fuzzy 𝑀-open and closed maps 

in Pythagorean fuzzy topological spaces and also discuss about some properties and 

characterization of Pythagorean fuzzy maps. Also one real life applications, we applied 

entropy measure for decision making problem of diet selection based on the performance. 

Keywords Pythagorean fuzzy M-open maps, Pythagorean fuzzy M-closed maps, 

Pythagorean Fuzzy Entropy. 

1. Introduction 

Considering the imprecision in decision-making, Zadeh [35] 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 nterval [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. The concept 

of fuzzy set theory seems to be inconclusive because of the exclusion of nonmembership function and 

the disregard for the possibility of hesitation margin. 

Atanassov critically studied these shortcomings and proposed a concept called intuitionistic fuzzy sets 

(𝐼𝐹𝑆s) [1, 2, 4, 5]. The construct (that is, 𝐼𝐹𝑆’s) incorporates both membership function, 𝜇 and 

nonmembership function, 𝜈 with hesitation margin, 𝜋 (that is, neither membership nor non-

membership functions), such that 𝜇 + 𝜈 ≤ 1 and 𝜇 + 𝜈 + 𝜋 = 1. Atanassov [3] introduced 

intuitionistic fuzzy sets of second type (𝐼𝐹𝑆𝑆𝑇) with the property that the sum of the square of the 

membership and non-membership degrees is less than or equal to one. This concept generalizes 𝐼𝐹𝑆’s 

in a way. The notion of 𝐼𝐹𝑆’s provides a flexible framework to elaborate uncertainty and vagueness. 



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The idea of 𝐼𝐹𝑆 seems to be resourceful in modelling many real-life situations like medical diagnosis 

[7, 8, 12, 28, 29], career determination [10], selection process [11], and multi-criteria decision-making 

[15, 16, 17], among others. 

There are situations where 𝜇 + 𝜈 ≥ 1 unlike the cases capture in 𝐼𝐹𝑆’s. This limitation in 𝐼𝐹𝑆 naturally 

led to a construct, called Pythagorean fuzzy sets (𝑝𝑓𝑠’s). Pythagorean fuzzy set (𝑝𝑓𝑠) proposed in [32, 

33, 34] is a new tool to deal with vagueness considering the membership grade, 𝜇 and non-membership 

grade, 𝜈 satisfying the conditions 𝜇 + 𝜈 ≤ 1 or 𝜇 + 𝜈 ≥ 1, and also, it follows that 𝜇2 + 𝜈2 + 𝜋2 = 1, 

where 𝜋 is the Pythagorean fuzzy set index. In fact, the origin of Pythagorean fuzzy sets emanated 

from 𝐼𝐹𝑆𝑆𝑇 earlier studied in the literature. As a generalized set, 𝑃𝐹𝑆 has close relationship with 𝐼𝐹𝑆. 

The construct of 𝑃𝐹𝑆’s can be used to characterize uncertain information more sufficiently and 

accurately than 𝐼𝐹𝑆. Garg [14] presented an improved score function for the ranking order of interval-

valued Pythagorean fuzzy sets (𝐼𝑉𝑃𝐹𝑆s). Based on it, a Pythagorean fuzzy technique for order of 

preference by similarity to ideal solution (𝑇𝑂𝑃𝑆𝐼𝑆) method by taking the preferences of the experts in 

the form of interval-valued Pythagorean fuzzy decision matrices was discussed. Other explorations of 

the theory of 𝑃𝐹𝑆’s can be found in [6, 9, 13, 18, 19, 25, 26]. Saha [27] defined 𝛿-open sets in 

topological spaces. Vadivel et al. [31] introduced 𝛿-open sets in a neutrosophic topological space. The 

notion of M-open sets in topological spaces were introduced by El-Maghrabi and Al-Juhani [23] in 

2011 and studied some of their properties. The class of sets namely, 𝑀-open sets are playing more 

important role in topological spaces, because of their applications in various fields of Mathematics and 

other real fields. Recently, Jeeva et al. [20, 21, 22] introduced neutrosophic soft 𝑀-open sets in 

neutrosophic topological spaces and developed the concepts of neutrosophic soft 𝑀-Continuity and 

𝑀-Irresolute maps. 

Entropy can be viewed as a gauge of the degree of uncertainty present in a set, regardless of how fuzzy, 

intuitionistic, ambiguous, etc. the set may be. Since the 𝑝𝑓𝑠 in this case can also handle uncertain data, 

it follows naturally that we are also interested in determining the entropy of an 𝑝𝑓𝑠. In 1965, Zadeh 

[35] made the first reference to entropy as a fuzziness metric. More recently, De Luca-Termini [8] 

axiomatized the entropy that is not probabilistic. 

The remainder of this paper is organized as follows. In section 2, some basic definitions of 𝑓𝑠’s, 𝐼𝐹𝑆’s 

and 𝑝𝑓𝑠’s are briefly reviewed. In sections 3 and 4, We develop the concept of some Pythagorean 

fuzzy open and closed maps in Pythagorean fuzzy topological space and also specialized some of their 

basic properties with examples. Finally, we presented an entropy measure for 𝑝𝑓𝑠’s and one real- 

world scenarios where this entropy measure can be used are mentioned in section 4. The paper is 

concluded in section 5. 

2 Preliminaries 

 We recall some basic notions of fuzzy sets, 𝐼𝐹𝑆’s and 𝑝𝑓𝑠’s . 

Definition 2.1  [35] Let 𝑋 be a nonempty set. A fuzzy set 𝐴 in 𝑋 is characterized by a membership 

function 𝜇𝐴: 𝑋 → [0,1]. That is:  



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 𝜇𝐴(𝑥) = {
1,    if        𝑥 ∈ 𝑋
0,    if        𝑥 ∉ 𝑋
(0,1)    if 𝑥 ispartlyin 𝑋.

 

Alternatively, a fuzzy set 𝐴 in 𝑋 is an object having the form 𝐴 = {< 𝑥, 𝜇𝐴(𝑥) > |𝑥 ∈ 𝑋} or 𝐴 =

{⟨
𝜇𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋}, where the function 𝜇𝐴(𝑥): 𝑋 → [0,1] defines the degree of membership of the 

element, 𝑥 ∈ 𝑋. 

The closer the membership value 𝜇𝐴(𝑥) to 1, the more 𝑥 belongs to 𝐴, where the grades 1 and 0 

represent full membership and full nonmembership. Fuzzy set is a collection of objects with graded 

membership, that is, having degree of membership. Fuzzy set is an extension of the classical notion of 

set. In classical set theory, the membership of elements in a set is assessed in a binary terms according 

to a bivalent condition; an element either belongs or does not belong to the set. Classical bivalent sets 

are in fuzzy set theory called crisp sets. Fuzzy sets are generalized classical sets, since the indicator 

function of classical sets is special cases of the membership functions of fuzzy sets, if the latter only 

take values 0 or 1. Fuzzy sets theory permits the gradual assessment of the membership of element in 

a set; this is described with the aid of a membership function valued in the real unit interval [0,1]. 

Let us consider two examples: 

(i) all employees of 𝑋𝑌𝑍 who are over 1.8𝑚 in height; (ii) all employees of 𝑋𝑌𝑍 who are tall. The first 

example is a classical set with a universe (all 𝑋𝑌𝑍 employees) and a membership rule that divides the 

universe into members (those over 1.8𝑚) and nonmembers. The second example is a fuzzy set, because 

some employees are definitely in the set and some are definitely not in the set, but some are borderline. 

This distinction between the ins, the outs, and the borderline is made more exact by the membership 

function, 𝜇. If we return to our second example and let 𝐴 represent the fuzzy set of all tall employees 

and 𝑥 represent a member of the universe 𝑋 (i.e. all employees), then 𝜇𝐴(𝑥) would be 𝜇𝐴(𝑥) = 1 if 𝑥 

is definitely tall or 𝜇𝐴(𝑥) = 0 if 𝑥 is definitely not tall or 0 < 𝜇𝐴(𝑥) < 1 for borderline cases.  

