Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1925 
https://internationalpubls.com 

More on Contra Continuous, Irresolute Maps in Pythagorean Fuzzy 

Nano Topological Spaces and Application in 𝐌𝐂𝐃𝐌 

 

K. Balasubramaniyan1, K. Nijanthan2, A. Vadivel3 and K. Shantha lakshmi4 

 1Department of Mathematics, Arignar Anna Government Arts College, Attur - 636 121, India. 

kgbalumaths@gmail.com 

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

nijanthanvdl1996@gmail.com 

   4Department of Mathematics, M.Kumarasamy College of Engineering, Karur - 639 113, India. 

avmaths@gmail.com 

  1,2,3,4Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India. 

kslakshmi20@gmail.com;   

Corresponding Author K. Shantha Lakshmi and K. Nijanthan 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: In our daily life we come across many situation which are non probabilistic 

and underestimation category. Since they are unpredictable, innumerable real life 

problems are in abundant condition for decision making from earlier 20th century. Fuzzy 

sets, intuitionstic fuzzy sets and Pythagorean fuzzy sets emerged one by one and many 

literature are adding on every day. The concept of Pythagorean fuzzy nano (resp. 𝛿, 𝛿𝒫, 

𝛿𝒮 , 𝛿𝛼  & 𝛿𝛽  or 𝑒∗ )-continuity in Pythagorean fuzzy nano topological spaces and 

specialize some of their basic properties with examples is our main contribution to those 

literature. Also, we discuss about properties and characterization of Pythagorean fuzzy 

irresolute maps and application of Multiple Criteria Decision Making (MCDM) 

techniques to the real-world problem using a proposed similarity measure in Pythagorean 

fuzzy nano topological spaces.  

Keywords: Pythagorean fuzzy nano topological spaces, 𝒫ℱ𝔑𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 , 

𝒫ℱ𝔑𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 , 𝒫ℱ𝔑  𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟 , 𝒫ℱ𝔑𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟  and Zhang similarity 

measure. 

AMS (2000) subject classification: 03E72, 54A40, 54C05, 94D05 

 

1  Introduction 

 In 1965, Zadeh [40] familiarized the concept of fuzzy set which has several applications in decision 

theory, artificial intelligence, operations research, expert systems, computer science, data analytics, 

pattern recognition, management science and robotics. In 1968, Chang and Warren [14, 34] defined 

fuzzy topological spaces, the basic philosophies of topology such as open set, closed set, 

neighbourhood, interior set, closure, continuity, compactness to fuzzy topological spaces (𝐹𝑇𝑆). 

Applications of fuzzy sets were studied [1, 13, 25, 30]. Later numerous fuzzy topological spaces raised 

which have unique properties. In 1997, Dogan Coker [9, 15, 19] introduced Intuitionistic fuzzy 

topological spaces and studied its continuity and compactness. Intuitionistic fuzzy sets have many 

applications [30, 27] and also flagged approach to study Pythagorean fuzzy sets. In both the sets 

membership and non-membership are incorporated in a different way. In Intuitionistic fuzzy set the 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1926 
https://internationalpubls.com 

membership 𝜇 and non-membership 𝜆 are incorporated in such a way that 𝜇 + 𝜆 ≤ 1 where as in 

Pythagorean fuzzy set it is 𝜇2 + 𝜆2 ≤ 1. In 2013, Yager [37] introduced the non-standard fuzzy sets 

called Pythagorean fuzzy sets in comparison with Intuitionistic fuzzy sets. He gave the basic definition 

of Pythagorean fuzzy set (𝑃𝐹𝑆) and its application in decision making [3, 39, 38]. 𝑃𝐹𝑆 has its 

applications in career placements based on academic performance [20], selection of mask during 

COVID-19 pandemic using Pythagorean TOPSIS technique [24], etc. Later Murat et.al [18] 

introduced the conception of Pythagorean fuzzy topological space (𝑃𝐹𝑇𝑆) by provoking from the 

conviction of 𝐹𝑇𝑆 [16, 17, 23]. He defined Pythagorean fuzzy continuous function between 𝑃𝐹𝑇𝑆. 

Saha [26] defined 𝛿-open sets in fuzzy topological spaces. In 2019, Acikgoz and Esenbel [2] defined 

neutrosophic soft 𝛿-topology. Aranganayagi et al., Surendra et al. and Vadivel et al. [7, 8, 28, 29, 32, 

33] introduced 𝛿 -open sets in neutrosophic, neutrosophic soft, neutrosophic hypersoft and 

neutrosophic nano topological spaces and studied its maps and separation axioms. 

Similarity measure is a significant means for measuring the uncertain information. The fuzzy 

similarity measure is a measure that depicts the closeness (difference) among fuzzy sets. Zhang [42] 

proposed the Pythagorean fuzzy similarity measures for dealing the multi-attribute decision-making 

problems. Peng et al. [21] 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 [35] presented some Pythagorean fuzzy cosine function for 

dealing with the decision-making problems. However, some existing similarity measures/distance 

measures cannot obey the 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 

counter-intuitive phenomena [35, 42, 21] of the existing similarity measures of 𝒫ℱ𝑠’s, they may be 

hard for 𝐷𝑀’s 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 for Pythagorean fuzzy set, 

which can be without counter intuitive phenomena. 

 Research Gap: No investigation on some stronger and weaker forms of Pythagorean fuzzy 

continuous and irresolute maps such as Pythagorean fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿 open map, Pythagorean 

fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿-semi open map, Pythagorean fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿-pre open map, Pythagorean 

fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼 open map and Pythagorean fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽 open maps on Pythagorean 

fuzzy nano topological space has been reported in the Pythagorean fuzzy nano literature. 

This leads to encompass the notion of 𝑃𝐹𝑁𝑡𝑠 by introducing Pythagorean fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿 

(resp. 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼 , 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮 , 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫  &  𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽  or 𝑐𝑜𝑛𝑡𝑟𝑎𝑒∗ )-continuous and discuss its 

properties. Also, we introduce the concept of Pythagorean fuzzy nano irresoluteness called 

Pythagorean fuzzy nano 𝑐𝑜𝑛𝑡𝑟𝑎𝛿  (resp. 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫, 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮, 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼  and 

𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽) -irresolute maps by using 𝒫ℱ𝒩𝒮𝑜𝑠  (resp. 𝒫ℱ𝒩𝛿𝑜𝑠 , 𝒫ℱ𝒩𝛿𝒫𝑜𝑠, 𝑝𝑓𝒩𝛿𝒮𝑜𝑠,

𝑝𝑓𝒩𝛿𝛼𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑜𝑠)’s and study some of their basic properties. 

 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1927 
https://internationalpubls.com 

2  Preliminaries 

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

Definition 2.1  [40] Let 𝑋 be a nonempty set. A fuzzy set 𝐴 in 𝑋 is characterized by a membership 

function 𝜇𝐴: 𝑋 → [0,1]. That is:  

 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  [9, 10, 11, 12] Let a nonempty set 𝑋 be fixed. An 𝐼𝐹𝑆 𝐴 in 𝑋 is an object having the 

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

𝑥
⟩ |𝑥 ∈ 𝑋},  where the functions 

𝜇𝐴(𝑥): 𝑋 → [0,1]  and 𝜆𝐴(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) + 𝜆𝐴(𝑥) ≤ 1. For each 𝐴 in 𝑋: 𝜋𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) − 𝜆𝐴(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) + 𝜆𝐴(𝑥) + 𝜋𝐴(𝑥) = 1.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1928 
https://internationalpubls.com 

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  [36, 37, 39] Let a non empty set 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), 𝜆𝐴(𝑥)|𝑥 ∈ 𝑋} 

or 𝐴 = {⟨
𝜇𝐴(𝑥),𝜆𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋},  where the functions 𝜇𝐴(𝑥): 𝑋 → [0,1]  and 𝜆𝐴(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 + (𝜆𝐴(𝑥))2 ≤ 1. Supposing (𝜇𝐴(𝑥))2 +

(𝜆𝐴(𝑥))2 ≤ 1 , then there is a degree of indeterminacy of 𝑥 ∈ 𝑋  to 𝐴  defined by 𝜋𝐴(𝑥) =

√1 − [(𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2]  and 𝜋𝐴(𝑥) ∈ [0,1] . In what follows, (𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2 +

(𝜋𝐴(𝑥))2 = 1. Otherwise, 𝜋𝐴(𝑥) = 0 whenever (𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2 = 1. We denote the set of all 

𝑃𝐹𝑆’s over 𝑋 by 𝑝𝑓𝑠(𝑋). 

Definition 2.4  [39] Let 𝐴 and 𝐵 be 𝑝𝑓𝑠’s of the forms 𝐴 = {< 𝑎, 𝜇𝐴(𝑎), 𝜆𝐴(𝑎) > |𝑎 ∈ 𝑋} and 

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

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

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

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

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

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

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

    7.  1̅𝑃 = 0𝑃 and 0̅𝑃 = 1𝑃. 

Definition 2.5  [4] Let 𝑈 be a non-empty set and 𝑅 be an equivalence relation on 𝑈. Let 𝐴 be a 

Pythagorean fuzzy set in 𝑈  with the membership function 𝜇𝐴(𝑥)  and non membership function 

𝜆𝐴(𝑥), ∀ 𝑥 ∈ 𝑈. The Pythagorean fuzzy nano lower, Pythagorean fuzzy nano upper approximation 

and Pythagorean fuzzy nano boundary of 𝐴  in the approximation (𝑈, 𝑅)  denoted by 

𝒫ℱ𝒩(𝐴), 𝒫ℱ𝒩(𝐴) and 𝐵𝒫ℱ𝒩(𝐴) are respectively defined as follows: [(i)]  

    1.  𝒫ℱ𝒩(𝐴) = {〈𝑥, 𝜇𝑅(𝐴)(𝑥), 𝜆𝑅(𝐴)(𝑥)〉/𝑦 ∈ [𝑥]𝑅 , 𝑥 ∈ 𝑈}  

    2.  𝒫ℱ𝒩(𝐹) = {〈𝑥, 𝜇𝑅(𝐴)(𝑥), 𝜆𝑅(𝐴)(𝑥)〉/𝑦 ∈ [𝑥]𝑅, 𝑥 ∈ 𝑈}  

    3.  𝐵𝒫ℱ𝒩(𝐹) = 𝒫ℱ𝒩(𝐹) − 𝒫ℱ𝒩(𝐹)  

  where 𝜇𝑅(𝐴)(𝑥) =∧𝑦∈[𝑥]𝑅
𝜇𝐴(𝑦) 

𝜆𝑅(𝐴)(𝑥) =∧𝑦∈[𝑥]𝑅
𝜆𝐴(𝑦), 

𝜇𝑅(𝐴)(𝑥) =∨𝑦∈[𝑥]𝑅
𝜇𝐴(𝑦), 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1929 
https://internationalpubls.com 

𝜆𝑅(𝐴)(𝑥) =∨𝑦∈[𝑥]𝑅
𝜆𝐴(𝑦). 

Definition 2.6  [4] Let 𝑈 be an universe of discourse, 𝑅 be an equivalence relation on 𝑈 and 𝐴 be 

a Pythagorean fuzzy set in 𝑈  and if the collection 𝜏ℛ(𝐴) = {0𝒫 , 1𝒫 , 𝒫ℱ𝒩(𝐴),

𝒫ℱ𝒩(𝐴),  𝐵𝒫ℱ𝒩(𝐴)} forms a topology then it is said to be a Pythagorean fuzzy nano topology. We 

call (𝑈, 𝜏ℛ(𝐴)) (or simply 𝑈) as the Pythagorean fuzzy nano topological space. The elements of 

𝜏ℛ(𝐴) are called Pythagorean fuzzy nano open (briefly, 𝒫ℱ𝒩𝑜) sets.  

Remark 2.1  [4] [𝜏ℛ(𝐴)]𝑐 is called the dual fuzzy nano topology of 𝜏ℛ(𝐴). Elements of [𝜏ℛ(𝐴)]𝑐 

are called Pythagorean fuzzy nano closed (briefly, 𝒫ℱ𝒩𝑐) sets. Thus, we note that a Pythagorean 

fuzzy set 𝐺 of 𝑈 is Pythagorean fuzzy nano closed in 𝜏ℛ(𝐴) if and only if 1𝑃 − 𝐺 is Pythagorean 

fuzzy nano open in 𝜏ℛ(𝐴).  

