Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2005 
https://internationalpubls.com 

Generalization of Open Sets in Pythagorean Fuzzy Nano Topological Spaces and its Real 

Application 

 

X. Arul Selvaraj1  and   N. Prabavathy2  

  1Department of Mathematics, DDE, Annamalai University, Annamalai Nagar - 608 002, India; 

 (Deputed to) Periyar Arts Collge, Cuddalore-607 001, Tamil Nadu, India. 

xaselvarajmaths@gmail.com 

  2 Department of Mathematics, Arignar Anna Gov. Arts College, Musiri-621 211, Trichy(D.t.), Tamil Nadu, India. 

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

online2020av@gmail.com , prabamarch23@gmail.com  

(Corresponding Author: N. Prabavathy) 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: The purpose of this paper is to define and study a new class of sets called 

Pythagorean fuzzy nano 𝑍 (resp. 𝛿, 𝛿𝒮 and pre)-open sets in Pythagorean fuzzy nano 

topological spaces. Basic properties of Pythagorean fuzzy nano 𝑍 (resp. 𝛿, 𝛿𝒮 and pre) 

-open and their respective closed sets are analysed. We also used them to introduce the 

new notions like Pythagorean fuzzy nano 𝑍  (resp. 𝛿 , 𝛿𝒮  and pre)-closure (resp. 

interior) and their relations with already existing well known sets are also investigated. 

Keywords: Pythagorean fuzzy nano open set, Pythagorean fuzzy nano pre open set, 

Pythagorean fuzzy nano 𝛿 semi open set, Pythagorean fuzzy nano 𝑍 open set. 

AMS(2000) Subject classification: 03E72, 54A05, 54A40.  

 

1  Introduction 

 Zadeh in [21] established the idea of fuzzy in 1965, which is a generalization of usual set using fuzzy 

where each element has a membership degree in [0,1]. The subsequent advanement of fuzzy subsets 

was the intuitionistic fuzzy set published by Atanassov, [4] in 1983, which has elements having 

membership and non-membership degree. In 1968, Chang in [5] defined fuzzy topological space and 

fundamental results such as continuity, open and closed set. Following this, Lowen in [7] defined 

fuzzy topological space in other form. Coker introduced the idea of intuitionistic fuzzy topological 

space with few properties, see [6]. The concept of Pythagorean fuzzy subset which is a typical fuzzy 

subset was presented by Yager, see [18, 19, 20]. Pythagorean fuzzy topological space was introduced 

by Olgun in [11] by taking the lead as from Chang. 

In 2013, a new topology called Nano topology was introduced by Lellis Thivagar [9] which is an 

extension of rough set theory. He also introduced Nano topological spaces which were defined in 

terms of approximations and boundary region of a subset of a universe using an equivalence relation 

on it. The elements of a Nano topological space are called the Nano open sets and its complements are 

called the Nano closed sets. Nano means something very small. Nano topology thus literally means the 

study of very small surface. The fundamental ideas in Nano topology are those of approximations and 

indiscernibility relation. Furthermore nano 𝛿 open sets in nano topological space was studied in [13]. 

mailto:prabamarch23@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2006 
https://internationalpubls.com 

Recently, Lellis Thivagar et. al [10] explored a new concept of neutrosophic nano topology, and also 

𝑍 -open sets in topological spaces by El-Magharabi and Mubarki [8], 𝑀 -open sets in a nano 

topological spaces by Padma et. al [12], 𝑍-open sets in nano topological spaces by Selvaraj and 

Balakrishna [3] and fuzzy 𝑍 -closed sets and generalized fuzzy 𝑍 -closed sets in double fuzzy 

topological spaces by Shiventhiradevi et. al in [14, 15]. Thangammal et. al [16] introduced fuzzy nano 

𝑍-open sets in fuzzy nano topological spaces. Vadivel et al. [17] discussed some open sets in fuzzy 

nano topological spaces. 