Definition 2.2  [1, 2, 4, 5] Let a nonempty set 𝑋 be fixed. An 𝐼𝐹𝑆 𝐴 in 𝑋 is an object having the form: 

𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥) > |𝑥 ∈ 𝑋} or 𝐴 = {⟨
𝜇𝐴(𝑥),𝜈𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋}, where the functions 𝜇𝐴(𝑥): 𝑋 →

[0,1] and 𝜈𝐴(𝑥): 𝑋 → [0,1] define the degree of membership and the degree of nonmembership, 

respectively, of the element 𝑥 ∈ 𝑋 to 𝐴, which is a subset of 𝑋, and for every 𝑥 ∈ 𝑋: 0 ≤ 𝜇𝐴(𝑥) +

𝜈𝐴(𝑥) ≤ 1. For each 𝐴 in 𝑋: 𝜋𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) − 𝜈𝐴(𝑥) is the intuitionistic fuzzy set index or 

hesitation margin of 𝑥 in 𝑋. The hesitation margin 𝜋𝐴(𝑥) is the degree of nondeterminacy of 𝑥 ∈ 𝑋 to 

the set 𝐴 and 𝜋𝐴(𝑥) ∈ [0,1]. The hesitation margin is the function that expresses lack of knowledge of 

whether 𝑥 ∈ 𝑋 or 𝑥 ∉ 𝑋. Thus: 𝜇𝐴(𝑥) + 𝜈𝐴(𝑥) + 𝜋𝐴(𝑥) = 1.  

Example 2.1  Let 𝑋 = {𝑥, 𝑦, 𝑧} be a fixed universe of discourse and 𝐴 = {⟨
0.6,0.1

𝑥
⟩ , ⟨

0.8,0.1

𝑦
⟩ , ⟨

0.5,0.3

𝑧
⟩}, be 

the intuitionistic fuzzy set in 𝑋. The hesitation margins of the elements 𝑥, 𝑦, 𝑧 to 𝐴 are as follows: 

𝜋𝐴(𝑥) = 0.3, 𝜋𝐴(𝑦) = 0.1 and 𝜋𝐴(𝑧) = 0.2.  

Definition 2.3  [32, 33, 34] Let 𝑋 be a universal set. Then, a Pythagorean fuzzy set 𝐴, which is a set 

of ordered pairs over 𝑋, is defined by the following: 𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥)|𝑥 ∈ 𝑋} or 𝐴 =



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{⟨
𝜇𝐴(𝑥),𝜈𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋}, where the functions 𝜇𝐴(𝑥): 𝑋 → [0,1] and 𝜈𝐴(𝑥): 𝑋 → [0,1] define the degree of 

membership and the degree of nonmembership, respectively, of the element 𝑥 ∈ 𝑋 to 𝐴, which is a 

subset of 𝑋, and for every 𝑥 ∈ 𝑋, 0 ≤ (𝜇𝐴(𝑥))2 + (𝜈𝐴(𝑥))2 ≤ 1. Supposing (𝜇𝐴(𝑥))2 + (𝜈𝐴(𝑥))2 ≤ 1, 

then there is a degree of indeterminacy of 𝑥 ∈ 𝑋 to 𝐴 defined by 𝜋𝐴(𝑥) = √1 − [(𝜇𝐴(𝑥))2 + (𝜈𝐴(𝑥))2] 

and 𝜋𝐴(𝑥) ∈ [0,1]. In what follows, (𝜇𝐴(𝑥))2 + (𝜈𝐴(𝑥))2 + (𝜋𝐴(𝑥))2 = 1. Otherwise, 𝜋𝐴(𝑥) = 0 

whenever (𝜇𝐴(𝑥))2 + (𝜈𝐴(𝑥))2 = 1. We denote the set of all 𝑃𝐹𝑆’s over 𝑋 by 𝑝𝑓𝑠(𝑋).  

Definition 2.4  [34] Let 𝐴 and 𝐵 be 𝑝𝑓𝑠’s of the forms 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋} and 𝐵 =

{< 𝑎, 𝜆𝐵(𝑎), 𝜇𝐵(𝑎) > |𝑎 ∈ 𝑋}. Then [(i)]  

    1.  𝐴 ⊆ 𝐵 if and only if 𝜆𝐴(𝑎) ≤ 𝜆𝐵(𝑎) and 𝜇𝐴(𝑎) ≥ 𝜇𝐵(𝑎) for all 𝑎 ∈ 𝑋.  

    2.  𝐴 = 𝐵 if and only if 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴.  

    3.  �̅� = {< 𝑎, 𝜇𝐴(𝑎), 𝜆𝐴(𝑎) > |𝑎 ∈ 𝑋}.  

    4.  𝐴 ∩ 𝐵 = {< 𝑎, 𝜆𝐴(𝑎) ∧ 𝜆𝐵(𝑎), 𝜇𝐴(𝑎) ∨ 𝜇𝐵(𝑎) > |𝑎 ∈ 𝑋}.  

    5.  𝐴 ∪ 𝐵 = {< 𝑎, 𝜆𝐴(𝑎) ∨ 𝜆𝐵(𝑎), 𝜇𝐴(𝑎) ∧ 𝜇𝐵(𝑎) > |𝑎 ∈ 𝑋}.  

    6.  0𝑋 = {< 𝑎, 0,1 > |𝑎 ∈ 𝑋} and 1𝑋 = {< 𝑎, 1,0 > |𝑎 ∈ 𝑋}.  

    7.  1̅ = 0 and 0̅ = 1.  

Definition 2.5  [24] An Pythagorean fuzzy topology by subsets of a non-empty set 𝑋 is a family 𝜏 of 

𝑝𝑓𝑠’s satisfying the following axioms. [(i)]  

    1.  𝜙, 𝑋 ∈ 𝜏.  

    2.  𝐺1 ∩ 𝐺2 ∈ 𝜏 for every 𝐺1, 𝐺2 ∈ 𝜏 and  

    3.  ⋃ 𝐺𝑖 ∈ 𝜏 for any arbitrary family {𝐺𝑖|𝑖 ∈ 𝑗} ⊆ 𝜏. The pair (𝑋, 𝜏) is called an Pythagorean fuzzy 

topological space (𝑝𝑓𝑡𝑠 in short) and any 𝑝𝑓𝑠 𝐺 in 𝜏 is called an Pythagorean fuzzy open set (𝑝𝑓𝑜𝑠 in 

short) in 𝑋. The complement �̅� of an Pythagorean fuzzy open set 𝐴 in an 𝑝𝑓𝑡𝑠(𝑋, 𝜏) is called an 

Pythagorean fuzzy closed set (𝑝𝑓𝑐𝑠 in short).  

Definition 2.6  [24] Let (𝑋, 𝜏) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋} be an 𝑝𝑓𝑠 in 𝑋. 

Then the interior and the closure of 𝐴 are denoted by 𝑝𝑓𝑖𝑛𝑡(𝐴) and 𝑝𝑓𝑐𝑙(𝐴) and are defined as 

follows: 𝑝𝑓𝑐𝑙(𝐴) =∩ {𝐾|𝐾 𝑖𝑠𝑎𝑛 𝑝𝑓𝑐𝑠 𝑎𝑛𝑑 𝐴 ⊆ 𝐾} and 𝑝𝑓𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 𝑖𝑠𝑎𝑛 𝑝𝑓𝑜𝑠 𝑎𝑛𝑑 𝐺 ⊆ 𝐴}. 