Definition 2.7  [4, 5] Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝒩𝑡𝑠 with respect to 𝐴 where 𝐴 is a Pythagorean 

fuzzy subset of 𝑈. Let 𝑆 be a Pythagorean fuzzy subset of 𝑈. Then Pythagorean fuzzy nano [(i)]  

    1.  interior of 𝑆  (briefly, 𝒫ℱ𝒩𝑖𝑛𝑡(𝑆) ) is defined by 𝒫ℱ𝒩𝑖𝑛𝑡(𝑆) =∪ {𝐼: 𝐼 ≤

𝑆 & 𝐼isa𝒫ℱ𝒩𝑜set in𝑈}.  

    2.  closure of 𝑆  (briefly, 𝒫ℱ𝒩𝑐𝑙(𝑆) ) is defined by 𝒫ℱ𝒩𝑐𝑙(𝑆) =∩ {𝐴: 𝑆 ≤

𝐴 & 𝐴isa𝒫ℱ𝒩𝑐set in𝑈}.  

    3.  regular open (briefly, 𝒫ℱ𝒩𝑟𝑜) set if 𝑆 = 𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙(𝑆)).  

    4.  regular closed (briefly, 𝒫ℱ𝒩𝑟𝑐) set if 𝑆 = 𝒫ℱ𝒩𝑐𝑙(𝒫ℱ𝒩𝑖𝑛𝑡(𝑆)).   

Definition 2.8  [6] Let (𝑈1, 𝜏𝑃(𝐴1)) and (𝑈2, 𝜏𝑃(𝐴2)) be two 𝑝𝑓𝒩𝑡𝑠’s. Then a function ℎ𝑃: 𝑈1 →

𝑈2 is said to be a Pythagorean fuzzy nano continuous (briefly, 𝒫ℱ𝒩𝐶𝑡𝑠) function if ℎ𝑃
−1(𝐺) is 

𝒫ℱ𝒩𝑜 set in 𝑈1 for all 𝒫ℱ𝒩𝑜 set 𝐺 in 𝑈2.  

Definition 2.9  [22] Let 𝑀, 𝑁  and 𝑂  be three 𝑝𝑓𝑠 ’s on 𝑋 . A similarity measure 𝑆(𝑀, 𝑁)  is 

mapping 𝑆: 𝑝𝑓𝑠(𝑋) × 𝑝𝑓𝑠(𝑋) → [0,1], possessing the following properties: [(S1)]  

    1.  0 ≤ 𝑆(𝑀, 𝑁) ≤ 1;  

    2.  𝑆(𝑀, 𝑁) = 𝑆(𝑁, 𝑀);  

    3.  𝑆(𝑀, 𝑁) = 1 iff 𝑀 = 𝑁;  

    4.  𝑆(𝑀, 𝑀𝑐) = 0 iff 𝑀 is a crisp set;  

    5.  If 𝑀 ⊆ 𝑁 ⊆ 𝑂, then 𝑆(𝑀, 𝑂) ≤ 𝑆(𝑀, 𝑁) and 𝑆(𝑀, 𝑂) ≤ 𝑆(𝑁, 𝑂). 

Let 𝑋 = 𝑥1, 𝑥2, … , 𝑥𝑛 be a finite universe of discourse, and 𝐴 and 𝐵 be two 𝑃𝐹𝑆’s in 𝑋, in which 

𝐴 = {< 𝑥𝑖 , 𝜇𝐴(𝑥𝑖), 𝜆𝐴(𝑥𝑖) > |𝑥𝑖 ∈ 𝑋} and 𝐵 = {< 𝑥𝑖 , 𝜇𝐵(𝑥𝑖), 𝜆𝐵(𝑥𝑖) > |𝑥𝑖 ∈ 𝑋}. 

Using the similarity measure in section 4, we have the Zhang [42] similarity measure are defined by 

𝑆𝑍(𝐴, 𝐵) = 

1

2
∑𝑛

𝑖=1
(|𝜇𝐴

2 (𝑥𝑖)−𝜈𝐵
2 (𝑥𝑖)|+|𝜈𝐴

2 (𝑥𝑖)−𝜇𝐵
2 (𝑥𝑖)|+|𝜋𝐴

2 (𝑥𝑖)−𝜋𝐵
2 (𝑥𝑖)|)

|𝜇𝐴
2 (𝑥𝑖)−𝜇𝐵

2 (𝑥𝑖)|+|𝜈𝐴
2 (𝑥𝑖)−𝜈𝐵

2 (𝑥𝑖)|+|𝜋𝐴
2 (𝑥𝑖)−𝜋𝐵

2 (𝑥𝑖)|+|𝜇𝐴
2 (𝑥𝑖)−𝜈𝐵

2 (𝑥𝑖)|+|𝜈𝐴
2 (𝑥𝑖)−𝜇𝐵

2 (𝑥𝑖)|+|𝜋𝐴
2 (𝑥𝑖)−𝜋𝐵

2 (𝑥𝑖)|
. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1930 
https://internationalpubls.com 

3  Pythagorean fuzzy nano contra 𝜹 (resp. 𝜹 pre, 𝜹 semi, 𝜹𝜶 and 𝜹𝜷)-continuous mappings 

 In this section, we introduce Pythagorean fuzzy nano contra 𝛿  (resp. 𝛿  pre, 𝛿  semi, 𝛿𝛼  and 

𝛿𝛽)-continuous mappings and discuss some of their properties. 

Definition 3.1  Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠 with respect to 𝐴 where 𝐴 is a 𝑝𝑓𝑠 of 𝑈. Let 𝑆 be a 

𝑝𝑓𝑠 of 𝑈. Then [(i)]  

    1.  Pythagorean fuzzy nano 𝛿  interior of 𝑆  (briefly, 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆) ) is defined by 

𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆) =∪ {𝐼: 𝐼 ⊆ 𝑆 & 𝐼isa𝒫ℱ𝔑𝑟𝑜 set in𝑈}.  

    2.  Pythagorean fuzzy nano 𝛿 closure of 𝑆 (briefly, 𝒫ℱ𝔑𝛿𝑐𝑙(𝑆)) is defined by 𝒫ℱ𝔑𝛿𝑐𝑙(𝑆) =

∩ {𝐴: 𝑆 ⊆ 𝐴 & 𝐴isa𝒫ℱ𝔑𝑟𝑐 set in 𝑈}.   

Definition 3.2   Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠 with respect to 𝐴 where 𝐴 is a 𝑝𝑓𝑠 of 𝑈. Then a 

𝒫ℱ𝑠 𝑆 in 𝑈 is said to be Pythagorean: [(i)]  

    1.  fuzzy nano 𝛿-open set (briefly, 𝒫ℱ𝔑𝛿𝑜𝑠) if 𝑆 = 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆).  

    2.  fuzzy nano 𝛿𝒫-open set (briefly, 𝒫ℱ𝔑𝛿𝒫𝑜𝑠) if 𝑆 ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆)).  

    3.  fuzzy nano 𝛿𝒮-open set (briefly, 𝒫ℱ𝔑𝛿𝒮𝑜𝑠) if 𝑆 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆)).  

    4.  fuzzy nano 𝛿𝛼  or 𝑎 -open set (briefly, 𝒫ℱ𝔑𝛿𝛼𝑜𝑠  or 𝒫ℱ𝔑𝑎𝑜𝑠 ) if 𝑆 ⊆

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆))).  

    5.  fuzzy nano 𝛿𝛽  or 𝑒∗  -open set (briefly, 𝒫ℱ𝔑𝛿𝛽𝑜𝑠  or 𝒫ℱ𝔑𝑒∗𝑜𝑠 ) if 𝑆 ⊆

𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆))).  

The complement of a 𝒫ℱ𝔑𝛿𝑜𝑠 (resp. 𝒫ℱ𝔑𝛿𝒫𝑜𝑠, 𝒫ℱ𝔑𝛿𝒮𝑜𝑠, 𝒫ℱ𝔑𝛿𝛼𝑜𝑠 & 𝒫ℱ𝔑𝛿𝛽𝑜𝑠) is called a 

Pythagorean fuzzy nano 𝛿  (resp. 𝛿𝒫 , 𝛿𝒮 , 𝛿𝛼  and 𝛿𝛽 ) closed set (briefly, 𝒫ℱ𝔑𝛿𝑐𝑠  (resp. 

𝒫ℱ𝔑𝛿𝒫𝑐𝑠, 𝒫ℱ𝔑𝛿𝒮𝑐𝑠, 𝒫ℱ𝔑𝛿𝛼𝑐𝑠 and 𝒫ℱ𝔑𝛿𝛽𝑐𝑠)) in 𝑈.  

The family of all 𝒫ℱ𝔑𝛿𝑜𝑠  (resp. 𝒫ℱ𝔑𝛿𝑐𝑠, 𝒫ℱ𝔑𝛿𝒫𝑜𝑠, 𝒫ℱ𝔑𝛿𝒫𝑐𝑠, 𝒫ℱ𝔑𝛿𝒮𝑜𝑠, 𝒫ℱ𝔑𝛿𝒮𝑐𝑠,

𝒫ℱ𝔑𝛿𝛼𝑜𝑠, 𝒫ℱ𝔑𝛿𝛼𝑐𝑠,   𝒫ℱ𝔑𝛿𝛽𝑜𝑠  and 𝒫ℱ𝔑𝛿𝛽𝑐𝑠)  of 𝑈  is denoted by 𝒫ℱ𝔑𝛿𝑂𝑆(𝑈)  (resp. 

𝒫ℱ𝔑𝛿𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒫𝑂𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒫𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒮𝑂𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒮𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝛼𝑂𝑆(𝑈),

𝒫ℱ𝔑𝛿𝛼𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝛽𝑂𝑆(𝑈) and 𝒫ℱ𝔑𝛿𝛽𝐶𝑆(𝑈)).    

Definition 3.3  Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠 with respect to 𝐴 where 𝐴 is a 𝑝𝑓𝑠 of 𝑈. Let 𝑆 be a 

𝑝𝑓𝑠 of 𝑈. Then Pythagorean fuzzy nano [(i)]  

    1.  𝛿  pre (resp. 𝛿  semi, 𝛿𝛼  and 𝛿𝛽 ) interior of 𝑆  (briefly, 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑆)  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝑆) , 𝒫ℱ𝔑𝛿𝛼𝑖𝑛𝑡(𝑆)  and 𝒫ℱ𝔑𝛿𝛽𝑖𝑛𝑡(𝑆) )) is defined by 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑆)  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝑆), 𝒫ℱ𝔑𝛿𝛼𝑖𝑛𝑡(𝑆)  and 𝒫ℱ𝔑𝛿𝛽𝑖𝑛𝑡(𝑆)) =∪ {𝐼: 𝐼 ⊆ 𝑆 & 𝐼isa𝒫ℱ𝔑𝛿𝒫𝑜  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑜, 𝒫ℱ𝔑𝛿𝛼𝑜 & 𝒫ℱ𝔑𝛿𝛽𝑜) set in𝑈}. 

    2.  𝛿 pre (resp. 𝛿 semi, 𝛿𝛼 and 𝛿𝛽) closure of 𝑆 (briefly, 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑆) (resp. 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝑆), 

𝒫ℱ𝔑𝛿𝛼𝑐𝑙(𝑆) and 𝒫ℱ𝔑𝛿𝛽𝑐𝑙(𝑆))) is defined by 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑆) (resp. 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝑆), 𝒫ℱ𝔑𝛿𝛼𝑐𝑙(𝑆) 

and 𝒫ℱ𝔑𝛿𝛽𝑐𝑙(𝑆)) =∩ {𝐴: 𝑆 ⊆ 𝐴 & 𝐴isa𝒫ℱ𝔑𝛿𝒫𝑐 (resp. 𝒫ℱ𝔑𝛿𝒮𝑐, 𝒫ℱ𝔑𝛿𝛼𝑐 & 𝒫ℱ𝔑𝛿𝛽𝑐)  set in  

𝑈}.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1931 
https://internationalpubls.com 

 Definition 3.4  Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠 with respect to 𝐴 where 𝐴 is a 𝑝𝑓𝑠 of 𝑈. Let 𝑆 be a 

𝑝𝑓𝑠 of 𝑈. Then [(i)]  

    1.  Pythagorean fuzzy nano 𝛿  interior of 𝑆  (briefly, 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆) ) is defined by 

𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆) =∪ {𝐼: 𝐼 ⊆ 𝑆 & 𝐼isa𝒫ℱ𝔑𝑟𝑜 set in𝑈}.  

    2.  Pythagorean fuzzy nano 𝛿 closure of 𝑆 (briefly, 𝒫ℱ𝔑𝛿𝑐𝑙(𝑆)) is defined by 𝒫ℱ𝔑𝛿𝑐𝑙(𝑆) =

∩ {𝐴: 𝑆 ⊆ 𝐴 & 𝐴isa𝒫ℱ𝔑𝑟𝑐 set in 𝑈}.   