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

open sets such as Pythagorean fuzzy nano 𝛿 open set, Pythagorean fuzzy nano 𝛿-semi open set, 

Pythagorean fuzzy nano pre open set and Pythagorean fuzzy nano 𝑍 open sets on Pythagorean fuzzy 

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

In this paper some preliminary concepts required in our work are briefly recalled in section 2. In 

section 3, we introduce the concept of 𝒫ℱ𝒩𝑜, 𝒫ℱ𝒩𝛿𝑜, 𝒫ℱ𝒩𝛿𝒮𝑜, 𝒫ℱ𝒩𝒫𝑜 and 𝒫ℱ𝒩𝑍𝑜 sets 

and studied some of their properties. Also, we discuss on Pythagorean fuzzy nano 𝑍-interior and 

Pythagorean fuzzy nano 𝑍-closure operators in Pythagorean fuzzy nano topological spaces. 

2  Preliminaries 

Definition 2.1  [21] A function 𝜆 from 𝑋 into the unit interval 𝐼 is called a fuzzy set in 𝑋. For every 

𝑥 ∈ 𝑋, 𝜆(𝑥) ∈ 𝐼 is called the grade of membership of 𝑥 in 𝜆. Some authors say that 𝜆 is a fuzzy 

subset of 𝑋 instead of saying that 𝜆 is a fuzzy set in 𝑋. The class of all fuzzy sets from 𝑋 into the 

closed unit interval 𝐼 will be denoted by 𝐼𝑋.  

Definition 2.2  [21] If 𝜆 and 𝜉 are any two fuzzy subsets of a set 𝑋, then 𝜆 is said to be included in 

𝜉 or 𝜆 is contained in 𝜉 or 𝜆 is less than or equal to 𝜉 iff 𝜆(𝑥) ≤ 𝜉(𝑥) for all 𝑥 in 𝑋 and is 

denoted by 𝜆 ≤ 𝜉. Equivalently, 𝜆 ≤ 𝜉 iff 𝜇𝜆(𝑥) ≤ 𝜇𝜉(𝑥) for all 𝑥 in 𝑋.  

Note that every fuzzy subset is included in itself and empty fuzzy subset is included in every fuzzy 

subset. 

Definition 2.3  [21] Two fuzzy subsets 𝜆 and 𝜇 of a set 𝑋 are said to be equal, written 𝜆 = 𝜇, if 

𝜆(𝑥) = 𝜇(𝑥) for every 𝑥 in 𝑋.  

Definition 2.4  [21] The complement of a fuzzy subset 𝜆 in a set 𝑋, denoted by 1 − 𝜆, is the fuzzy 

subset of 𝑋 defined by 1 − 𝜆(𝑥) for all 𝑥 in 𝑋. Note that 1 − (1 − 𝜆) = 𝜆.  

Definition 2.5  [21] The union of two fuzzy subsets 𝜆 and 𝜇 in a set 𝑋, denoted by 𝜆 ∨ 𝜇, is fuzzy 

subset in 𝑋 defined by (𝜆 ∨ 𝜇)(𝑥) = 𝑚𝑎𝑥{𝜆(𝑥), 𝜇(𝑥)}, for all 𝑥 in 𝑋.  

In general, the union of a family of fuzzy subsets {𝜉𝑖: 𝑖 ∈ 𝐼} is a fuzzy subset denoted by ∨𝑖∈𝐼 𝜉𝑖 and 

defined by (∨𝑖∈𝐼 𝜉𝑖)(𝑥) = sup{𝜉𝑖(𝑥): 𝑖 ∈ 𝐼}, for all 𝑥 in 𝑋. 

Definition 2.6  [21] The intersection of two fuzzy subsets 𝜆 and 𝜇 in a set 𝑋, denoted by 𝜆 ∧ 𝜇, is 

fuzzy subset in 𝑋 defined by (𝜆 ∧ 𝜇)(𝑥) = 𝑚𝑖𝑛{𝜆(𝑥), 𝜇(𝑥)}, for all 𝑥 in 𝑋. 