Also, it can be established that 𝑝𝑓𝑐𝑙(𝐴) is an 𝑝𝑓𝑐𝑠 and 𝑝𝑓𝑖𝑛𝑡(𝐴) is an 𝑝𝑓𝑜𝑠, 𝐴 is an 𝑝𝑓𝑐𝑠 if and only 

if 𝑝𝑓𝑐𝑙(𝐴) = 𝐴 and 𝐴 is an 𝑝𝑓𝑜𝑠 if and only if 𝑝𝑓𝑖𝑛𝑡(𝐴) = 𝐴. We say that 𝐴 is 𝑝𝑓-dense if 𝑝𝑓𝑐𝑙(𝐴) =

𝑋.  

Lemma 2.1  [30]  For any Pythagorean fuzzy set 𝐴 in (𝑋, 𝜏), we have 𝑋 − 𝑝𝑓𝑖𝑛𝑡(𝐴) = 𝑝𝑓𝑐𝑙(𝑋 − 𝐴) 

and 𝑋 − 𝑝𝑓𝑐𝑙(𝐴) = 𝑝𝑓𝑖𝑛𝑡(𝑋 − 𝐴).  

Definition 2.7  [30] Let (𝑋, 𝜏) be an 𝑝𝑓𝑡𝑠 and 𝐴 be an 𝑝𝑓𝑠. Then 𝐴 is said to be an Pythagorean fuzzy 

(i) regular open set (𝑝𝑓𝑟𝑜𝑠 in short) if 𝐴 = 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝑐𝑙(𝐴)). (ii) regular closed set (𝑝𝑓𝑟𝑐𝑠 in short) if 

𝐴 = 𝑝𝑓𝑐𝑙(𝑝𝑓𝑖𝑛𝑡(𝐴)). By Lemma 2.1, it follows that 𝐴 is an 𝑝𝑓𝑟𝑜𝑠 iff �̅� is an 𝑝𝑓𝑟𝑐𝑠.  



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3 Pythagorean fuzzy 𝑴-open mappings 

Definition 3.1   Let (𝑋1, 𝛤𝑃) (or 𝑋1) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 

𝑋1. Then the (i) 𝑝𝑓𝛿-interior of 𝐴 are denoted by 𝑝𝑓𝛿𝑖𝑛𝑡(𝐴) and are defined as follows. 𝑝𝑓𝛿𝑖𝑛𝑡(𝐴) =

∪ {𝐺|𝐺 is an 𝑝𝑓𝑟𝑜𝑠 and 𝐺 ⊆ 𝐴}. (ii) 𝑝𝑓𝛿-closure of 𝐴 are denoted by 𝑝𝑓𝛿𝑐𝑙(𝐴) and are defined as 

follows. 𝑝𝑓𝛿𝑐𝑙(𝐴) =∩ {𝐾|𝐾 is an 𝑝𝑓𝑟𝑐𝑠 and 𝐴 ⊆ 𝐾}.  

Definition 3.2  Let (𝑋1, 𝛤𝑃) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 𝑋1. A set 

𝐴 is said to be 𝑝𝑓 [(i)]  

    1.  𝛿-open set (briefly, 𝑝𝑓𝛿𝑜𝑠) if 𝐴 = 𝑝𝑓𝛿𝑖𝑛𝑡(𝐴),  

    2.  𝛿-pre open set (briefly, 𝑝𝑓𝛿𝒫𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝛿𝑐𝑙(𝐴)),  

    3.  𝛿-semi open set (briefly, 𝑝𝑓𝛿𝒮𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝛿𝑖𝑛𝑡(𝐴)),  

    4.  𝑒 open set (briefly, 𝑝𝑓𝑒𝑜𝑠 ) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝛿𝑖𝑛𝑡(𝐴)) ∪ 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝛿𝑐𝑙(𝐴)),  

    5.  𝛿 (resp. 𝛿-pre, 𝛿-semi and 𝑒) dense if 𝑝𝑓𝛿𝑐𝑙(𝐴) (resp. 𝑝𝑓𝛿𝒫𝑐𝑙(𝐴), 𝑝𝑓𝛿𝒮𝑐𝑙(𝐴) and 𝑝𝑓𝑒𝑐𝑙(𝐴)) =

𝑋1. 

The complement of an 𝑝𝑓𝛿𝑜𝑠 (resp. 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠 and 𝑝𝑓𝑒𝑜𝑠) is called an 𝑝𝑓𝛿 (resp. 𝑝𝑓𝛿𝒫, 𝑝𝑓𝛿𝒮 

and 𝑝𝑓𝑒) closed set (briefly, 𝑝𝑓𝛿𝑐𝑠 (resp. 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠 and 𝑝𝑓𝑒𝑐𝑠)) in 𝑋1. 

The family of all 𝑝𝑓𝛿𝑜𝑠 (resp. 𝑝𝑓𝛿𝑐𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝑒𝑜𝑠 and 𝑝𝑓𝑒𝑐𝑠) of 𝑋1 is 

denoted by 𝑝𝑓𝛿𝑂𝑆(𝑋1), (resp. 𝑝𝑓𝛿𝐶𝑆(𝑋1), 𝑝𝑓𝛿𝒫𝑂𝑆(𝑋1), 𝑝𝑓𝛿𝒫𝐶𝑆(𝑋1), 𝑝𝑓𝛿𝒮𝑂𝑆(𝑋1), 𝑝𝑓𝛿𝒮𝐶𝑆(𝑋1), 

𝑝𝑓𝑒𝑂𝑆(𝑋1) and 𝑝𝑓𝑒𝐶𝑆(𝑋1)). 

Definition 3.3  Let (𝑋, 𝜏) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 𝑋1. Then 

the (i) 𝑝𝑓𝛿-pre (resp. 𝑝𝑓𝛿-semi and 𝑝𝑓𝑒)-interior of 𝐴 are denoted by 𝑝𝑓𝛿𝒫𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑖𝑛𝑡(𝐴) 

and 𝑝𝑓𝑒𝑖𝑛𝑡(𝐴)) and are defined as follows: 𝑝𝑓𝛿𝒫𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑖𝑛𝑡(𝐴) and 𝑝𝑓𝑒𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 

in a 𝑝𝑓𝛿𝒫𝑜𝑠 (resp. 𝑝𝑓𝛿𝒮𝑜𝑠 and 𝑝𝑓𝑒𝑜𝑠) and 𝐺 ⊆ 𝐴}, (ii) 𝑝𝑓𝛿-pre (resp. 𝑝𝑓𝛿-semi and 𝑝𝑓𝑒)-closure 

of 𝐴 are denoted by 𝑝𝑓𝛿𝒫𝑐𝑙(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑐𝑙(𝐴) and 𝑝𝑓𝑒𝑐𝑙(𝐴)) and are defined as follows: 

𝑝𝑓𝛿𝒫𝑐𝑙(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑐𝑙(𝐴) and 𝑝𝑓𝑒𝑐𝑙(𝐴)) =∩ {𝐾|𝐾 is an 𝑝𝑓𝛿𝒫𝑐𝑠 (resp. 𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝑒𝑐𝑠) and 𝐴 ⊆

𝐾}.  

Definition 3.4   Let (𝑋1, 𝛤𝑃) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 𝑋1. A 

set 𝐴 is said to be 𝑝𝑓  

    1.  𝜃-interior of 𝐴 (briefly, 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴)) is defined by 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴) =∪ {𝑝𝑓𝑖𝑛𝑡(𝐵): 𝐵 ⊆

𝐴 & 𝐵 isa 𝑝𝑓𝑐𝑠 in 𝑋1}.  

    2.  𝜃-open set (briefly, 𝑝𝑓𝜃𝑜𝑠) if 𝐴 = 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴).  

    3.  𝜃 -semi open set (briefly, 𝑝𝑓𝜃𝒮𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝜃𝑖𝑛𝑡(𝐴)).  

    4.  𝑀-open set (briefly, 𝑝𝑓𝑀𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝜃𝑖𝑛𝑡(𝐴)) ∪ 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝛿𝑐𝑙(𝐴)).  