Definition 3.5   Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠 with respect to 𝐴 where 𝐴 is a 𝑝𝑓𝑠 of 𝑈. Then a 

𝒫ℱ𝑠 𝑆 in 𝑈 is said to be Pythagorean: [(i)]  

    1.  fuzzy nano 𝛿-open set (briefly, 𝒫ℱ𝔑𝛿𝑜𝑠) if 𝑆 = 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆).  

    2.  fuzzy nano 𝛿𝒫-open set (briefly, 𝒫ℱ𝔑𝛿𝒫𝑜𝑠) if 𝑆 ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆)).  

    3.  fuzzy nano 𝛿𝒮-open set (briefly, 𝒫ℱ𝔑𝛿𝒮𝑜𝑠) if 𝑆 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆)).  

    4.  fuzzy nano 𝛿𝛼  or 𝑎 -open set (briefly, 𝒫ℱ𝔑𝛿𝛼𝑜𝑠  or 𝒫ℱ𝔑𝑎𝑜𝑠 ) if 𝑆 ⊆

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆))).  

    5.  fuzzy nano 𝛿𝛽  or 𝑒∗  -open set (briefly, 𝒫ℱ𝔑𝛿𝛽𝑜𝑠  or 𝒫ℱ𝔑𝑒∗𝑜𝑠 ) if 𝑆 ⊆

𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆))).  

The complement of a 𝒫ℱ𝔑𝛿𝑜𝑠 (resp. 𝒫ℱ𝔑𝛿𝒫𝑜𝑠, 𝒫ℱ𝔑𝛿𝒮𝑜𝑠, 𝒫ℱ𝔑𝛿𝛼𝑜𝑠 & 𝒫ℱ𝔑𝛿𝛽𝑜𝑠) is called a 

Pythagorean fuzzy nano 𝛿  (resp. 𝛿𝒫 , 𝛿𝒮 , 𝛿𝛼  and 𝛿𝛽 ) closed set (briefly, 𝒫ℱ𝔑𝛿𝑐𝑠  (resp. 

𝒫ℱ𝔑𝛿𝒫𝑐𝑠, 𝒫ℱ𝔑𝛿𝒮𝑐𝑠, 𝒫ℱ𝔑𝛿𝛼𝑐𝑠 and 𝒫ℱ𝔑𝛿𝛽𝑐𝑠)) in 𝑈.  

The family of all 𝒫ℱ𝔑𝛿𝑜𝑠  (resp. 𝒫ℱ𝔑𝛿𝑐𝑠, 𝒫ℱ𝔑𝛿𝒫𝑜𝑠, 𝒫ℱ𝔑𝛿𝒫𝑐𝑠, 𝒫ℱ𝔑𝛿𝒮𝑜𝑠, 𝒫ℱ𝔑𝛿𝒮𝑐𝑠,

𝒫ℱ𝔑𝛿𝛼𝑜𝑠, 𝒫ℱ𝔑𝛿𝛼𝑐𝑠,   𝒫ℱ𝔑𝛿𝛽𝑜𝑠  and 𝒫ℱ𝔑𝛿𝛽𝑐𝑠)  of 𝑈  is denoted by 𝒫ℱ𝔑𝛿𝑂𝑆(𝑈),  (resp. 

𝒫ℱ𝔑𝛿𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒫𝑂𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒫𝐶𝑆(𝑈),  𝒫ℱ𝔑𝛿𝒮𝑂𝑆(𝑈), 𝒫ℱ𝔑𝛿𝒮𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝛼𝑂𝑆(𝑈),

𝒫ℱ𝔑𝛿𝛼𝐶𝑆(𝑈), 𝒫ℱ𝔑𝛿𝛽𝑂𝑆(𝑈) and 𝒫ℱ𝔑𝛿𝛽𝐶𝑆(𝑈)). 

Definition 3.6  Let (𝑈, 𝜏𝒫(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠 with respect to 𝐴 where 𝐴 is a 𝑝𝑓𝑠 of 𝑈. Let 𝑆 be a 

𝑝𝑓𝑠 of 𝑈. Then Pythagorean fuzzy nano [(i)]  

    1.  𝛿  pre (resp. 𝛿  semi, 𝛿𝛼  and 𝛿𝛽 ) interior of 𝑆  (briefly, 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑆)  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝑆) , 𝒫ℱ𝔑𝛿𝛼𝑖𝑛𝑡(𝑆)  and 𝒫ℱ𝔑𝛿𝛽𝑖𝑛𝑡(𝑆) )) is defined by 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑆)  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝑆), 𝒫ℱ𝔑𝛿𝛼𝑖𝑛𝑡(𝑆)  and 𝒫ℱ𝔑𝛿𝛽𝑖𝑛𝑡(𝑆)) =∪ {𝐼: 𝐼 ⊆ 𝑆 & 𝐼isa𝒫ℱ𝔑𝛿𝒫𝑜  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑜, 𝒫ℱ𝔑𝛿𝛼𝑜 & 𝒫ℱ𝔑𝛿𝛽𝑜) set in𝑈}. 

    2.  𝛿 pre (resp. 𝛿 semi, 𝛿𝛼 and 𝛿𝛽) closure of 𝑆 (briefly, 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑆) (resp. 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝑆), 

𝒫ℱ𝔑𝛿𝛼𝑐𝑙(𝑆) and 𝒫ℱ𝔑𝛿𝛽𝑐𝑙(𝑆))) is defined by 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑆) (resp. 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝑆), 𝒫ℱ𝔑𝛿𝛼𝑐𝑙(𝑆) 

and 𝒫ℱ𝔑𝛿𝛽𝑐𝑙(𝑆)) =∩ {𝐴: 𝑆 ⊆ 𝐴 & 𝐴isa𝒫ℱ𝔑𝛿𝒫𝑐 (resp. 𝒫ℱ𝔑𝛿𝒮𝑐, 𝒫ℱ𝔑𝛿𝛼𝑐 & 𝒫ℱ𝔑𝛿𝛽𝑐)  set in  

𝑈}.  

 Definition 3.7   Let (𝑈1, 𝜏𝑃(𝐴1))  and (𝑈2, 𝜏𝑃(𝐴2))  be two 𝑃𝐹𝒩𝑡𝑠 ’s. Then a function 

ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑝(𝐴2)) is said to be a Pythagorean fuzzy nano 𝛿 (resp. 𝛿 pre, 𝛿 semi, 𝛿𝛼 

and 𝛿𝛽 ) continuous (briefly, 𝒫ℱ𝒩𝛿𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝛿𝒫𝐶𝑡𝑠 , 𝒫ℱ𝒩𝛿𝒮𝐶𝑡𝑠 , 𝒫ℱ𝒩𝛿𝛼𝐶𝑡𝑠  and 

𝒫ℱ𝒩𝛿𝛽𝐶𝑡𝑠 )) function if ℎ𝑃
−1(𝐺)  is 𝒫ℱ𝒩𝛿𝑜  (resp. 𝒫ℱ𝒩𝛿𝒫𝑜 , 𝒫ℱ𝒩𝛿𝒮𝑜 , 𝒫ℱ𝒩𝛿𝛼𝑜  & 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1932 
https://internationalpubls.com 

𝒫ℱ𝒩𝛿𝛽𝑜) set in 𝑈1 for all 𝒫ℱ𝒩𝑜 set 𝐺 in 𝑈2.  

Definition 3.8  Let (𝑈1, 𝜏𝑃(𝐴1))  and (𝑈2, 𝜏𝑃(𝐴2))  be two 𝑃𝐹𝒩𝑡𝑠 ’s. Then a function 

ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑝(𝐴2)) is said to be a Pythagorean fuzzy nano contra 𝛿 (resp. 𝛿 pre, 𝛿 

semi, 𝛿𝛼  and 𝛿𝛽 ) continuous (briefly, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 , 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠)) function if ℎ𝑃
−1(𝐺) is 𝒫ℱ𝒩𝛿𝑐 

(resp. 𝒫ℱ𝒩𝛿𝒫𝑐, 𝒫ℱ𝒩𝛿𝒮𝑐, 𝒫ℱ𝒩𝛿𝛼𝑐 & 𝒫ℱ𝒩𝛿𝛽𝑐) set in 𝑈1 for all 𝒫ℱ𝒩𝑜 set 𝐺 in 𝑈2.  

Lemma 3.1   Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be a function. Then the following statements 

hold. 

    1.  If 𝑆 and 𝑇 are 𝑝𝑓𝑠’s of 𝑈1 such that 𝑆 ⊆ 𝑇, then ℎ𝑃(𝑆) ⊆ ℎ𝑃(𝑇).  

    2.  If 𝑆 and 𝑇 are 𝑝𝑓𝑠’s of 𝑈2 such that 𝑆 ⊆ 𝑇, then ℎ𝑃
−1(𝑆) ⊆ ℎ𝑃

−1(𝑇).  

 Lemma 3.2   Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be a function. If 𝑆 is a 𝑝𝑓𝑠 of 𝑈1 and 𝑇 is a 

𝑝𝑓𝑠 of 𝑈2. Then  

    1.  ℎ𝑃(ℎ𝑃
−1(𝑆)) ⊆ 𝑆  

    2.  ℎ𝑃(ℎ𝑃
−1(𝑆)) = 𝑆 ⇔ ℎ𝑃 is surjective.  

    3.  ℎ𝑃
−1(ℎ𝑃(𝑆)) ⊇ 𝑆  

    4.  ℎ𝑃
−1(ℎ𝑃(𝑆)) = 𝑆 whenever ℎ𝑃 is injective.   

Theorem 3.1   Let (𝑈1, 𝜏𝑃(𝐴1)) and (𝑈2, 𝜏𝑃(𝐴2)) be two 𝑃𝑁𝒩𝑡𝑠’s and let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) →

(𝑈2, 𝜏𝑃(𝐴2)), then  

    (i)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠.  

    (ii)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠.  

    (iii)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠.  

    (iV)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠.  

    (V)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠.  

    (Vi)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠.  

    (Vii)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠.  

 But not converse.  

  Proof. (i) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠. Let 𝑆 be a 𝒫ℱ𝒩𝑜 set in 

(𝑈2, 𝜏𝑃(𝐴2)). Then ℎ𝑃
−1(𝑆) is 𝒫ℱ𝒩𝛿𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Since every 𝒫ℱ𝒩𝛿𝑐 set is 𝒫ℱ𝒩𝑐𝑠, 

ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 function. 

(ii) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 . Let 𝑆  be a 𝒫ℱ𝒩𝑜  set in 

(𝑈2, 𝜏𝑃(𝐴2)). Then ℎ𝑃
−1(𝑆) is 𝒫ℱ𝒩𝛿𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Since every 𝒫ℱ𝒩𝛿𝑐 set is 𝒫ℱ𝒩𝛿𝒫𝑐𝑠, 

ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝛿𝒫𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 function. 

(iii) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 . Let 𝑆  be a 𝒫ℱ𝒩𝑜  set in 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1933 
https://internationalpubls.com 

(𝑈2, 𝜏𝑃(𝐴2)). Then ℎ𝑃
−1(𝑆) is 𝒫ℱ𝒩𝛿𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Since every 𝒫ℱ𝒩𝛿𝑐 set is 𝒫ℱ𝒩𝛿𝒮𝑐𝑠, 

ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝛿𝒮𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 function. 

(iv) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 . Let 𝑆  be a 𝒫ℱ𝒩𝑜  set in 

(𝑈2, 𝜏𝑃(𝐴2)) . Then ℎ𝑃
−1(𝑆)  is 𝒫ℱ𝒩𝛿𝒮𝑐  set in (𝑈1, 𝜏𝑃(𝐴1)) . Since every 𝒫ℱ𝒩𝛿𝒮𝑐  set is 

𝒫ℱ𝒩𝛿𝛽𝑐𝑠, ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝛿𝛽𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 function. 

(v) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 . Let 𝑆  be a 𝒫ℱ𝒩𝑐  set in 

(𝑈2, 𝜏𝑃(𝐴2)) . Then ℎ𝑃
−1(𝑆)  is 𝒫ℱ𝒩𝛿𝒫𝑐  set in (𝑈1, 𝜏𝑃(𝐴1)) . Since every 𝒫ℱ𝒩𝛿𝒫𝑐  set is 

𝒫ℱ𝒩𝛿𝛽𝑐𝑠, ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝛿𝛽𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 function. 

(vi) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 . Let 𝑆  be a 𝒫ℱ𝒩𝑜  set in 

(𝑈2, 𝜏𝑃(𝐴2)) . Then ℎ𝑃
−1(𝑆)  is 𝒫ℱ𝒩𝛿𝛼𝑐  set in (𝑈1, 𝜏𝑃(𝐴1)) . Since every 𝒫ℱ𝒩𝛿𝛼𝑐  set is 

𝒫ℱ𝒩𝛿𝒫𝑐𝑠, ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝛿𝒫𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 function. 