In general, the intersection of a family of fuzzy subsets {𝜉𝑖: 𝑖 ∈ 𝐼} is a fuzzy subset denoted by ∧𝑖∈𝐼 𝜉𝑖 

and defined by (∧𝑖∈𝐼 𝜉𝑖)(𝑥) = inf{𝜉𝑖(𝑥): 𝑖 ∈ 𝐼}, for all 𝑥 in 𝑋. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2007 
https://internationalpubls.com 

Definition 2.7  [18, 19, 20] 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), 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 2.8  [20] 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   

    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 2.9  [1] 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:  

    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), 

𝜆𝑅(𝐴)(𝑥) =∨𝑦∈[𝑥]𝑅
𝜆𝐴(𝑦). 

Definition 2.10  [1] 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),



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2008 
https://internationalpubls.com 

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  [1] [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 2.11  [1, 2] Let (𝑈, 𝜏ℛ(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  

    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)). 

3  Pythagorean fuzzy nano 𝒁 (resp. 𝜹, 𝜹𝓢 and pre)-open sets 

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

𝑝𝑓𝑠 of 𝑈. Then Pythagorean   

    1.  fuzzy nano 𝛿 interior of 𝑆 (briefly, 𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆)) is defined by 𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆) =∪ {𝐼: 𝐼 ⊆

𝑆 & 𝐼isa𝒫ℱ𝒩𝑟𝑜 set in𝑈}.  

    2.  fuzzy nano 𝛿  closure of 𝑆 (briefly, 𝒫ℱ𝒩𝛿𝑐𝑙(𝑆)) is defined by 𝒫ℱ𝒩𝛿𝑐𝑙(𝑆) =∩ {𝐴: 𝑆 ⊆

𝐴 & 𝐴isa𝒫ℱ𝒩𝑟𝑐set in 𝑈}.  

Definition 3.2  Let (𝑈, 𝜏ℛ(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 :  

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

    2.  fuzzy nano 𝛿-𝛼-open (or) fuzzy nano 𝑎-open (briefly, 𝒫ℱ𝒩𝛿𝛼𝑜 (or) 𝒫ℱ𝒩𝑎𝑜) set if 𝑆 ⊆

𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙 (𝒫ℱ𝒩𝛿𝑖𝑛𝑡 (𝑆))).  

    3.  fuzzy nano 𝛿-semi open (briefly, 𝒫ℱ𝒩𝛿𝒮𝑜) set if 𝑆 ⊆ 𝒫ℱ𝒩𝑐𝑙(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆)).  

    4.  fuzzy nano pre open (briefly, 𝒫ℱ𝒩𝒫𝑜) set if 𝑆 ⊆ 𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙(𝑆)).  

The complement of an 𝒫ℱ𝒩𝛿𝑜  (resp. 𝒫ℱ𝒩𝛿𝛼𝑜 , 𝒫ℱ𝒩𝛿𝒮𝑜  &  𝒫ℱ𝒩𝒫𝑜 ) set is called a 

Pythagorean fuzzy nano 𝛿 (resp. Pythagorean fuzzy nano 𝛿-𝛼, Pythagorean fuzzy nano 𝛿-semi & 

Pythagorean fuzzy nano pre) closed (briefly, 𝒫ℱ𝒩𝛿𝑐 (resp. 𝒫ℱ𝒩𝛿𝛼𝑐, 𝒫ℱ𝒩𝛿𝒮𝑐 & 𝒫ℱ𝒩𝒫𝑐)) in 

𝑈.  

Definition 3.3  Let (𝑈, 𝜏ℛ(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  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2009 
https://internationalpubls.com 

    1.  𝛿  semi interior of 𝑆  (briefly, 𝒫ℱ𝒩𝛿𝒮𝑖𝑛𝑡(𝑆) ) is defined by 𝒫ℱ𝒩𝛿𝒮𝑖𝑛𝑡(𝑆) =∪ {𝐼: 𝐼 ⊆

𝑆 & 𝐼isa𝒫ℱ𝒩𝛿𝒮𝑜 set in𝑈}.  

    2.  𝛿  semi closure of 𝑆  (briefly, 𝒫ℱ𝒩𝛿𝒮𝑐𝑙(𝑆) ) is defined by 𝒫ℱ𝒩𝛿𝒮𝑐𝑙(𝑆) =∩ {𝐴: 𝑆 ⊆

𝐴 & 𝐴isa𝒫ℱ𝒩𝛿𝒮𝑐set in 𝑈}.  