The complement of a 𝑝𝑓𝑀𝑜𝑠 (resp. 𝑝𝑓𝜃𝑜𝑠 & 𝑝𝑓𝜃𝒮𝑜𝑠) is called an 𝑝𝑓𝑀 (resp. 𝑝𝑓𝜃 & 𝑝𝑓𝜃𝒮) closed 

set (briefly, 𝑝𝑓𝑀𝑐𝑠 (resp. 𝑝𝑓𝜃𝑐𝑠 & 𝑝𝑓𝜃𝒮𝑐𝑠)) in 𝑋1. 



Communications on Applied Nonlinear Analysis 

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The family of all 𝑝𝑓𝜃𝑜𝑠 (resp. 𝑝𝑓𝜃𝑐𝑠, 𝑝𝑓𝜃𝒮𝑜𝑠, 𝑝𝑓𝜃𝒮𝑐𝑠, 𝑝𝑓𝑀𝑜𝑠 and 𝑝𝑓𝑀𝑐𝑠) of 𝑋1 is denoted by 

𝑝𝑓𝜃𝑂𝑆(𝑋1), (resp. 𝑝𝑓𝜃𝐶𝑆(𝑋1), 𝑝𝑓𝜃𝒮𝑂𝑆(𝑋1), 𝑝𝑓𝜃𝒮𝐶𝑆(𝑋1), 𝑝𝑓𝑀𝑂𝑆(𝑋1) and 𝑝𝑓𝑀𝐶𝑆(𝑋1)). 

Definition 3.5  Let (𝑋1, 𝛤𝑃) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 𝑋1. Then 

the 𝑝𝑓  

    1.  𝑀 (resp. 𝑝𝑓𝜃-semi )-interior of 𝐴 (briefly,  𝑝𝑓𝑀𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝜃𝒮𝑖𝑛𝑡(𝐴)) is defined by 

𝑝𝑓𝑀𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴) and 𝑝𝑓𝜃𝒮𝑖𝑛𝑡(𝐴)) =∪ {𝐵: 𝐵 ⊆ 𝐴 and 𝐵 is a 𝑝𝑓𝑀𝑜𝑠 (resp. 𝑝𝑓𝜃𝒮𝑜𝑠) in 

𝑋1}.  

    2.  𝑀 (resp. 𝜃-semi )-closure of 𝐴 (briefly, 𝑝𝑓𝑀𝑐𝑙(𝐴) (resp. 𝑝𝑓𝜃𝒮𝑐𝑙(𝐴)) is defined by 𝑝𝑓𝑀𝑐𝑙(𝐴) 

(resp. 𝑝𝑓𝜃𝒮𝑐𝑙(𝐴)) =∩ {𝐵: 𝐴 ⊆ 𝐵 and 𝐴 is a 𝑝𝑓𝑀𝑐𝑠 (resp. 𝑝𝑓𝜃𝒮𝑐𝑠) in 𝑋1}.  

Definition 3.6  Let (𝑋1, 𝛤𝑃) and (𝑋2, 𝛹𝑃) be any two 𝑝𝑓𝑡𝑠’s. A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is said 

to be a Pythagorean fuzzy (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 𝑒, 𝜃, 𝜃𝒮 and 𝑀 )-continuous (briefly, 𝑝𝑓𝐶𝑡𝑠 (resp. 𝑝𝑓𝛿𝐶𝑡𝑠, 

𝑝𝑓𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝑒𝐶𝑡𝑠, 𝑝𝑓𝜃𝐶𝑡𝑠, 𝑝𝑓𝜃𝒮𝐶𝑡𝑠 and 𝑝𝑓𝑀𝐶𝑡𝑠)) if the inverse image of every 𝑝𝑓𝑜𝑠 in 

(𝑋2, 𝛹𝑃) is a 𝑝𝑓𝑜𝑠 (resp. 𝑝𝑓𝛿𝑜𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝑒𝑜𝑠, 𝑝𝑓𝜃𝑜𝑠, 𝑝𝑓𝜃𝒮𝑜𝑠 and 𝑝𝑓𝑀𝑜𝑠) in (𝑋1, 𝛤𝑃).  

Definition 3.7  Let (𝑋1, 𝛤𝑃) and (𝑋2, 𝛹𝑃) be any two 𝑝𝑓𝑡𝑠’s. A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is said 

to be a Pythagorean fuzzy (resp. 𝜃, 𝜃𝒮, 𝛿, 𝛿𝒫, 𝛿𝒮, 𝑀 and 𝑒 )-open (briefly, 𝑝𝑓𝑂 (resp. 𝑝𝑓𝜃𝑂, 𝑝𝑓𝜃𝒮𝑂, 

𝑝𝑓𝛿𝑂, 𝑝𝑓𝛿𝒫𝑂, 𝑝𝑓𝛿𝒮𝑂, 𝑝𝑓𝑀𝑂 and 𝑝𝑓𝑒𝑂)) mapping if the image of every 𝑝𝑓𝑜𝑠 in (𝑋1, 𝛤𝑃) is a 𝑝𝑓𝑜𝑠 

(resp. 𝑝𝑓𝜃𝑜𝑠, 𝑝𝑓𝜃𝒮𝑜𝑠, 𝑝𝑓𝛿𝑜𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝑀𝑜𝑠 and 𝑝𝑓𝑒𝑜𝑠) in (𝑋2, 𝛹𝑃).  

Proposition 3.1  Let (𝑋1, 𝛤𝑃) & (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑡𝑠’s. Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a mapping. Then 

the following statements are hold for 𝑝𝑓𝑡𝑠, but not conversely.  

    1.  Every 𝑝𝑓𝜃𝑂 is a 𝑝𝑓𝑂.  

    2.  Every 𝑝𝑓𝜃𝑂 is a 𝑝𝑓𝜃𝒮𝑂.  

    3.  Every 𝑝𝑓𝜃𝒮𝑂 is a 𝑝𝑓𝑀𝑂.  

    4.  Every 𝑝𝑓𝛿𝑂 is a 𝑝𝑓𝛿𝒮𝑂.  

    5.  Every 𝑝𝑓𝛿𝑂 is a 𝑝𝑓𝛿𝒫𝑂.  

    6.  Every 𝑝𝑓𝛿𝒮𝑂 is a 𝑝𝑓𝑒𝑂.  

    7.  Every 𝑝𝑓𝛿𝒫𝑂 is a 𝑝𝑓𝑀𝑂.  

    8.  Every 𝑝𝑓𝑀𝑂 is a 𝑝𝑓𝑒𝑂.  

    9.  Every 𝑝𝑓𝛿𝑂 is a 𝑝𝑓𝑂.  

  Proof.  

    1.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝜃𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝜃𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝜃𝑜𝑠 

is a 𝑝𝑓𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑂. 

    2.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝜃𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝜃𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝜃𝑜𝑠 

is a 𝑝𝑓𝜃𝒮𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝜃𝒮𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝜃𝒮𝑂. 



Communications on Applied Nonlinear Analysis 

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    3.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝜃𝒮𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝜃𝒮𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 

𝑝𝑓𝜃𝒮𝑜𝑠 is a 𝑝𝑓𝑀𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑀𝑂. 

    4.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝛿𝑜𝑠 

is a 𝑝𝑓𝛿𝒮𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝛿𝒮𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝛿𝒮𝑂. 

    5.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝛿𝑜𝑠 

is a 𝑝𝑓𝛿𝒫𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝛿𝒫𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝛿𝒫𝑂. 

    6.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝒮𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝒮𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 

𝑝𝑓𝛿𝒮𝑜𝑠 is a 𝑝𝑓𝑒𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑒𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑒𝑂. 

    7.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝒫𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝒫𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 

𝑝𝑓𝛿𝒫𝑜𝑠 is a 𝑝𝑓𝑀𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑀𝑂. 

    8.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑀𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝑀𝑜𝑠 

is a 𝑝𝑓𝑒𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑒𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑒𝑂.  

    9.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝑂, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝑜𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝛿𝑜𝑠 

is a 𝑝𝑓𝑜𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑂   

Remark 3.1  We obtain the following diagram from the results are discussed above.  