(vii) Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 . Let 𝑆  be a 𝒫ℱ𝒩𝑜  set in 

(𝑈2, 𝜏𝑃(𝐴2)) . Then ℎ𝑃
−1(𝑆)  is 𝒫ℱ𝒩𝛿𝛼𝑐  set in (𝑈1, 𝜏𝑃(𝐴1)) . Since every 𝒫ℱ𝒩𝛿𝛼𝑐  set is 

𝒫ℱ𝒩𝛿𝒮𝑐𝑠, ℎ𝑃
−1(𝑆) is 𝑝𝑓𝒩𝛿𝒮𝑐 set in (𝑈1, 𝜏𝑃(𝐴1)). Hence ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 function.              

Remark 3.1  The following Figure shows the relations among the different types of Pythagorean 

fuzzy 𝛿 continuous mappings that were studied in this section. 

 

Figure : 𝓟𝓕𝓝𝒄𝒐𝒏𝒕𝒓𝒂𝜹𝑪𝒕𝒔 mappings in 𝓟𝓕𝓝𝒄𝒐𝒏𝒕𝒓𝒂𝑪𝒕𝒔 

Example 3.1   Assume 𝑈1 = 𝑈2 = 𝑈 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}  be the universe set and the equivalence 

relation is 𝑈/𝑅 = {{𝑠1, 𝑠4}, {𝑠2}, {𝑠3}}.  

Let 𝐴 = {⟨
𝑠1

0.3,0.1
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩ , ⟨

𝑠4

0.4,0.25
⟩} be a Pythagorean fuzzy subset of 𝑈.  

 𝒫ℱ𝔑(𝐴) = {⟨
𝑠1,𝑠4

0.3,0.25
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}, 

 𝒫ℱ𝔑(𝐴) = {⟨
𝑠1,𝑠4

0.4,0.1
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴) = {⟨
𝑠1,𝑠4

0.25,0.3
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}. 

Now 𝜏𝑃(𝐴1) = {0𝒫 , 1𝒫 , 𝒫ℱ𝔑(𝐴), 𝒫ℱ𝔑(𝐴), 𝐵𝒫ℱ𝔑(𝐴)}  and 𝜏𝑃(𝐴2) = {0𝒫 , 1𝒫 , (𝒫ℱ𝔑(𝐴))𝑐,



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1934 
https://internationalpubls.com 

(𝒫ℱ𝔑(𝐴))𝑐, (𝐵𝒫ℱ𝔑(𝐴))𝑐}. Let ℎ𝑃: (𝑈, 𝜏𝑃(𝐴1)) → (𝑈, 𝜏𝑃(𝐴2)) be an identity function, Then ℎ𝑃 is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠  but not 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 . Since, 𝒫ℱ𝔑(𝐴)  is a 𝒫ℱ𝒩𝑜  set in 𝑈2  but 

ℎ𝑃
−1(𝒫ℱ𝔑(𝐴)) = 𝒫ℱ𝔑(𝐴) is not 𝒫ℱ𝒩𝛿𝑐 set in 𝑈1. 

Example 3.2   Let 𝑈1 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}  , 𝑈2 = {𝑡1, 𝑡2, 𝑡3, 𝑡4}  are the universe sets and the 

equivalence relations are 𝑈1/𝑅 = {{𝑠1, 𝑠3}, {𝑠2, 𝑠4}}  and 𝑈2/𝑅 = {{𝑡1, 𝑡3}, {𝑡2, 𝑡4}} . Let 𝐴1 =

{⟨
𝑠1

0.4,0.8
⟩ , ⟨

𝑠2

0.5,0.4
⟩ , ⟨

𝑠3

0.6,0.6
⟩ , ⟨

𝑠4

0.7,0.6
⟩}  and 𝐴2 = {⟨

𝑡1

0.4,0.8
⟩ , ⟨

𝑡2

0.5,0.7
⟩ , ⟨

𝑡3

0.4,0.6
⟩ , ⟨

𝑡4

0.3,0.7
⟩}  be a Pythagorean 

fuzzy subsets of 𝑈1 and 𝑈2 respectively.  

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.8
⟩ , ⟨

𝑠2,𝑠4

0.5,0.6
⟩} 

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.6,0.6
⟩ , ⟨

𝑠2,𝑠4

0.7,0.4
⟩} 

 𝐵𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.6,0.6
⟩ , ⟨

𝑠2,𝑠4

0.6,0.5
⟩} 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.4,0.7
⟩ , ⟨

𝑡2,𝑡4

0.3,0.7
⟩} 

 𝐵𝒫ℱ𝔑(𝐴2) = 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.4,0.6
⟩ , ⟨

𝑡2,𝑡4

0.5,0.7
⟩}. 

Now 𝜏𝑃(𝐴1) = {0𝒫 , 1𝒫 ,  𝒫ℱ𝔑(𝐴1), 𝒫ℱ𝔑(𝐴1), 𝐵𝒫ℱ𝔑(𝐴1)} , 𝜏𝑃(𝐴2) = {0𝒫 ,  1𝒫 , 𝒫ℱ𝔑(𝐴2),

𝒫ℱ𝔑(𝐴2) = 𝐵𝒫ℱ𝔑(𝐴2)}. Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be an identity function, then ℎ𝑃  is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠  and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 ) but not 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠  and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 ). Since, 𝒫ℱ𝔑(𝐴2)  is a 

𝒫ℱ𝒩𝑜  set in 𝑈2  but ℎ𝑃
−1(𝒫ℱ𝔑(𝐴2)) = 𝒫ℱ𝔑(𝐴2)  is not 𝒫ℱ𝒩𝛿𝑐  (resp. 𝒫ℱ𝒩𝛿𝒮𝑐  and 

𝒫ℱ𝒩𝛿𝛼𝑐) set in 𝑈1. 

Example 3.3   Let 𝑈1 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}  , 𝑈2 = {𝑡1, 𝑡2, 𝑡3, 𝑡4}  are the universe sets and the 

equivalence relations are 𝑈1/𝑅 = {{𝑠1, 𝑠4}, {𝑠2}, {𝑠3}}  and 𝑈2/𝑅 = {{𝑡1, 𝑡4}, {𝑡2}, {𝑡3}} . Let 𝐴1 =

{⟨
𝑠1

0.3,0.7
⟩ , ⟨

𝑠2

0.1,0.6
⟩ , ⟨

𝑠3

0.4,0.6
⟩ , ⟨

𝑠4

0.4,0.6
⟩}  and 𝐴2 = {⟨

𝑡1

0.6,0.4
⟩ , ⟨

𝑡2

0.6,0.2
⟩ , ⟨

𝑡3

0.6,0.4
⟩ , ⟨

𝑡4

0.6,0.4
⟩}  be a Pythagorean 

fuzzy subsets of 𝑈1 and 𝑈2 respectively.  

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠4

0.3,0.7
⟩ , ⟨

𝑠2

0.1,0.6
⟩ , ⟨

𝑠3

0.4,0.6
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴1) = 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠4

0.4,0.6
⟩ , ⟨

𝑠2

0.1,0.6
⟩ , ⟨

𝑠3

0.4,0.6
⟩}, 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡4

0.6,0.4
⟩ , ⟨

𝑡2

0.6,0.2
⟩ , ⟨

𝑡3

0.6,0.4
⟩}, 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡4

0.6,0.4
⟩ , ⟨

𝑡2

0.6,0.2
⟩ , ⟨

𝑡3

0.6,0.4
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡4

0.4,0.6
⟩ , ⟨

𝑡2

0.2,0.6
⟩ , ⟨

𝑡3

0.4,0.6
⟩}. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1935 
https://internationalpubls.com 

Now 𝜏𝑃(𝐴1) = {0𝒫 ,  1𝒫 , 𝒫ℱ𝔑(𝐴1), 𝒫ℱ𝔑(𝐴1) = 𝐵𝒫ℱ𝔑(𝐴1)} , 𝜏𝑃(𝐴2) = {0𝒫 ,  1𝒫 , 𝒫ℱ𝔑(𝐴2),

𝒫ℱ𝔑(𝐴2), 𝐵𝒫ℱ𝔑(𝐴2)} . Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be an identity function, then ℎ𝑃  is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 ) but not 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠  (resp. 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠). Since, 𝒫ℱ𝔑(𝐴2) is a 𝒫ℱ𝒩𝑜 set in 𝑈2 but ℎ𝑃
−1(𝒫ℱ𝔑(𝐴2)) = 𝒫ℱ𝔑(𝐴2) is 

not 𝒫ℱ𝒩𝛿𝑐 (resp. 𝒫ℱ𝒩𝛿𝛼𝑐) set in 𝑈1. 

Example 3.4  Let 𝑈1 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}, 𝑈2 = {𝑡1, 𝑡2, 𝑡3, 𝑡4} are the universe sets and the equivalence 

relations are 𝑈1/𝑅 = {{𝑠1, 𝑠3}, {𝑠2, 𝑠4}}  and 𝑈2/𝑅 = {{𝑡1, 𝑡3}, {𝑡2, 𝑡4}} . Let 𝐴1 =

{⟨
𝑠1

0.4,0.7
⟩ , ⟨

𝑠2

0.5,0.7
⟩ , ⟨

𝑠3

0.4,0.6
⟩ , ⟨

𝑠4

0.3,0.7
⟩}  and 𝐴2 = {⟨

𝑡1

0.4,0.8
⟩ , ⟨

𝑡2

0.5,0.4
⟩ , ⟨

𝑡3

0.6,0.6
⟩ , ⟨

𝑡4

0.7,0.6
⟩}  be a Pythagorean 

fuzzy subsets of 𝑈1 and 𝑈2 respectively.  

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.7
⟩ , ⟨

𝑠2,𝑠4

0.3,0.7
⟩} 

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.6
⟩ , ⟨

𝑠2,𝑠4

0.5,0.7
⟩} 

 𝐵𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.6
⟩ , ⟨

𝑠2,𝑠4

0.5,0.7
⟩} 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.4,0.8
⟩ , ⟨

𝑡2,𝑡4

0.5,0.6
⟩} 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.6,0.6
⟩ , ⟨

𝑡2,𝑡4

0.7,0.4
⟩} 

 𝐵𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.6,0.6
⟩ , ⟨

𝑡2,𝑡4

0.6,0.5
⟩}. 

Now 𝜏𝑃(𝐴1) = {0𝒫 , 1𝒫 , 𝒫ℱ𝔑(𝐴1), 𝒫ℱ𝔑(𝐴1),  𝐵𝒫ℱ𝔑(𝐴1)} , 𝜏𝑃(𝐴2) = {0𝒫 ,  1𝒫 , 𝒫ℱ𝔑(𝐴2),

𝒫ℱ𝔑(𝐴2),  𝐵𝒫ℱ𝔑(𝐴2)} . Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be an identity function, then ℎ𝑃  is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠  but not 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 . Since, 𝒫ℱ𝔑(𝐴2)  is a 𝒫ℱ𝒩𝑜  set in 𝑈2  but 

ℎ𝑃
−1(𝒫ℱ𝔑(𝐴2)) = 𝒫ℱ𝔑(𝐴2) is not 𝒫ℱ𝒩𝛿𝒫𝑐 set in 𝑈1. 

Theorem 3.2  Let (𝑈1, 𝜏𝑃(𝐴1)) & (𝑈2, 𝜏𝑃(𝐴2)) be a 𝒫ℱ𝒩𝑡𝑠’s. A mapping ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) →

(𝑈2, 𝜏𝑃(𝐴2)) satisfies the following conditions are equivalent.  

    (i)  ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠;  

    (ii)  The inverse ℎ𝑃
−1(𝐾) of all 𝒫ℱ𝒩𝑐𝑠 set 𝐾 in 𝑈2 is 𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑈1.  

  Proof. (i) →  (ii): Consider a 𝒫ℱ𝒩𝑐𝑠  𝐾  in 𝑈2 . Then 𝐾𝑐  is 𝒫ℱ𝒩𝑜𝑠  in 𝑈2 . As ℎ𝑃  is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 , ℎ𝑃
−1(𝐾𝑐)  is 𝒫ℱ𝒩𝛿𝛽𝑐𝑠  in 𝑈1 . As ℎ𝑃

−1(𝐾𝑐) = (ℎ𝑃
−1(𝐾))𝑐 , ℎ𝑃

−1(𝐾)  is a 

𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑈1. 