    3.  pre interior of 𝑆  (briefly, 𝒫ℱ𝒩𝒫𝑖𝑛𝑡(𝑆) ) is defined by 𝒫ℱ𝒩𝒫𝑖𝑛𝑡(𝑆) =∪ {𝐼: 𝐼 ⊆

𝑆 & 𝐼isa𝒫ℱ𝒩𝒫𝑜 set in𝑈}.  

    4.  pre closure of 𝑆  (briefly, 𝒫ℱ𝒩𝒫𝑐𝑙(𝑆) ) is defined by 𝒫ℱ𝒩𝒫𝑐𝑙(𝑆) =∩ {𝐴: 𝑆 ⊆

𝐴 & 𝐴isa𝒫ℱ𝒩𝒫𝑐set in 𝑈}.  

 Definition 3.4   Let (𝑈, 𝜏ℛ(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 a Pythagorean fuzzy nano  

1.  𝑍-open (briefly, 𝒫ℱ𝔑𝑍𝑜) set if 𝑆 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆)) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝑆)),  

2.  𝑍-closed (briefly, 𝒫ℱ𝔑𝑍𝑐) set if 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆)) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝑆)) ⊆ 𝑆.  

  The family of all 𝒫ℱ𝔑𝑍𝑜 (resp. 𝒫ℱ𝔑𝑍𝑐) sets of a space (𝑈, 𝜏ℛ(𝐴)) will be as always denoted by 

𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴) (resp. 𝒫ℱ𝔑𝑍𝐶(𝑈, 𝐴)). 

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

𝑝𝑓𝑠 𝐾 in 𝑈, then the Pythagorean   

    1.  fuzzy nano 𝑍-interior of 𝐾 is the union of all 𝒫ℱ𝔑𝑍𝑜 sets contained in 𝐾 and denoted by 

𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾).  

    2.  fuzzy nano 𝑍-closure of 𝐾 is the intersection of all 𝒫ℱ𝔑𝑍𝑐 sets containing 𝐾 and denoted 

by 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾).  

 Remark 3.1  Let 𝐾  be a subset of a 𝒫ℱ𝒩𝑡𝑠  (𝑈, 𝜏ℛ(𝐴)).  Then (𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))𝑐 =

𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾𝑐), (𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾))𝑐 = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾𝑐).  

Theorem 3.1   Let (𝑈, 𝜏ℛ(𝐴)) be a 𝒫ℱ𝒩𝑡𝑠. Then,   

    (i)  Every 𝒫ℱ𝔑𝛿𝑜 set is 𝒫ℱ𝔑𝑜 set.  

    (ii)  Every 𝒫ℱ𝔑𝑜 set is 𝒫ℱ𝔑𝒫𝑜 set.  

    (iii)  Every 𝒫ℱ𝔑𝛿𝑜 set is 𝒫ℱ𝔑𝛿𝒮𝑜 set.  

    (iV)  Every 𝒫ℱ𝔑𝛿𝒮𝑜 set is 𝒫ℱ𝔑𝑍𝑜 set.  

    (V)  Every 𝒫ℱ𝔑𝒫𝑜 set is 𝒫ℱ𝔑𝑍𝑜 set.  

   Proof. (i) If 𝐾 is a 𝒫ℱ𝔑𝛿𝑜𝑠 in 𝑈, then 𝐾 = 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾). Therefore, 𝐾 is a 

𝒫ℱ𝔑𝑜𝑠. 

(ii) If 𝐾 is a 𝒫ℱ𝔑𝑜𝑠 in 𝑈, then 𝐾 = 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾). So, 𝐾 = 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾) ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾)). 

Therefore, 𝐾 is a 𝒫ℱ𝔑𝒫𝑜𝑠. 

(iii) If 𝐾  is a 𝒫ℱ𝔑𝛿𝑜𝑠  in 𝑈,  then 𝐾  is 𝒫ℱ𝔑𝑜𝑠  by (𝑖) . So, 𝐾 ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾) ⊆

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)). Therefore, 𝐾 is a 𝒫ℱ𝔑𝛿𝒫𝑜𝑠. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2010 
https://internationalpubls.com 

(iv) 𝐾  is a 𝒫ℱ𝔑𝛿𝒮𝑜𝑠 , then 𝐾 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾))  and so 𝐾 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ⊆

𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾)). ∴ 𝐾 is a 𝒫ℱ𝔑𝑍𝑜𝑠. 