 

Note: 𝐴 → 𝐵 denotes 𝐴 implies 𝐵, but not conversely. 

Example 3.1   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3  & 𝐴4 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 



Communications on Applied Nonlinear Analysis 

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 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 Now, we have Γ𝑃 = Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an identity 

mapping. Then, ℎ𝑃 is 𝑝𝑓𝑂 but not 𝑝𝑓𝜃𝑂, because the set 𝐴1 is 𝑝𝑓𝑜𝑠 in 𝑋1 but ℎ𝑃(𝐴1) = 𝐴1 is not 

𝑝𝑓𝜃𝑜𝑠 in 𝑋2.  

Example 3.2   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.60,0.40 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝜃𝒮𝑂 (resp. 𝑝𝑓𝛿𝒮𝑂) but not 𝑝𝑓𝜃𝑂 (resp. 𝑝𝑓𝛿𝑂), because the set 𝐵1 is 

𝑝𝑓𝑜𝑠 in 𝑋1 but ℎ𝑃(𝐵1) = 𝐵1 is not 𝑝𝑓𝜃𝑜𝑠 (resp. 𝑝𝑓𝛿𝑜𝑠) in 𝑋2.  

Example 3.3   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = 𝐵1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑀𝑂 but not 𝑝𝑓𝜃𝒮𝑂, because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 𝑋1 but ℎ𝑃(𝐵1) = 𝐵1 

is not 𝑝𝑓𝜃𝒮𝑜𝑠 in 𝑋2.  

Example 3.4   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.40,0.20 >, < 𝑥2, 0.40,0.40 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑒𝑂 but not 𝑝𝑓𝑀𝑂, because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 𝑋1 but ℎ𝑃(𝐵1) = 𝐵1 

is not 𝑝𝑓𝑀𝑜𝑠 in 𝑋2.  

Example 3.5   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 4s (2025) 

 

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 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐵1 = 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑂 (resp. 𝑝𝑓𝑒𝑂 and 𝑝𝑓𝛿𝒫𝑂 ) but not 𝑝𝑓𝛿𝑂 (resp. 𝑝𝑓𝛿𝒮𝑂 and 𝑝𝑓𝛿𝑂), 

because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 𝑋1 but ℎ𝑃(𝐵1) = 𝐵1 is not 𝑝𝑓𝛿𝑜𝑠 (resp. 𝑝𝑓𝛿𝒮𝑜𝑠 and 𝑝𝑓𝛿𝑜𝑠) in 𝑋2.  

Example 3.6   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.60,0.30 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑀𝑂 but not 𝑝𝑓𝛿𝒫𝑂, because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 𝑋1 but ℎ𝑃(𝐵1) =

𝐵1 is not 𝑝𝑓𝛿𝒫𝑜𝑠 in 𝑋2.  

Theorem 3.1  A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝑂 iff for every 𝑝𝑓𝑠 𝐾 of (𝑋1, 𝛤𝑃), 

ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(𝐾)) ⊆ 𝑝𝑓𝑀𝑖𝑛𝑡(ℎ𝑃(𝐾)).  

 Proof. Necessity: Let ℎ𝑃 be a 𝑝𝑓𝑀𝑂 mapping and 𝐾 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Now, 𝑝𝑓𝑖𝑛𝑡(𝐾) ⊆ 𝐾 

implies ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(𝐾)) ⊆ ℎ𝑃(𝐾). Since ℎ𝑃 is a 𝑝𝑓𝑀𝑂 mapping, ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(𝐾)) is 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃) 

such that ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(𝐾)) ⊆ ℎ𝑃(𝐾). Therefore ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(𝐾)) ⊆ 𝑝𝑓𝑀𝑖𝑛𝑡(ℎ𝑃(𝐾)). 

Sufficiency: Assume 𝐾 is a 𝑝𝑓𝑜𝑠 of (𝑋1, Γ𝑃). Then ℎ𝑃(𝐾) = ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(𝐾)) ⊆ 𝑝𝑓𝑀𝑖𝑛𝑡(ℎ𝑃(𝐾)). But 

𝑝𝑓𝑀𝑖𝑛𝑡ℎ𝑃(𝐾) ⊆ ℎ𝑃(𝐾). So ℎ𝑃(𝐾) = 𝑝𝑓𝑀𝑖𝑛𝑡(𝐾) which implies ℎ𝑃(𝐾) is a 𝑝𝑓𝑀𝑜𝑠 of (𝑋2, Ψ𝑃) and 

hence ℎ𝑃 is a 𝑝𝑓𝑀𝑂 

Theorem 3.2  If ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is a 𝑝𝑓𝑀𝑂 mapping, then 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃
−1(𝐾)) ⊆

ℎ𝑃
−1(𝑝𝑓𝑀𝑖𝑛𝑡(𝐾)) for every 𝑝𝑓𝑠 𝐾 of (𝑋2, 𝛹𝑃).  

 Proof. Let 𝐾 be a 𝑝𝑓𝑠 of (𝑋2, Ψ𝑃). Then 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃
−1(𝐾)) is a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑀𝑂, 

ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(ℎ𝑃
−1(𝐾))) is 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃) and hence ℎ𝑃(𝑝𝑓𝑖𝑛𝑡(ℎ𝑃

−1(𝐾))) ⊆ 𝑝𝑓𝑀𝑖𝑛𝑡(ℎ𝑃(ℎ𝑃
−1(𝐾))) ⊆

𝑝𝑓𝑀𝑖𝑛𝑡(𝐾). Thus 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃
−1(𝐾)) ⊆ ℎ𝑃

−1(𝑝𝑓𝑀𝑖𝑛𝑡(𝐾)).              

Theorem 3.3   A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝑂 iff for each 𝑝𝑓𝑠 𝐺 of (𝑋2, 𝛹𝑃) and for each 

𝑝𝑓𝑐𝑠 𝐾 of (𝑋1, 𝛤𝑃) containing ℎ𝑃
−1(𝐺), there is a 𝑝𝑓𝑀𝑐𝑠 𝐻 of (𝑋2, 𝛹𝑃) such that 𝐺 ⊆ 𝐻 and ℎ𝑃

−1(𝐻) ⊆

𝐾.  

 Proof. Necessity: Assume ℎ𝑃 is a 𝑝𝑓𝑀𝑂 mapping. Let 𝐺 be the 𝑝𝑓𝑐𝑠 of (𝑋2, Ψ𝑃) and 𝐾 is a 𝑝𝑓𝑐𝑠 of 

(𝑋1, Γ𝑃) such that ℎ𝑃
−1(𝐺) ⊆ 𝐾. Then 𝐻 = (ℎ𝑃

−1(𝐾𝑐))𝑐 is 𝑝𝑓𝑀𝑐𝑠 of (𝑋2, Ψ𝑃) such that ℎ𝑃
−1(𝐻) ⊆ 𝐾. 



Communications on Applied Nonlinear Analysis 

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Sufficiency: Assume 𝐾 is a 𝑝𝑓𝑜𝑠 of (𝑋1, Γ𝑃). Then ℎ𝑃
−1((ℎ𝑃(𝐾))𝑐) ⊆ 𝐾𝑐 and 𝐾𝑐 is 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). 

By hypothesis, there is a 𝑝𝑓𝑀𝑐𝑠 𝐻 of (𝑋2, Ψ𝑃) such that (ℎ𝑃(𝐾))𝑐 ⊆ 𝐻 and ℎ𝑃
−1(𝐻) ⊆ 𝐾𝑐. Therefore 

𝐾 ⊆ (ℎ𝑃
−1(𝐻))𝑐. Hence 𝐻𝑐 ⊆ ℎ𝑃(𝐾) ⊆ ℎ𝑃((ℎ𝑃

−1(𝐻))𝑐) ⊆ 𝐻𝑐 which implies ℎ𝑃(𝐾) = 𝐻𝑐. Since 𝐻𝑐 is 

𝑝𝑓𝑀𝑜𝑠 of (𝑋2, Ψ𝑃), ℎ𝑃(𝐾) is 𝑝𝑓𝑀𝑂 in (𝑋2, Ψ𝑃) and thus ℎ𝑃 is 𝑝𝑓𝑀𝑂 mapping.              