(ii) → (i): Consider a 𝒫ℱ𝒩𝑐𝑠 𝐾 in 𝑈2. So 𝐾𝑐 is a 𝒫ℱ𝒩𝑜𝑠 in 𝑈2. By presumption, ℎ𝑃
−1(𝐾𝑐) is 

𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1. As ℎ𝑃
−1(𝐾𝑐) = (ℎ𝑃

−1𝐾)𝑐, (ℎ𝑃
−1(𝐾))𝑐 is a 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1. Hence ℎ𝑃

−1(𝐾) is a 

𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑈1. Thus ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠.              

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1936 
https://internationalpubls.com 

Theorem 3.3  Let (𝑈1, 𝜏𝑃(𝐴1)) & (𝑈2, 𝜏𝑃(𝐴2)) be a 𝒫ℱ𝒩𝑡𝑠’s. A mapping ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) →

(𝑈2, 𝜏𝑃(𝐴2)) is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 satisfies the following conditions are hold.   

    (i)  ℎ𝑃(𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝐿)) ⊇ 𝒫ℱ𝒩𝛿𝑖𝑛𝑡(ℎ𝑃(𝐿)), for all 𝑝𝑓𝑠 𝐿 in 𝑈1.  

    (ii)  𝒫ℱ𝒩𝛿𝛽𝑐𝑙(ℎ𝑃
−1(𝐾)) ⊇ ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝐾)), for all 𝑝𝑓𝑠 𝐾 in 𝑈2.  

   Proof. (i) Since 𝒫ℱ𝒩𝛿𝑐𝑙(ℎ𝑃(𝐿))  is a 𝒫ℱ𝒩𝛿𝑐𝑠  in 𝑈2  and ℎ𝑃  is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 , then 

ℎ𝑃
−1(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(  ℎ𝑃(𝐿)))  is 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 𝑈1 . Now, since 𝐿 ⊇ ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(ℎ𝑃(𝐿))) , 

𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝐿)  ⊇ ℎ𝑃
−1(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(ℎ𝑃(𝐿))). Therefore, ℎ𝑃(𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝐿)) ⊇  𝒫ℱ𝒩𝛿𝑖𝑛𝑡(ℎ𝑃(𝐿)). 

(ii) By replacing 𝐿 with 𝐾 in (i),  

we obtain ℎ𝑃 (𝒫ℱ𝒩𝛿𝛽𝑐𝑙(ℎ𝑃
−1(𝐾)))  ⊇  𝒫ℱ𝒩𝛿𝑖𝑛𝑡(ℎ𝑃(ℎ𝑃

−1(𝐾))) ⊇ 𝒫ℱ𝒩𝛿𝑖𝑛(𝐾) . Hence, 

𝒫ℱ𝒩𝛿𝛽𝑐𝑙(ℎ𝑃
−1(𝐾)) ⊇ ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝐾)).              

Remark 3.2  Let (𝑈1, 𝜏𝑃(𝐴1))  &  (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑡𝑠 ’s. Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) →

(𝑈2, 𝜏𝑃(𝐴2)) be a mapping. If ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠, then  

    1.  ℎ𝑃(𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝐿)) is not necessarily equal to 𝒫ℱ𝒩𝛿𝑖𝑛𝑡(ℎ𝑃(𝐿)) where 𝐿 ∈ 𝑈1.  

    2.  𝒫ℱ𝒩𝛿𝛽𝑐𝑙(ℎ𝑃
−1(𝐾)) is not necessarily equal to ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝐾)) where 𝐾 ∈ 𝑈2.  

Example 3.5 Assume 𝑈1 = 𝑈2 = 𝑈 = {𝑠1, 𝑠2, 𝑠3, 𝑠4} be the universe set and the equivalence relation 

is 𝑈/𝑅 = {{𝑠1, 𝑠4}, {𝑠2}, {𝑠3}}.  

Let 𝐴 = {⟨
𝑠1

0.3,0.1
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩ , ⟨

𝑠4

0.4,0.25
⟩} be a Pythagorean fuzzy subset of 𝑈.  

 𝒫ℱ𝔑(𝐴) = {⟨
𝑠1,𝑠4

0.3,0.25
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}, 

 𝒫ℱ𝔑(𝐴) = {⟨
𝑠1,𝑠4

0.4,0.1
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴) = {⟨
𝑠1,𝑠4

0.25,0.3
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}. 

Now 𝜏𝑃(𝐴1) = 𝜏𝑃(𝐴2) = 𝜏𝑃(𝐴) = {0𝒫 , 1𝒫 , 𝒫ℱ𝔑(𝐴), 𝒫ℱ𝔑(𝐴), 𝐵𝒫ℱ𝔑(𝐴)}  is a 𝒫ℱ𝒩𝑡𝑠 . Let 

ℎ𝑃: (𝑈, 𝜏𝑃(𝐴1)) → (𝑈, 𝜏𝑃(𝐴2)) be an identity function, then ℎ𝑃 is 𝒫ℱ𝒩𝛿𝛽𝐶𝑡𝑠.     

     1.  ℎ𝑃(𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝒫ℱ𝔑(𝐴))) = 𝐵𝒫ℱ𝔑(𝐴). But 𝒫ℱ𝒩𝛿𝑐𝑙 (ℎ𝑃 (𝒫ℱ𝔑(𝐴))) = 𝐵𝒫ℱ𝔑(𝐴)𝑐. Thus 

ℎ𝑃(𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝒫ℱ𝔑(𝐴))) ≠ 𝒫ℱ𝒩𝛿𝑐𝑙(ℎ𝑃(𝒫ℱ𝔑(𝐴))).  

    2.  𝒫ℱ𝒩𝛿𝛽𝑐𝑙(ℎ𝑃
−1(𝒫ℱ𝔑(𝐴))) = 𝐵𝒫ℱ𝔑(𝐴). But ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝑐𝑙(𝒫ℱ𝔑(𝐴))) = 𝐵𝒫ℱ𝔑(𝐴)𝑐 . Thus 

𝒫ℱ𝒩𝛿𝛽𝑐𝑙(ℎ𝑃
−1(𝒫ℱ𝔑(𝐴))) ≠ ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝑐𝑙(𝒫ℱ𝔑(𝐴))).  

4  Pythagorean fuzzy nano contra 𝜹 (resp. 𝜹 pre, 𝜹 semi, 𝜹𝜶 and 𝜹𝜷)-irresolute maps 

 In this section, we introduce the concept of Pythagorean fuzzy nano contra irresoluteness called 

Pythagorean fuzzy nano contra (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 𝛿𝛼 and 𝛿𝛽)-irresolute maps by using 𝒫ℱ𝒩𝒮𝑜𝑠 

(resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝑜𝑠 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝑜𝑠, 𝒫ℱ𝒩  𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝑜𝑠, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝑜𝑠  and 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1937 
https://internationalpubls.com 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝑜𝑠)’s and study some of their basic properties. This definition enables us to obtain 

conditions under which maps and inverse maps preserve respective open sets. 

Definition 4.1  A map ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) is said to be Pythagorean fuzzy nano contra 

(resp. 𝛿 , 𝛿𝒫, 𝛿𝒮, 𝛿𝛼  and 𝛿𝛽) -irresolute (in short, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟 , 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟,   𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟 )) map if 

ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝒮𝑐𝑠 (resp. 𝒫ℱ𝒩𝛿𝑐𝑠, 𝒫ℱ𝒩𝛿𝒫𝑐𝑠, 𝒫ℱ𝒩𝛿𝒮𝑐𝑠, 𝒫ℱ𝒩𝛿𝛼𝑐𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑐𝑠) in 

(𝑈1, 𝜏𝑃(𝐴1))  for each 𝒫ℱ𝒩𝒮𝑜𝑠  (resp. 𝒫ℱ𝒩𝛿𝑜𝑠 ,  𝒫ℱ𝒩𝛿𝒫𝑜𝑠, 𝒫ℱ𝒩𝛿𝒮𝑜𝑠, 𝒫ℱ𝒩𝛿𝛼𝑜𝑠  and 

𝒫ℱ𝒩𝛿𝛽𝑜𝑠) 𝐾 of (𝑈2, 𝜏𝑃(𝐴2)).  

Theorem 4.1  Let (𝑈1, 𝜏𝑃(𝐴1))  &  (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑡𝑠 ’s. Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) →

(𝑈2, 𝜏𝑃(𝐴2)) be a mapping. Then the following statements are hold for 𝒫ℱ𝒩𝑡𝑠, but not conversely.  

    (i)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟 map is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝒮𝐶𝑡𝑠.  

    (ii)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟 map is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠.  

    (iii)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 map is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠.  

    (iv)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟 map is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠.  

    (V)  Every 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟 map is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠. 

   Proof. (i) Consider a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟 map ℎ𝑃 and a 𝒫ℱ𝒩𝑜𝑠 𝐾 in 𝑈2. As each 𝒫ℱ𝒩𝑜𝑠 is a 

𝒫ℱ𝒩𝒮𝑜𝑠, 𝐾 is a 𝒫ℱ𝒩𝒮𝑜𝑠 in 𝑈2 . By presumption, ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝒮𝑐𝑠 in 𝑈1. Thus ℎ𝑃  is a 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝒮𝐶𝑡𝑠 map. 

(ii) Consider a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟  map ℎ𝑃  and a 𝒫ℱ𝒩𝛿𝑜𝑠  𝐾  in 𝑈2 . As each 𝒫ℱ𝒩𝛿𝑜𝑠  is a 

𝒫ℱ𝒩𝑜𝑠 and 𝒫ℱ𝒩𝛿𝒮𝑜𝑠, 𝐾 is a 𝒫ℱ𝒩𝛿𝑜𝑠 and 𝒫ℱ𝒩𝛿𝒮𝑜𝑠 in 𝑈2 . By presumption, ℎ𝑃
−1(𝐾) is a 

𝒫ℱ𝒩𝛿𝒮𝑐𝑠 in 𝑈1. Thus ℎ𝑃 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 map. 

(iii) Consider a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 map ℎ𝑃  and a 𝒫ℱ𝒩𝛿𝑜𝑠  𝐾  in 𝑈2 . As each 𝒫ℱ𝒩𝛿𝑜𝑠  is a 

𝒫ℱ𝒩𝑜𝑠 and 𝒫ℱ𝒩𝛿𝒫𝑜𝑠, 𝐾 is a 𝒫ℱ𝒩𝛿𝑜𝑠 and 𝒫ℱ𝒩𝛿𝒫𝑜𝑠 in 𝑈2. By presumption, ℎ𝑃
−1(𝐾) is a 

𝒫ℱ𝒩𝛿𝒫𝑐𝑠 in 𝑈1. Thus ℎ𝑃 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 map. 

(iv) Consider a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  map ℎ𝑃  and a 𝒫ℱ𝒩𝛿𝑜𝑠  𝐾  in 𝑈2 . As each 𝒫ℱ𝒩𝛿𝑜𝑠  is a 

𝒫ℱ𝒩𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛼𝑜𝑠, 𝐾 is a 𝒫ℱ𝒩𝛿𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛼𝑜𝑠 in 𝑈2 . By presumption, ℎ𝑃
−1(𝐾) is a 

𝒫ℱ𝒩𝛿𝛼𝑐𝑠 in 𝑈1. Thus ℎ𝑃 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 map. 

(v) Consider a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟  map ℎ𝑃  and a 𝒫ℱ𝒩𝛿𝑜𝑠  𝐾  in 𝑈2 . As each 𝒫ℱ𝒩𝛿𝑜𝑠  is a 

𝒫ℱ𝒩𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑜𝑠, 𝐾 is a 𝒫ℱ𝒩𝛿𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑈2 . By presumption, ℎ𝑃
−1(𝐾) is a 

𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1. Thus ℎ𝑃 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 map.              

Example 4.1  Let 𝑈1 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}, 𝑈2 = {𝑡1, 𝑡2, 𝑡3, 𝑡4} are the universe sets and the equivalence 

relations are 𝑈1/𝑅 = {{𝑠1, 𝑠3}, {𝑠2, 𝑠4}}  and 𝑈2/𝑅 = {{𝑡1, 𝑡3}, {𝑡2, 𝑡4}} . Let 𝐴1 =

{⟨
𝑠1

0.1,0.8
⟩ , ⟨

𝑠2

0.3,0.7
⟩ , ⟨

𝑠3

0.2,0.9
⟩ , ⟨

𝑠4

0.4,0.6
⟩}  and 𝐴2 = {⟨

𝑡1

0.4,0.8
⟩ , ⟨

𝑡2

0.5,0.4
⟩ , ⟨

𝑡3

0.6,0.6
⟩ , ⟨

𝑡4

0.7,0.6
⟩}  be a Pythagorean 

fuzzy subsets of 𝑈1 and 𝑈2 respectively.  

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.1,0.9
⟩ , ⟨

𝑠2,𝑠4

0.3,0.7
⟩}, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1938 
https://internationalpubls.com 

 𝐵𝒫ℱ𝔑(𝐴1) = 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.2,0.8
⟩ , ⟨

𝑠2,𝑠4

0.4,0.6
⟩}, 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.4,0.8
⟩ , ⟨

𝑡2,𝑡4

0.5,0.6
⟩}, 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.6,0.6
⟩ , ⟨

𝑡2,𝑡4

0.7,0.4
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.6,0.6
⟩ , ⟨

𝑡2,𝑡4

0.6,0.5
⟩}. 

Here 𝜏𝑝(𝐴1) = {0𝑃,  1𝑃, 𝒫ℱ𝔑(𝐴1), 𝒫ℱ𝔑(𝐴1) = 𝐵𝒫ℱ𝔑(𝐴1)}  and 𝜏𝑝(𝐴2) = {0𝑃, 1𝑃, 𝒫ℱ𝔑(𝐴2),

𝒫ℱ𝔑(𝐴2),  𝐵𝒫ℱ𝔑(𝐴2)} are the 𝒫ℱ𝒩𝑡𝑠′𝑠 on 𝑈1 and 𝑈2 respectively. 

Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be an identity function, then ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝒮𝐶𝑡𝑠 but not 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟 , because the set (𝒫ℱ𝔑(𝐴2))𝑐  is a 𝒫ℱ𝒩𝒮𝑜𝑠  in 𝑈2  but ℎ𝑃
−1((𝒫ℱ𝔑(𝐴2))𝑐) =

(𝒫ℱ𝔑(𝐴2))𝑐 is not 𝒫ℱ𝒩𝒮𝑐𝑠 in 𝑈1.  

Example 4.2  Let 𝑈1 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}, 𝑈2 = {𝑡1, 𝑡2, 𝑡3, 𝑡4} are the universe sets and the equivalence 

relations are 𝑈1/𝑅 = {{𝑠1, 𝑠3}, {𝑠2, 𝑠4}}  and 𝑈2/𝑅 = {{𝑡1, 𝑡3}, {𝑡2, 𝑡4}} . Let 𝐴1 =

{⟨
𝑠1

0.4,0.7
⟩ , ⟨

𝑠2

0.5,0.7
⟩ , ⟨

𝑠3

0.4,0.6
⟩ , ⟨

𝑠4

0.3,0.7
⟩}  and 𝐴2 = {⟨

𝑡1

0.4,0.8
⟩ , ⟨

𝑡2

0.5,0.4
⟩ , ⟨

𝑡3

0.6,0.6
⟩ , ⟨

𝑡4

0.7,0.6
⟩}  be a Pythagorean 

fuzzy subsets of 𝑈1 and 𝑈2 respectively.  

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.7
⟩ , ⟨

𝑠2,𝑠4

0.3,0.7
⟩} 

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.6
⟩ , ⟨

𝑠2,𝑠4

0.5,0.7
⟩} 

 𝐵𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠3

0.4,0.6
⟩ , ⟨

𝑠2,𝑠4

0.5,0.7
⟩} 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.4,0.8
⟩ , ⟨

𝑡2,𝑡4

0.5,0.6
⟩} 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.6,0.6
⟩ , ⟨

𝑡2,𝑡4

0.7,0.4
⟩} 

 𝐵𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡3

0.6,0.6
⟩ , ⟨

𝑡2,𝑡4

0.6,0.5
⟩} 

Now 𝜏𝑃(𝐴1) = {0𝒫 ,  1𝒫 , 𝒫ℱ𝔑(𝐴1), 𝒫ℱ𝔑(𝐴1),  𝐵𝒫ℱ𝔑(𝐴1)} , 𝜏𝑃(𝐴2) =

{0𝒫 ,  1𝒫 ,  𝒫ℱ𝔑(𝐴2),  𝒫ℱ𝔑(𝐴2),  𝐵𝒫ℱ𝔑(𝐴2)} . Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be an identity 

function, then ℎ𝑃  is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠  but not 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 . Since, (𝒫ℱ𝔑(𝐴2))𝑐  is a 

𝒫ℱ𝒩𝛿𝒫𝑜 set in 𝑈2 but ℎ𝑃
−1((𝒫ℱ𝔑(𝐴2))𝑐) = (𝒫ℱ𝔑(𝐴2))𝑐 is not 𝒫ℱ𝒩𝛿𝒫𝑐 set in 𝑈1. 

Example 4.3  Let 𝑈1 = {𝑠1, 𝑠2, 𝑠3, 𝑠4}, 𝑈2 = {𝑡1, 𝑡2, 𝑡3, 𝑡4} are the universe sets and the equivalence 

relations are 𝑈1/𝑅 = {{𝑠1, 𝑠4}, {𝑠2}, {𝑠3}}  and 𝑈2/𝑅 = {{𝑡1, 𝑡4}, {𝑡2}, {𝑡3}}  . Let 𝐴1 =

{⟨
𝑠1

0.4,0.3
⟩ , ⟨

𝑠2

0.4,0.2
⟩ , ⟨

𝑠3

0.5,0.5
⟩ , ⟨

𝑠4

0.5,0.2
⟩} and 𝐴2 = {⟨

𝑡1

0.3,0.1
⟩ , ⟨

𝑡2

0.1,0.5
⟩ , ⟨

𝑡3

0.2,0.45
⟩ , ⟨

𝑡4

0.4,0.25
⟩} be a Pythagorean 

fuzzy subsets of 𝑈1 and 𝑈2 respectively.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1939 
https://internationalpubls.com 

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠4

0.4,0.3
⟩ , ⟨

𝑠2

0.4,0.2
⟩ , ⟨

𝑠3

0.5,0.3
⟩}, 

 𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠4

0.5,0.2
⟩ , ⟨

𝑠2

0.4,0.2
⟩ , ⟨

𝑠3

0.5,0.3
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴1) = {⟨
𝑠1,𝑠4

0.3,0.4
⟩ , ⟨

𝑠2

0.2,0.4
⟩ , ⟨

𝑠3

0.3,0.5
⟩}, 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡4

0.3,0.25
⟩ , ⟨

𝑡2

0.1,0.5
⟩ , ⟨

𝑡3

0.2,0.45
⟩}, 

 𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡4

0.4,0.1
⟩ , ⟨

𝑡2

0.1,0.5
⟩ , ⟨

𝑡3

0.2,0.45
⟩}, 

 𝐵𝒫ℱ𝔑(𝐴2) = {⟨
𝑡1,𝑡4

0.25,0.3
⟩ , ⟨

𝑡2

0.1,0.5
⟩ , ⟨

𝑡3

0.2,0.45
⟩}. 

Here 𝜏𝑝(𝐴1) = {0𝑃,  1𝑃, 𝒫ℱ𝔑(𝐴1), 𝒫ℱ𝔑(𝐴1),  𝐵𝒫ℱ𝔑(𝐴1)}  and 𝜏𝑝(𝐴2) = {0𝑃, 1𝑃 , 𝒫ℱ𝔑(𝐴2),

𝒫ℱ𝔑(𝐴2),  𝐵𝒫ℱ𝔑(𝐴2)} are the 𝒫ℱ𝒩𝑡𝑠′𝑠 on 𝑈1 and 𝑈2 respectively. 

Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be an identity function, then ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 but not 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟 , because the set 𝐵 = {⟨
𝑡1,𝑡4

0.4,0.1
⟩ , ⟨

𝑡2

0.4,0.2
⟩ , ⟨

𝑡3

0.2,0.3
⟩}  is 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 𝑈2  but 

ℎ𝑃
−1(𝐵) = 𝐵 is not 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1. 

Definition 4.2  A 𝒫ℱ𝒩𝑡𝑠 (𝑈1, 𝜏𝑃(𝐴1)) is known as a Pythagorean fuzzy nano 𝛿𝒮𝑈1

2

 (resp. 𝛿𝒫𝑈1

2

, 

𝛿𝛼𝑈1

2

 and 𝛿𝛽𝑈1

2

) (in short, 𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 𝒫ℱ𝒩𝛿𝛽𝑈1

2

))-space, if 

each 𝒫ℱ𝒩𝛿𝒮𝑜𝑠 (resp. 𝒫ℱ𝒩𝛿𝒫𝑜𝑠, 𝒫ℱ𝒩𝛿𝛼𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑜𝑠) in 𝑋 is 𝒫ℱ𝒩𝑜𝑠 in 𝑈1.  

Theorem 4.2  Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  be a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟  (resp. 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟  ) map. Then ℎ𝑃  is a 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠  map if 𝑋  is a 𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 

𝒫ℱ𝒩𝛿𝛽𝑈1

2

)-space.  

   Proof. (i) Consider a 𝒫ℱ𝒩𝑜𝑠 𝐾 in 𝑈2. Then 𝐾 is a 𝒫ℱ𝒩𝛿𝒮𝑜𝑠 in 𝑈2. Therefore ℎ𝑃
−1(𝐾) is a 

𝒫ℱ𝒩𝛿𝒮𝑐𝑠 in 𝑈1 . Since 𝑈1  is a 𝒫ℱ𝒩𝛿𝒮𝑈1

2

-space, ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝑐𝑠 in 𝑈1 . Hence ℎ𝑃  is a 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map. 

(ii) Consider a 𝒫ℱ𝒩𝑜𝑠  𝐾  in 𝑈2 . Then 𝐾  is a 𝒫ℱ𝒩𝛿𝒫𝑜𝑠  in 𝑈2 . Therefore ℎ𝑃
−1(𝐾)  is a 

𝒫ℱ𝒩𝛿𝒫𝑐𝑠 in 𝑈1 . Since 𝑋 is a 𝒫ℱ𝒩𝛿𝒫𝑈1

2

-space, ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝑐𝑠 in 𝑈1 . Hence ℎ𝑃  is a 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map. 

(iii) Consider a 𝒫ℱ𝒩𝑜𝑠  𝐾  in 𝑈2 . Then 𝐾  is a 𝒫ℱ𝒩𝛿𝛼𝑜𝑠  in 𝑈2 . Therefore ℎ𝑃
−1(𝐾)  is a 

𝒫ℱ𝒩𝛿𝛼𝑐𝑠 in 𝑈1 . Since 𝑈1  is a 𝒫ℱ𝒩𝛿𝛼𝑈1

2

-space, ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝑐𝑠 in 𝑈1 . Hence ℎ𝑃  is a 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map. 

(iv) Consider a 𝒫ℱ𝒩𝑜𝑠  𝐾  in 𝑈2 . Then 𝐾  is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 𝑈2 . Therefore ℎ𝑃
−1(𝐾)  is a 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1940 
https://internationalpubls.com 

𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1 . Since 𝑈1  is a 𝒫ℱ𝒩𝛿𝛽𝑈1

2

-space, ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝑐𝑠 in 𝑈1 . Hence ℎ𝑃  is a 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map.              

Theorem 4.3  Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1))  →  (𝑈2, 𝜏𝑃(𝐴2))  and 𝑔𝑃: (𝑈2, 𝜏𝑃(𝐴2)) →  (𝑈3, 𝜏𝑃(𝐴3))  be 

mappings. Then 𝑔𝑃 ∘ ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈3, 𝜏𝑃(𝐴3)) is 

    (i)  𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠,  𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 

and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠)  if ℎ𝑃  is 𝒫ℱ𝒩𝛿𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝛿𝒮𝐼𝑟𝑟 , 𝒫ℱ𝒩𝛿𝒫𝐼𝑟𝑟 , 𝒫ℱ𝒩𝛿𝛼𝐼𝑟𝑟  and 

𝒫ℱ𝒩𝛿𝛽𝐼𝑟𝑟) maps and 𝑔𝑃  is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 , 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠 and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠). 

    (ii)  𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟,

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟 and 𝒫ℱ𝒩  𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟) if ℎ𝑃  is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟 (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟 ) and 𝑔𝑃  is 

𝒫ℱ𝒩𝛿𝐼𝑟𝑟 (resp. 𝒫ℱ𝒩𝛿𝒮𝐼𝑟𝑟, 𝒫ℱ𝒩𝛿𝒫𝐼𝑟𝑟, 𝒫ℱ𝒩𝛿𝛼𝐼𝑟𝑟 and 𝒫ℱ𝒩𝛿𝛽𝐼𝑟𝑟). 

    (iii)  𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟,

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟 and 𝒫ℱ𝒩 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟) if ℎ𝑃 is 𝒫ℱ𝒩𝛿𝐼𝑟𝑟 (resp. 𝒫ℱ𝒩𝛿𝒮𝐼𝑟𝑟, 𝒫ℱ𝒩𝛿𝒫𝐼𝑟𝑟, 

𝒫ℱ𝒩𝛿𝛼𝐼𝑟𝑟  and 𝒫ℱ𝒩𝛿𝛽𝐼𝑟𝑟  ) and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟  (resp. 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟,     𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟,      𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟,      𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  & 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟).  