Theorem 3.4  A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝑂 iff ℎ𝑃
−1(𝑝𝑓𝑀𝑐𝑙(𝐺)) ⊆ 𝑝𝑓𝑐𝑙(ℎ𝑃

−1(𝐺)) for 

every 𝑝𝑓𝑠 𝐺 of (𝑋2, 𝛹𝑃).  

 Proof. Necessity: Assume ℎ𝑃 is a 𝑝𝑓𝑀𝑂 mapping. For any 𝑝𝑓𝑠 𝐺 of (𝑋2, Ψ𝑃), ℎ𝑃
−1(𝐺) ⊆

𝑝𝑓𝑐𝑙(ℎ𝑃
−1(𝐺)). Therefore by Theorem 3.3, there exists a 𝑝𝑓𝑀𝑐𝑠 𝐾 in (𝑋2, Ψ𝑃) such that 𝐺 ⊆ 𝐾 and 

ℎ𝑃
−1(𝐾) ⊆ 𝑝𝑓𝑐𝑙(ℎ𝑃

−1(𝐺)). Therefore we obtain that ℎ𝑃
−1(𝑝𝑓𝑀𝑐𝑙(𝐺)) ⊆ ℎ𝑃

−1(𝐾) ⊆ 𝑝𝑓𝑐𝑙(ℎ𝑃
−1(𝐺)). 

Sufficiency: Assume 𝐺 is a 𝑝𝑓𝑠 of (𝑋2, Ψ𝑃) and 𝐾 is a 𝑝𝑓𝑐𝑠 of (𝑋1, Γ𝑃) containing ℎ𝑃
−1(𝐺). Put 𝐻 =

𝑝𝑓𝑐𝑙(𝐺), then 𝐺 ⊆ 𝐻 and 𝐻 is 𝑝𝑓𝑀𝑐𝑠 and ℎ𝑃
−1(𝐻) ⊊ 𝑝𝑓𝑐𝑙(ℎ𝑃

−1(𝐺)) ⊆ 𝐾. Then by Theorem 3.3, ℎ𝑃 is 

𝑝𝑓𝑀𝑂 mapping.              

Theorem 3.5  If ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) and 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) be two 𝑝𝑓 mappings and 𝑔𝑃 ∘

ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝑂. If 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝐼𝑟𝑟, then ℎ𝑃: (𝑋1, 𝛤𝑃) →

(𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝑂 mapping.  

 Proof. Let 𝐾 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Then (𝑔𝑃 ∘ ℎ𝑃)(𝐾) is 𝑝𝑓𝑀𝑜𝑠 of (𝑋3, Φ𝑃) because 𝑔𝑝 ∘ ℎ𝑃 is 

𝑝𝑓𝑀𝑂 mapping. Since 𝑔𝑃 is 𝑝𝑓𝑀𝐼𝑟𝑟 and (𝑔𝑃 ∘ ℎ𝑃)(𝐾) is 𝑝𝑓𝑀𝑜𝑠 of (𝑋3, Φ𝑃), 𝑔𝑃
−1((𝑔𝑃 ∘ ℎ𝑃)(𝐾)) =

ℎ𝑃(𝐾) is 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃). Hence ℎ𝑃 is 𝑝𝑓𝑀𝑂 mapping 

Theorem 3.6  If ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑂 and 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝑂 mappings, then 

𝑔𝑃 ∘ ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝑂.  

 Proof. Let 𝐾 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). Then ℎ𝑃(𝐾) is a 𝑝𝑓𝑜𝑠 of (𝑋2, Ψ𝑃) because ℎ𝑃 is a 𝑝𝑓𝑂 mapping. 

Since 𝑔𝑃 is 𝑝𝑓𝑀𝑂, 𝑔𝑃(ℎ𝑃(𝐾)) = (𝑔𝑃 ∘ ℎ𝑃)(𝐾) is a 𝑝𝑓𝑀𝑜𝑠 of (𝑋3, Φ𝑃). Hence 𝑔𝑃 ∘ ℎ𝑃 is 𝑝𝑓𝑀𝑂 

mapping.              

4   Pythagorean fuzzy 𝑴-closed mapping 

Definition 4.1  Let (𝑋1, 𝛤𝑃) and (𝑋2, 𝛹𝑃) be any two 𝑝𝑓𝑡𝑠’s. A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is said 

to be a Pythagorean fuzzy (resp. 𝜃, 𝜃𝒮, 𝛿, 𝛿𝒫, 𝛿𝒮, 𝑀 and 𝑒 )-closed (briefly, 𝑝𝑓𝐶 (resp. 𝑝𝑓𝜃𝐶, 𝑝𝑓𝜃𝒮𝐶, 

𝑝𝑓𝛿𝐶, 𝑝𝑓𝛿𝒫𝐶, 𝑝𝑓𝛿𝒮𝐶, 𝑝𝑓𝑀𝐶 and 𝑝𝑓𝑒𝐶)) mapping if the image of every 𝑝𝑓𝑐𝑠 in (𝑋1, 𝛤𝑃) is a 𝑝𝑓𝑐𝑠 

(resp. 𝑝𝑓𝜃𝑐𝑠, 𝑝𝑓𝜃𝒮𝑐𝑠, 𝑝𝑓𝛿𝑐𝑠, 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝑀𝑐𝑠 and 𝑝𝑓𝑒𝑐𝑠) in (𝑋2, 𝛹𝑃).  

Proposition 4.1  Let (𝑋1, 𝛤𝑃) & (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑡𝑠’s. Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a mapping. Then 

the following statements are hold for 𝑝𝑓𝑡𝑠, but not conversely.  

    1.  Every 𝑝𝑓𝜃𝐶 is a 𝑝𝑓𝐶.  

    2.  Every 𝑝𝑓𝜃𝐶 is a 𝑝𝑓𝜃𝒮𝐶.  

    3.  Every 𝑝𝑓𝜃𝒮𝐶 is a 𝑝𝑓𝑀𝐶.  

    4.  Every 𝑝𝑓𝛿𝐶 is a 𝑝𝑓𝛿𝒮𝐶.  

    5.  Every 𝑝𝑓𝛿𝐶 is a 𝑝𝑓𝛿𝒫𝐶.  



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    6.  Every 𝑝𝑓𝛿𝒮𝐶 is a 𝑝𝑓𝑒𝐶.  

    7.  Every 𝑝𝑓𝛿𝒫𝐶 is a 𝑝𝑓𝑀𝐶.  

    8.  Every 𝑝𝑓𝑀𝐶 is a 𝑝𝑓𝑒𝐶.  

    9.  Every 𝑝𝑓𝛿𝐶 is a 𝑝𝑓𝐶.  

  Proof.  

    1.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝜃𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝜃𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝜃𝑐𝑠 

is a 𝑝𝑓𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝐶. 

    2.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝜃𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝜃𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝜃𝑐𝑠 

is a 𝑝𝑓𝜃𝒮𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝜃𝒮𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝜃𝒮𝐶. 

    3.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝜃𝒮𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝜃𝒮𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 

𝑝𝑓𝜃𝒮𝑐𝑠 is a 𝑝𝑓𝑀𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑀𝐶. 

    4.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝛿𝑐𝑠 

is a 𝑝𝑓𝛿𝒮𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝛿𝒮𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝛿𝒮𝐶. 

    5.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝛿𝑐𝑠 

is a 𝑝𝑓𝛿𝒫𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝛿𝒫𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝛿𝒫𝐶. 

    6.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝒮𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝒮𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 

𝑝𝑓𝛿𝒮𝑐𝑠 is a 𝑝𝑓𝑒𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑒𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑒𝐶. 