   Proof. (i) Let 𝐾  be a 𝒫ℱ𝒩𝑜𝑠  in 𝑈3 . Then 𝑔𝑃
−1(𝐾)  is a 𝒫ℱ𝒩𝛿𝛽𝑐𝑠  in 𝑈2 . As 𝑔𝑃  is a 

𝒫ℱ𝒩𝛿𝛽𝐼𝑟𝑟, ℎ𝑃
−1(𝑔𝑃

−1(𝐾)) is a 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1. Thus 𝑔𝑃 ∘ ℎ𝑃  is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map. The 

other cases are similar.              

Theorem 4.4  Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)) be a mapping.   

    (i)  If (𝑈1, 𝜏𝑃(𝐴1))  is 𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 𝒫ℱ𝒩𝛿𝛽𝑈1

2

)-space, 

then the concepts of 𝒫ℱ𝒩  𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠  and 𝒫ℱ𝒩𝛿𝒮𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝛿𝒫𝐶𝑡𝑠 , 𝒫ℱ𝒩𝛿𝛼𝐶𝑡𝑠  and 

𝒫ℱ𝒩𝛿𝛽𝐶𝑡𝑠) are equivalent. 

    (ii)  If 𝑔𝑃  is 𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 𝒫ℱ𝒩𝛿𝛽𝑈1

2

)-space, then the 

concepts of 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠  and 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠) and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  and 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟) are equivalent.  

    (iii)  If ℎ𝑃  and 𝑔𝑃  are 𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 𝒫ℱ𝒩𝛿𝛽𝑈1

2

)-spaces, 

then the concepts of 𝑁𝑠𝑆𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 , 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠  and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 ) and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 , 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟 and 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟) are equivalent.  

   Proof.  

(i) Let 𝐾 be a 𝒫ℱ𝒩𝑐𝑠 in 𝑈2. Then ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑈1 if ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠. 

As 𝑈1  is a 𝒫ℱ𝒩𝛿𝛽𝑈1

2

-space, ℎ𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝑆𝑜𝑠 in 𝑈1 . Hence ℎ𝑃  is also 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1941 
https://internationalpubls.com 

map. The other cases are similar.   

Theorem 4.5  Let ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2))  and 𝑔𝑃: (𝑈2, 𝜏𝑃(𝐴2)) → (𝑈3, 𝜏𝑃(𝐴3))  be 

𝒫ℱ𝒩𝛿𝒮𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝛿𝒫𝐶𝑡𝑠 , 𝒫ℱ𝒩𝛿𝛼𝐶𝑡𝑠  and 𝒫ℱ𝒩𝛿𝛽𝐶𝑡𝑠 ) mappings and 𝑈2  be a 

𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 𝒫ℱ𝒩𝛿𝛽𝑈1

2

)-spaces. Then 𝑔𝑃 ∘

ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈3, 𝜏𝑃(𝐴3))  is 𝒫ℱ𝒩𝛿𝒮𝐶𝑡𝑠  (resp. 𝒫ℱ𝒩𝛿𝒫𝐶𝑡𝑠 , 𝒫ℱ𝒩𝛿𝛼𝐶𝑡𝑠  and 

𝒫ℱ𝒩𝛿𝛽𝐶𝑡𝑠).  

  Proof. Let 𝐾  be a 𝒫ℱ𝒩𝑐𝑠  in 𝑈3 . Then 𝑔𝑃
−1(𝐾)  is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 𝑈2  since 𝑔𝑃  is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠. As 𝑈2 is a 𝒫ℱ𝒩𝛿𝛽𝑈1

2

-space, 𝑔𝑃
−1(𝐾) is a 𝒫ℱ𝒩𝑜𝑠 in 𝑈2. Then, 𝑔𝑃(ℎ𝑃

−1(𝐾)) 

is 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1 because ℎ𝑃 is 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠. Hence, 𝑔𝑃 ∘ ℎ𝑃 is a 𝒫ℱ𝒩𝛿𝛽𝐶𝑡𝑠 map.              

Theorem 4.6   Let a map ℎ𝑃: (𝑈1, 𝜏𝑃(𝐴1)) → (𝑈2, 𝜏𝑃(𝐴2)). Then the following conditions are 

equivalent if 𝑈1 and 𝑈2 are 𝒫ℱ𝒩𝛿𝒮𝑈1

2

 (resp. 𝒫ℱ𝒩𝛿𝒫𝑈1

2

, 𝒫ℱ𝒩𝛿𝛼𝑈1

2

 and 𝒫ℱ𝒩𝛿𝛽𝑈1

2

)-spaces.  

    (i)  ℎ𝑃  is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟  (resp. 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟  and 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟) map. 

    (ii)  ℎ𝑃
−1(𝐵) is a 𝒫ℱ𝒩𝛿𝒮𝑜𝑠 (resp. 𝒫ℱ𝒩𝛿𝒫𝑜𝑠, 𝒫ℱ𝒩𝛿𝛼𝑜𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑜𝑠) in 𝑈1, for each 

𝒫ℱ𝒩𝛿𝒮𝑐𝑠 (resp. 𝒫ℱ𝒩𝛿𝒫𝑐𝑠, 𝒫ℱ𝒩𝛿𝛼𝑐𝑠 and 𝒫ℱ𝒩𝛿𝛽𝑐𝑠) 𝐵 in 𝑈2. 

    (iii) 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐼𝑟𝑟 

𝒫ℱ𝒩𝑐𝑙(ℎ𝑃
−1(𝐵)) ⊇ ℎ𝑃

−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵)), for each 𝑝𝑓𝑠 𝐵 of 𝑈2.  

   Proof. (i) → (ii): Let 𝐵 be any 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈2. Then, 𝐵𝑐 is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑈2. Since ℎ𝑃 is 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟 , ℎ𝑃
−1(𝐵𝑐)  is a 𝒫ℱ𝒩𝛿𝛽𝑐𝑠  in 𝑈1 . But ℎ𝑃

−1(𝐵𝑐) = (ℎ𝑃
−1(𝐵))𝑐 . Therefore, 

ℎ𝑃
−1(𝐵) is a 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈1. 

(ii) → (iii) : Let 𝐵 be any 𝑝𝑓𝑠 in 𝑈2 and 𝒫ℱ𝒩𝑖𝑛𝑡(𝐵) ⊆ 𝐵. Then, ℎ𝑃
−1(𝒫ℱ𝒩𝑐𝑙(𝐵)) ⊆ ℎ𝑃

−1(𝐵). 

Since 𝒫ℱ𝒩𝑖𝑛𝑡(𝐵)  is a 𝒫ℱ𝒩𝑜𝑠  in 𝑈2 , 𝒫ℱ𝒩𝑖𝑛𝑡(𝐵)  is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 𝑈2 . Therefore, 

(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵))𝑐  is a 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈2 . By hypothesis, ℎ𝑃
−1((𝒫ℱ𝒩𝑖𝑛𝑡(𝐵))𝑐) is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 

𝑈1 . Since, ℎ𝑃
−1((𝒫ℱ𝒩𝑖𝑛𝑡(𝐵))𝑐) = (ℎ𝑃

−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵)))𝑐, ℎ𝑃
−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵))  is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠  in 

𝑈1 . Since, 𝑈1  is a 𝒫ℱ𝒩𝛿𝛽𝑈1/2  -space, ℎ𝑃
−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵))  is a 𝒫ℱ𝒩𝑜𝑠  in 𝑈1 . Hence, 

𝒫ℱ𝒩𝑐𝑙(ℎ𝑃
−1(𝐵)) ⊇ 𝒫ℱ𝒩𝑐𝑙(ℎ𝑃

−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵))) = ℎ𝑃
−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵)) . That is, 

𝒫ℱ𝒩𝑐𝑙(ℎ𝑃
−1(𝐵)) ⊇ ℎ𝑃

−1(𝒫ℱ𝒩𝑖𝑛𝑡(𝐵)). 

(iii) → (i) : Let 𝐵 be any 𝒫ℱ𝒩𝛿𝛽𝑐𝑠 in 𝑈2. Since 𝑈2 is a 𝒫ℱ𝒩𝛿𝛽𝑈1/2-space, 𝐵 is a 𝒫ℱ𝒩𝑐𝑠 in 

𝑈2  and 𝒫ℱ𝒩𝑐𝑙(𝐵) = 𝐵.  Hence, ℎ𝑃
−1(𝐵) = ℎ𝑃

−1(𝒫ℱ𝒩𝛿𝛽𝑐𝑙(𝐵)) ⊇ 𝒫ℱ𝒩𝛿𝛽𝑖𝑛𝑡(ℎ𝑃
−1(𝐵)) . But 

clearly, ℎ𝑃
−1(𝐵) ⊇ 𝒫ℱ𝒩𝑖𝑛𝑡(ℎ𝑃

−1(𝐵)) . Therefore, 𝒫ℱ𝒩𝑖𝑛𝑡(ℎ𝑃
−1(𝐵)) = ℎ𝑃

−1(𝐵) . This implies, 

ℎ𝑃
−1(𝐵) is a 𝒫ℱ𝒩𝑜𝑠 and hence, it is a 𝒫ℱ𝒩𝛿𝛽𝑜𝑠 in 𝑋1. Thus, ℎ𝑃 is a 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐼𝑟𝑟 map. 

The proof of the others are similar.             width 0.22 true cm height 0.22 true cm depth 0pt 

5  Application 

Measures of similarity is a real-valued function which quantifies the similarity between two 

pythagoren fuzzy sets and its value is always expressed as a number between 0 and 1. 0, a low level of 

similarity and 1, a high level of similarity. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1942 
https://internationalpubls.com 

Example 5.1  

Gemstones within a species can exhibit distinct colors or optical characteristics, which are prized by 

gemologists and enthusiasts alike. These unique features play a crucial role in determining the 

gemstone’s beauty, rarity, and ultimately, its value. A vast array of vibrant and colorful gemstones, 

representing various types, can be found in different regions of particular area. A panel of experts 

specializing in gemology aims to rank several of these gemstone varieties in order of preference based 

on a set of predetermined criteria. After thorough discussion and deliberation, the experts have 

identified five gemstone varieties 𝑆1, 𝑆2, 𝑆3, 𝑆4 and 𝑆5  as alternatives for ranking. The ranking 

decision will be based on the following four key criteria’s 𝐶𝑗, 𝑗 = 1, 2, 3, 4: Color and clarity is 𝐶1, 

Cut and polish is 𝐶2, Carat and luster or originality is 𝐶3 and hardness is 𝐶4. Now, the experts 

evaluate the alternatives 𝑆𝑖, 𝑖 = 1,2,3,4 under the criteria𝐶𝑗, 𝑗 = 1,2,3,4 which can be represented by 

the following 𝑝𝑓𝑠’s. 

𝑆1 = {< 𝐶1; 0.8,0.4 >, < 𝐶2; 0.3,0.8 >, < 𝐶3; 0.6,0.3 >, < 𝐶4; 0.6,0.4 >} 

𝑆2 = {< 𝐶1; 0.4,0.7 >, < 𝐶2; 0.3,0.8 >, < 𝐶3; 0.5,0.6 >, < 𝐶4; 0.6,0.5 >} 

𝑆3 = {< 𝐶1; 0.6,0.5 >, < 𝐶2; 0.7,0.4 >, < 𝐶3; 0.3,0.8 >, < 𝐶4; 0.3,0.5 >} 

𝑆4 = {< 𝐶1; 0.7,0.4 >, < 𝐶2; 0.4,0.7 >, < 𝐶3; 0.5,0.7 >, < 𝐶4; 0.6,0.4 >} 

𝑆5 = {< 𝐶1; 0.9,0.2 >, < 𝐶2; 0.4,0.8 >, < 𝐶3; 0.6,0.3 >, < 𝐶4; 0.7,0.2 >} 

These 𝑝𝑓𝑠s are shown with the help of decision matrix in Table 1. The 𝑝𝑓𝑠 of positive index set and 

𝑝𝑓𝑠 of negative index set are constructed respectively as follows: 

𝑆𝑝 = {< 𝐶1; 1,0 >, < 𝐶2; 1,0 >, < 𝐶1; 1,0 >, < 𝐶2; 1,0 >} 

𝑆𝑛 = {< 𝐶1; 0,1 >, < 𝐶2; 0,1 >, < 𝐶1; 0,1 >, < 𝐶2; 0,1 >} 

Next, calculating the Zhang similarity between each alternatives 𝑆𝑖 to 𝑝𝑓𝑠 of positive index set and 

𝑝𝑓𝑠 of negative index set respectively. The the results are shown in the Table 2.The similarity between 

each alternative 𝑆𝑖  to 𝑝𝑓𝑠  of positive index set and 𝑝𝑓𝑠  of negative index set respectively are 

utilized to calculate the degree of closeness [41] 𝒟𝒞(𝑆𝑖)  is calculated as follows. 𝒟𝒞(𝑆𝑖) =
𝒮𝑍(𝑆𝑛,𝑆𝑖)

𝒮𝑍(𝑆𝑝,𝑆𝑖)+𝒮𝑍(𝑆𝑛,𝑆𝑖)
 and the results are display in the Table 3. 