    7.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝒫𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝒫𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 

𝑝𝑓𝛿𝒫𝑐𝑠 is a 𝑝𝑓𝑀𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑀𝐶. 

    8.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑀𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝑀𝑐𝑠 

is a 𝑝𝑓𝑒𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑒𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑒𝐶.  

    9.  Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Since ℎ𝑃 is 𝑝𝑓𝛿𝐶, ℎ𝑃(𝐵) is 𝑝𝑓𝛿𝑐𝑠 in (𝑋2, Ψ𝑃). Since every 𝑝𝑓𝛿𝑐𝑠 

is a 𝑝𝑓𝑐𝑠, ℎ𝑃(𝐵) is a 𝑝𝑓𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝐶  

Remark 4.1  We obtain the following diagram from the results are discussed above.  

 

Note: 𝐴 → 𝐵 denotes 𝐴 implies 𝐵, but not conversely. 



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Example 4.1   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3  & 𝐴4 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 Now, we have Γ𝑃 = Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an identity 

mapping. Then, ℎ𝑃 is 𝑝𝑓𝐶 but not 𝑝𝑓𝜃𝐶, because the set 𝐴1
𝑐 is 𝑝𝑓𝑐𝑠 in 𝑋1 but ℎ𝑃(𝐴1

𝑐) = 𝐴1
𝑐 is not 

𝑝𝑓𝜃𝑐𝑠 in 𝑋2.  

Example 4.2   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.60,0.40 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝜃𝒮𝐶 (resp. 𝑝𝑓𝛿𝒮𝐶) but not 𝑝𝑓𝜃𝐶 (resp. 𝑝𝑓𝛿𝐶), because the set 𝐵1
𝑐 is 

𝑝𝑓𝑐𝑠 in 𝑋1 but ℎ𝑃(𝐵1
𝑐) = 𝐵1

𝑐 is not 𝑝𝑓𝜃𝑐𝑠 (resp. 𝑝𝑓𝛿𝑐𝑠) in 𝑋2.  

Example 4.3   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = 𝐵1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑀𝐶 but not 𝑝𝑓𝜃𝒮𝐶, because the set 𝐵1
𝑐 is 𝑝𝑓𝑐𝑠 in 𝑋1 but ℎ𝑃(𝐵1

𝑐) =

𝐵1
𝑐 is not 𝑝𝑓𝜃𝒮𝑐𝑠 in 𝑋2.  

Example 4.4   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.40,0.20 >, < 𝑥2, 0.40,0.40 >} 



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 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑒𝐶 but not 𝑝𝑓𝑀𝐶, because the set 𝐵1
𝑐 is 𝑝𝑓𝑐𝑠 in 𝑋1 but ℎ𝑃(𝐵1

𝑐) = 𝐵1
𝑐 

is not 𝑝𝑓𝑀𝑐𝑠 in 𝑋2.  

Example 4.5   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐵1 = 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝐶 (resp. 𝑝𝑓𝑒𝐶 and 𝑝𝑓𝛿𝒫𝐶 ) but not 𝑝𝑓𝛿𝐶 (resp. 𝑝𝑓𝛿𝒮𝐶 and 𝑝𝑓𝛿𝐶), 

because the set 𝐵1
𝑐 is 𝑝𝑓𝑐𝑠 in 𝑋1 but ℎ𝑃(𝐵1

𝑐) = 𝐵1
𝑐 is not 𝑝𝑓𝛿𝑐𝑠 (resp. 𝑝𝑓𝛿𝒮𝑐𝑠 and 𝑝𝑓𝛿𝑐𝑠) in 𝑋2.  

Example 4.6   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋2  & 𝐵1 in 𝑋1 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.60,0.30 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐵1} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑀𝐶 but not 𝑝𝑓𝛿𝒫𝐶, because the set 𝐵1
𝑐 is 𝑝𝑓𝑐𝑠 in 𝑋1 but ℎ𝑃(𝐵1

𝑐) =

𝐵1
𝑐 is not 𝑝𝑓𝛿𝒫𝑐𝑠 in 𝑋2.  

Theorem 4.1  A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝐶 iff for each 𝑝𝑓𝑠 𝐺 of (𝑋2, 𝛹𝑃) and for 

each 𝑝𝑓𝑜𝑠 𝐾 of (𝑋1, 𝛤𝑃) containing ℎ𝑃
−1(𝐺), there is a 𝑝𝑓𝑀𝑜𝑠 𝐿 of (𝑋2, 𝛹𝑃) such that 𝐺 ⊆ 𝐿 and 

ℎ𝑃
−1(𝐿) ⊆ 𝐾.  

 Proof. Necessity: Assume ℎ𝑃 is a 𝑝𝑓𝑀𝐶 mapping. Let 𝐺 be the 𝑝𝑓𝑐𝑠 of (𝑋2, Ψ𝑃) and 𝐾 is a 𝑝𝑓𝑜𝑠 of 

(𝑋1, Γ𝑃) such that ℎ𝑃
−1(𝐺) ⊆ 𝐾. Then 𝐿 = 1𝑋 − ℎ𝑃

−1(𝐾𝑐) is 𝑝𝑓𝑀𝑜𝑠 of (𝑋2, Ψ𝑃) such that ℎ𝑃
−1(𝐿) ⊆ 𝐾. 

Sufficiency: Assume 𝐾 is a 𝑝𝑓𝑐𝑠 of (𝑋1, Γ𝑃). Then (ℎ𝑃(𝐾))𝑐 is a 𝑝𝑓𝑠 of (𝑋2, Ψ𝑃) and 𝐾𝑐 is 𝑝𝑓𝑜𝑠 in 

(𝑋1, Γ𝑃) such that ℎ𝑃
−1((ℎ𝑃(𝐾))𝑐) ⊆ 𝐾𝑐. By hypothesis, there is a 𝑝𝑓𝑀𝑜𝑠 𝐿 of (𝑋2, Ψ𝑃) such that 

(ℎ𝑃(𝐾))𝑐 ⊆ 𝐿 and ℎ𝑃
−1(𝐿) ⊆ 𝐾𝑐. Therefore 𝐾 ⊆ (ℎ𝑃

−1(𝐿))𝑐. Hence 𝐿𝑐 ⊆ ℎ𝑃(𝐿) ⊆ ℎ𝑃((ℎ𝑃
−1(𝐿))𝑐) ⊆ 𝐿𝑐 

which implies ℎ𝑃(𝐾) = 𝐿𝑐. Since 𝐿𝑐 is 𝑝𝑓𝑀𝑐𝑠 of (𝑋2, Ψ𝑃), ℎ𝑃(𝐾) is 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃) and thus ℎ𝑃 

is 𝑝𝑓𝑀𝐶 mapping.              

Theorem 4.2  If ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝐶 and 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝐶. Then 𝑔𝑃 ∘

ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝐶.  



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 Proof. Let 𝐾 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Then ℎ𝑃(𝐾) is 𝑝𝑓𝑐𝑠 of (𝑋2, Ψ𝑃) because ℎ𝑃 is 𝑝𝑓𝐶 mapping. Now 

(𝑔𝑃 ∘ ℎ𝑃)(𝐾) = 𝑔𝑃(ℎ𝑃(𝐾)) is 𝑝𝑓𝑀𝑐𝑠 in (𝑋3, Φ𝑃) because 𝑔𝑃 is 𝑝𝑓𝑀𝐶 mapping. Thus 𝑔𝑃 ∘ ℎ𝑃 is 

𝑝𝑓𝑀𝐶 mapping 

Theorem 4.3  If ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝐶 map, then 𝑝𝑓𝑀𝑐𝑙(ℎ𝑃(𝐾)) ⊆ ℎ𝑃(𝑝𝑓𝑐𝑙(𝐾)).  

 Proof. Obvious.              

Theorem 4.4  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) and 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) are 𝑝𝑓𝑀𝐶 mappings. If 

every 𝑝𝑓𝑀𝑐𝑠 of (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑐𝑠, then 𝑔𝑃 ∘ ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋3, 𝛷𝑃) is 𝑝𝑓𝑀𝐶.  