Table 3, shows the degree of closeness of each alternative 𝑆𝑖  over the criteria 𝐶𝑗 :The degree of 

closeness is used to rank the alternative 𝑆𝑖 in preference order. The results are exhibited in Table 4. 

Table 4, shows the rank of alternative 𝑆𝑖 according to the degree of closeness in preferred order.The 

alternative 𝑆𝑖 with the largest degree of closeness is considered as the best alternative. 

Table 1:𝒑𝒇𝒔’s of alternatives 

Alternative Criteria 1 (𝐶1) Criteria 2 (𝐶2) Criteria 3 (𝐶3) Criteria 4 (𝐶4) 

𝑆1 < 𝑆1, 𝐶1; 0.8,0.4 > < 𝑆1, 𝐶2; 0.3,0.8 > < 𝑆1, 𝐶3; 0.6,0.3 > < 𝑆1, 𝐶4; 0.6,0.4 > 

𝑆2 < 𝑆2, 𝐶1; 0.4,0.7 > < 𝑆2, 𝐶2; 0.3,0.8 > < 𝑆2, 𝐶3; 0.5,0.6 > < 𝑆2, 𝐶4; 0.6,0.5 > 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1943 
https://internationalpubls.com 

𝑆3 < 𝑆3, 𝐶1; 0.6,0.5 > < 𝑆3, 𝐶2; 0.7,0.4 > < 𝑆3, 𝐶3; 0.3,0.8 > < 𝑆3, 𝐶4; 0.3,0.5 > 

𝑆4 < 𝑆4, 𝐶1; 0.7,0.4 > < 𝑆4, 𝐶2; 0.4,0.7 > < 𝑆4, 𝐶3; 0.5,0.7 > < 𝑆4, 𝐶4; 0.6,0.4 > 

𝑆5 < 𝑆5, 𝐶1; 0.9,0.2 > < 𝑆5, 𝐶2; 0.4,0.8 > < 𝑆5, 𝐶3; 0.6,0.3 > < 𝑆5, 𝐶4; 0.7,0.2 > 

 

Table 2: Zhang similarity measure of each alternative with 𝑺𝒑 and 𝑺𝒏 

Alternative 𝒮𝑍(𝑆𝑝, 𝑆𝑖) 𝒮𝑍(𝑆𝑛, 𝑆𝑖) 

𝑆1 0.476 1.000 

𝑆2 0.618 1.000 

𝑆3 0.536 1.000 

𝑆4 0.495 0.181 

𝑆5 0.588 0.090 

 

Table 3: Degree of closeness of each alternative 

Alternative 𝒟𝒞(𝑆𝑖) 

𝑆1 0.68 

𝑆2 0.62 

𝑆3 0.65 

𝑆4 0.27 

𝑆5 0.13 

 

Table 4:Ranking of each alternative 

Similarity Ranking 

𝒮𝑍(𝑆𝑖) 𝒟𝒞(𝑆5) < 𝒟𝒞(𝑆4) < 𝒟𝒞(𝑆2) < 𝒟𝒞(𝑆3) < 𝒟𝒞(𝑆1) 

 

Hence, the alternative 𝑆1 is selected as the best alternative. 

6  Conclusion 

 In this paper, 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 , 𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 , 

𝒫ℱ𝒩𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛼𝐶𝑡𝑠, and 𝒫ℱ𝒩 𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝛽𝐶𝑡𝑠 respective irresolute map is defined using 𝒫ℱ𝒩𝛿𝑜, 

𝒫ℱ𝒩𝛿𝒮𝑜 , 𝒫ℱ𝒩𝛿𝒫𝑜 , 𝒫ℱ𝒩𝛿𝛼𝑜  and 𝒫ℱ𝒩𝛿𝛽𝑜  set and its properties are analyzed with the 

examples. Also we extended the concept of Pythagorean fuzzy contra irresolute maps in Pythagorean 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1944 
https://internationalpubls.com 

fuzzy topological spaces using above mentioned open sets. Some examples and basic relationships 

between the contra irresolute mappings were also discussed. In future, these can be extended to 

Pythagorean fuzzy open, closed, homeomorphism and contra maps. Application for Economical 

decision making problem was solved with the proposed similarity measure. In future, MCDM to the 

field of medical diagnostic can be develope to the 𝒫ℱ𝒩𝑡𝑠. 

References  

[1]  S. E. Abbas (2012),  Weaker Forms of Fuzzy Contra-continuity, The Journal of Fuzzy 

Mathematics, 2010. 

[2]  A. Acikgoz and F. Esenbel,  Neutrosophic soft 𝛿-topology and neutrosophic soft compactness, 

AIP Conference Proceedings 2183,  030002, (2019). 

[3]  M. Adabitabar Firozja, B. Agheli and E. Baloui Jamkhaneh (2019),  A new similarity measure 

for Pythagorean fuzzy sets, Complex and Intelligent Systems. 

[4]  D. Ajay and J. Joseline Charisma,  Pythagorean nano topological space, International Journal of 

Recent Technology and Engineering, 8 (2020), 3415-3419. 

[5]  D. Ajay and J. Joseline Charisma,  On weak forms of Pythagorean nano open sets, Advances in 

Mathematics: Scientific Journal, 9 (2020), 5953-5963. 

[6]  D. Ajay and J. Joseline Charisma,  Pythagorean nano continuity, Advances in Mathematics: 

Scientific Journal, 9 (8) (2020), 6291-6298. 

[7]  S. Aranganayagi, M. Saraswathi and K. Chitirakala,  More on open maps and closed maps in 

fuzzy hypersoft topological spaces and application in Covid-19 diagnosis using cotangent 

similarity measure, International Journal of Neutrosophic Science,  21(2), (2023), 32-58. 

[8]  S. Aranganayagi, M. Saraswathi, K. Chitirakala and A. Vadivel,  The 𝑒 -open sets in 

neutrosophic hypersoft topologial spaces and application in Covid-19 diagnosis using 

normalized hamming distance, Journal of the Indonesian Mathematical Society,  29(2), (2023), 

177-196. 

[9]  K. T. Atanassov (1983),  Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia. 

[10]  K. T. Atanassov (1986),  Intuitionistic fuzzy sets, Fuzzy Sets Syst.  20, 87-96. 

[11]  K. T. Atanassov (1999),  Intuitionistic fuzzy sets: theory and applications, Physica, Heidelberg. 

[12]  K. T. Atanassov (2012),  On intuitionistic fuzzy sets theory, Springer, Berlin. 

[13]  K. K. Azad (1981),  On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly 

continuity, J. Math. Anal. App, (82), 14-32. 

[14]  C. L. Chang (1968),  Fuzzy topological spaces, J. Math. Anal. Appi. (24), 182-190. 

[15]  Dogan Coker (1997),  An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and 

Systems, (88), 81-89. 

[16]  N. B. Gnanachristy and G. K. Revathi (2020),  Analysis of Various Fuzzy Topological Spaces, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1945 
https://internationalpubls.com 

Journal of Critical Reviews (7), 2394-5125. 

[17]  N. B. Gnanachristy and G. K. Revathi, (2021)  A View on Pythagorean Fuzzy Contra 𝒢  

Continuous Function, Journal of Physics Conference Series, (2115), 012041. 

[18]  Murat Olgun, Mehmet Unver and Seyhmus Yardimci (2019),  Pythagorean fuzzy topological 

spaces, Complex & Intelligent Systems. https://doi.org/10.1007/s40747-019-0095-2. 

[19]  Necla Turanli and Dogan coker (2000),  Fuzzy connectedness in intuitionistic fuzzy topological 

spaces, Fuzzy Sets and Systems, (116) 369-375. 

[20]  Paul Augustine Ejegwa (2019),  Pythagorean fuzzy set and its application in career placements 

based on academic performance using max-min-max composition Complex and Intelligent 

Systems. 

[21]  X. Peng and Y. Yang (2015),  Some results for Pythagorean fuzzy sets, Int. J Intell Syst.  30, 

1133-1160. 

[22]  X. Peng and G. Selvachandran (2017),  Pythagorean fuzzy set state of the art and future 

directions, Artif Intell Rev. https://doi.org/10.1007/s10462-017-9596-9. 

[23]  N. Preethi and G. K. Revathi (2020),  A conceptual View on 𝑃𝐹𝐷 functions and its Properties, 

Test Engineering and Management, 0913-4120. 

[24]  Rana Muhammad Zulqarnain et al (2021),  Development of TOPSIS Technique under 

Pythagorean Fuzzy Hypersoft Environment Based on Correlation Coefficient and Its 

Application towards the Selection of Antivirus Mask in COVID-19 Pandemic Hindawi 

Complexity. 

[25]  G. K. Revathi, E. Roja and M. K. Uma (2010)  Fuzzy Contra G continuous functions, 

International Review of Fuzzy mathematics, (5), 81-91. 

[26]  S. Saha,  Fuzzy 𝛿-continuous mappings, Journal of Mathematical Analysis and Applications,  

126 (1987), 130-142.  

[27]  R. Santhi and K. Arul Prakash (2011),  Intuitionistic fuzzy contra semi-generalised continuous 

mappings, (3), 30-40. 

[28]  P. Surendra, K. Chitirakala and A. Vadivel,  𝛿-open sets in neutrosophic hypersoft topological 

spaces, International Journal of Neutrosophic Science,  20 (4), (2023), 93-105. 

[29]  P. Surendra, A. Vadivel and K. Chitirakala,  𝛿 -separation axioms on fuzzy hypersoft 

topological spaces, International Journal of Neutrosophic Science,  23 (1), (2024), 17-26. 

[30]  M. Shukla (2013),  On Fuzzy Contra 𝑔∗ Semi-Continuous Functions, International Journal of 

Scientific and Engineering Research (4). 

[31]  M. Udhaya Shalini and A. Stanis Arul Mary (2022),  Generalized pre-closed sets in 

Pythagorean fuzzy topological spaces, International Journal of Creative Research Thoughts 

(IJCRT),  10 (30), e142-e147. 

[32]  A. Vadivel, M. Seenivasan and C. John Sundar,  An Introduction to 𝛿 -open sets in a 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1946 
https://internationalpubls.com 

Neutrosophic Topological Spaces, Journal of Physics: Conference Series, 1724 (2021), 012011. 

[33]  A. Vadivel, C. John Sundar, K. Kirubadevi and S. Tamilselvan,  More on Neutrosophic Nano 

Open Sets, International Journal of Neutrosophic Science (IJNS),  18 (4) (2022), 204-222. 

[34]  R. H. Warren (1978),  Neighborhoods, Bases and Continuity in Fuzzy Topological Spaces, 

Rocky Mountain Journal of Mathematics, (8). 

[35]  GW. Wei and G. Lan Grey (2008),  relational analysis method for interval valued intuitionistic 

fuzzy multiple attribute decision making, In Fifth international conference on fuzzy systems and 

knowledge discovery, 291-295. 

[36]  R. R. Yager (2013),  Pythagorean membership grades in multicriteria decision making, In: 

Technical report 𝑀𝐼𝐼-3301. Machine Intelligence Institute, Iona College, New Rochelle. 

[37]  R. R. Yager (2013),  Pythagorean fuzzy subsets, In: Proceedings of the joint 𝐼𝐹𝑆𝐴 world 

congress 𝑁𝐴𝐹𝐼𝑃𝑆 annual meeting, 57-61. 

[38]  R. R. Yager and A. M. Abbasov (2013),  Pythagorean membership grades, complex numbers, 

and decision making, Int J Intell Syst (28), 436-452. 

[39]  R. R. Yager (2014),  Pythagorean membership grades in multicriteria decision making, 𝐼𝐸𝐸𝐸 

Trans Fuzzy Syst.  22 (4), 958-965. 

[40]  L. A. Zadeh (1965),  Fuzzy sets, Inf. Control,  8, 338-353. 

[41]  Zahid Hussain, Sherbaz Alam, Rashid Hussian and Shams ur Rahman (2024),  New Similarity 

measure of Pythagorean fuzzy sets based on the Jaccard index with its application to clustering, 

Ain Shams Engineering Journal,  15, 102294. 

[42]  X. Zhang (2016),  A novel approach based on similarity measure for Pythagorean fuzzy 

multiple criteria group decision making, Int J Intell Syst (31), 593-611.