 Proof. Let 𝐾 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Then ℎ𝑃(𝐾) is 𝑝𝑓𝑀𝑐𝑠 of (𝑋2, Ψ𝑃) because ℎ𝑃 is 𝑝𝑓𝑀𝐶 mapping. 

By hypothesis, ℎ𝑃(𝐾) is 𝑝𝑓𝑐𝑠 of (𝑋2, Ψ𝑃). Now 𝑔𝑃(ℎ𝑃(𝐾)) = (𝑔𝑃 ∘ ℎ𝑃)(𝐾) is 𝑝𝑓𝑀𝑐𝑠 in (𝑋3, Φ𝑃) 

because 𝑔𝑃 is 𝑝𝑓𝑀𝐶 mapping. Thus 𝑔𝑃 ∘ ℎ𝑃 is 𝑝𝑓𝑀𝐶 mapping 

Theorem 4.5  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a bijective mapping. Then the following statements are 

equivalent: [(i)]  

    1.  ℎ𝑃 is a 𝑝𝑓𝑀𝑂 mapping.  

    2.  ℎ𝑃 is a 𝑝𝑓𝑀𝐶 mapping.  

    3.  ℎ𝑃
−1

 is 𝑝𝑓𝑀𝐶𝑡𝑠 mapping.  

  

 Proof. (i) ⇒ (ii): Let us assume that ℎ𝑃 is a 𝑝𝑓𝑀𝑂 mapping. By definition, 𝐾 is a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃), 

then ℎ𝑃(𝐾) is a 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃). Here, 𝐾 is 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Then 1𝑋 − 𝐾 is a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). 

By assumption, ℎ𝑃(1𝑋 − 𝐾) is a 𝑝𝑓𝑀𝑜𝑠 in (𝑋2, Ψ𝑃). Hence, 1𝑌 − ℎ𝑃(1𝑋 − 𝐾) is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃). 

Therefore, ℎ𝑃 is a 𝑝𝑓𝑀𝐶 mapping. 

(ii) ⇒ (iii): Let 𝐾 be a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃) By (ii), ℎ𝑃(𝐾) is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃(𝐾) =

(ℎ𝑃
−1)−1(𝐾). So ℎ𝑃

−1
 is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋2, Ψ𝑃). Hence, ℎ𝑃

−1
 is 𝑝𝑓𝑀𝐶𝑡𝑠. 

(iii) ⇒ (i): Let 𝐾 be a 𝑝𝑓𝑜𝑠 in (𝑋1, Γ𝑃). By (iii), (ℎ𝑃
−1)−1(𝐾) = ℎ𝑃(𝐾) is a 𝑝𝑓𝑀𝑂 mapping.              

5 Application 

Entropy as a measure of fuzziness was first proposed by Zadeh [35]. Later many mathematicians 

defined several entropy measures. In this section, we focus on defining an entropy measure for 𝑝𝑓𝑠 

that connects the degree of membership and non-membership. As an example, we have applied the 

proposed entropy measure in the field of seasons. 

Definition 5.1  Let 𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥)|𝑥 ∈ 𝑋} be a 𝑝𝑓𝑠 in 𝑋. The new entropy measure for 𝐴 

denoted by 휀𝑝𝑓𝑠(𝐴), is a function, 휀𝑝𝑓𝑠: 𝜏𝑝𝑓𝑠(𝑋) → [0,1] and is defined as 휀𝑝𝑓𝑠(𝐴) = 1 −
1

𝑛
∑𝑛

𝑖=1 (𝛼𝐴 − 𝛾𝐴)2;  𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦`𝑥𝑖 ∈ 𝐴, where 𝜏𝑝𝑓𝑠(𝑋) denote the family of all 𝑝𝑓𝑠’s on 𝑋.  

Example 5.1  The development of the each nation is measured by the economical development, 

Technical development, Educational growth, AI implementation, Import and Export, Defence status, 

Sports and etc. In Olympics and commonwealth games the developed countries are the toppers of the 

list, so the developing countries are also making lot of efforts and allot funds for sports. Even-though 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 4s (2025) 

 

40 
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practice and coaching plays the major role, the diet supplement also plays a vital role in athlete’s 

performance. Each sports academy, there are well trained nutritions to plan for the diet of the 

athlete. 

In a private sports academy named as 𝑋 has three diet plans for athlete who is on the training of the 

national games. Each diet was scheduled for a week, implemented at the gap of one week difference, 

the performance was rated by four trainers, for each day on the diet week. The rating of the trainers 

was converted as 𝑝𝑓𝑠 and entropy measure is used to select the best diet for the athlete . The following 

table displays 𝑝𝑓𝑠 of performance of each diet week. Here we use the notations for Diet 1, Diet 2 and 

Diet 3 are 𝐷𝑖1, 𝐷𝑖2 and 𝐷𝑖3. 

Table 1. Reviews of the Hotels based on the Criteria   

   Day 1 (𝐷1)   Day 2 (𝐷2)   Day 3 (𝐷3)   Day 4 (𝐷4)  

𝐷𝑖1  < 𝐷𝑖1, 𝐷1; 0.8,0.0 >   < 𝐷𝑖1, 𝐷2; 0.6,0.6 >  < 𝐷𝑖1, 𝐷3; 0.4,0.6 >  < 𝐷𝑖1, 𝐷4; 0.3,0.3 >  

𝐷𝑖2   < 𝐷𝑖2, 𝐷1; 0.7,0.3 >   < 𝐷𝑖2, 𝐷2; 0.8,0.6 >   < 𝐷𝑖2, 𝐷3; 0.7,0.3 >   < 𝐷𝑖2, 𝐷4; 0.8,0.2 >  

𝐷𝑖3  < 𝐷𝑖3, 𝐷1; 0.7,0.1 >   < 𝐷𝑖3, 𝐷2; 0.8,0.0 >   < 𝐷𝑖3, 𝐷3; 0.7,0.4 >   < 𝐷𝑖3, 𝐷4; 0.6,0.3 >  

 Day 5 (𝐷5) Day 6 (𝐷6) Day 7 (𝐷7) 

𝐷𝑖1  < 𝐷𝑖1, 𝐷5; 0.8,0.2 >  < 𝐷𝑖1, 𝐷6; 0.7,0.7 > < 𝐷𝑖1, 𝐷7; 0.7,0.5 >  

𝐷𝑖2 < 𝐷𝑖2, 𝐷5; 0.2,0.4 >   < 𝐷𝑖2, 𝐷6; 0.6,0.2 >  < 𝐷𝑖2, 𝐷7; 0.7,0.5 >  

𝐷𝑖3  < 𝐷𝑖3, 𝐷5; 0.7,0.5 >  < 𝐷𝑖3, 𝐷6; 0.7,0.2 >  < 𝐷𝑖3, 𝐷7; 0.8,0.5 >  

Clearly, all values in the Table 1 are 𝑝𝑓𝑠’s. Now we calculate the 휀𝑝𝑓𝑠 of each value. 

Table 2. Entropy measure of each Diet. 

 entropy measure 

𝐷𝑖1  0.73  

𝐷𝑖2  0.76  

𝐷𝑖3   0.61  

From Table 2, Clearly that 휀𝑝𝑓𝑠(𝐷𝑖3) < 휀𝑝𝑓𝑠(𝐷𝑖1) < 휀𝑝𝑓𝑠(𝐷𝑖2). 

Hence we conclude that 𝐷𝑖3 is the best for the athlete. 

6 Conclusion 

 In this paper, we have studied a new class of maps called Pythagorean fuzzy 𝑀 open and Pythagorean 

fuzzy 𝑀 closed and their properties are discussed. Also we applied entropy measure for decision 

making problem of calculation of diet selection based on the performance. In future, we decide to apply 

entropy measure for decision making in various fields. 

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