(v) 𝐾  is a 𝒫ℱ𝔑𝒫𝑜𝑠 , then 𝐾 ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))  and so 𝐾 ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾)) ⊆

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾)) ∪ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)). ∴ 𝐾 is a 𝒫ℱ𝔑𝑍𝑜𝑠. 

The converse of the above propositions need not to be true. The following examples show it. 

Example 3.1   Assume 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 𝑝𝑓𝑠 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
⟩}. 

Thus 𝜏ℛ(𝐴) = {0𝒫, 1𝒫 , 𝒫ℱ𝔑(𝐴), 𝒫ℱ𝔑(𝐴), 𝐵𝒫ℱ𝔑(𝐴)}. Then  

    1.  {⟨
𝑠1,𝑠4

0.4,0.1
⟩ , ⟨

𝑠2

0.1,0.5
⟩ , ⟨

𝑠3

0.2,0.45
⟩}  is a 𝒫ℱ𝔑𝑜  (resp. 𝒫ℱ𝔑𝑍𝑜  ) set but not 𝒫ℱ𝔑𝛿𝑜  (resp. 

𝒫ℱ𝔑𝛿𝒮𝑜) set.  

    2.  {⟨
𝑠1,𝑠4

0.25,0.3
⟩ , ⟨

𝑠2

0.5,0.1
⟩ , ⟨

𝑠3

0.45,0.2
⟩}  is a 𝒫ℱ𝔑𝛿𝒮𝑜  (resp. 𝒫ℱ𝔑𝑍𝑜 ) set but not 𝒫ℱ𝔑𝛿𝑜  (resp. 

𝒫ℱ𝔑𝒫𝑜) set.  

    3.  {⟨
𝑠1,𝑠4

0.45,0.35
⟩ , ⟨

𝑠2

0.25,0.45
⟩ , ⟨

𝑠3

0.3,0.2
⟩} is a 𝒫ℱ𝔑𝒫𝑜 set but not 𝒫ℱ𝔑𝑜 set. 

Remark 3.2   According to Definition 3.4 and Theorem 3.1, the following diagram holds for any set 

in 𝒫ℱ𝒩𝑡𝑠.  

 

Lemma 3.1   Let (𝑈, 𝜏ℛ(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠. Then the following statements are hold.  

    (i)  The union of arbitrary 𝒫ℱ𝔑𝑍𝑜 sets is 𝒫ℱ𝔑𝑍𝑜,  

    (ii)  The intersection of arbitrary 𝒫ℱ𝔑𝑍𝑐 sets is 𝒫ℱ𝔑𝑍𝑐.  

  Proof. (i) Let {𝐾𝑖 , 𝑖 ∈ 𝐼} be a family of 𝒫ℱ𝔑𝑍𝑜 sets.  

Then 𝐾𝑖 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾𝑖)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾𝑖))  and hence ∪𝑖 𝐾𝑖 ⊆

∪𝑖 (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾𝑖))  ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾𝑖))) ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(∪𝑖 𝐾𝑖)) ∩



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2011 
https://internationalpubls.com 

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(∪𝑖 𝐾𝑖)), for all 𝑖 ∈ 𝐼. Thus ∪𝑖 𝐾𝑖 is 𝒫ℱ𝔑𝑍𝑜. 

(ii) It follows from (i).              

Remark 3.3   By the following we show that the intersection of any two 𝒫ℱ𝔑𝑍𝑜 sets is not 𝒫ℱ𝔑𝑍𝑜. 

Example 3.2  In Example 3.1, let 𝐴 = {⟨
𝑠1,𝑠4

0.3
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.5
⟩}  and 𝐵 = {⟨

𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.2
⟩ , ⟨

𝑠3

0.7
⟩}  are 

𝒫ℱ𝒩𝑍𝑜 sets but 𝐴 ∩ 𝐵 = {⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.2
⟩ , ⟨

𝑠3

0.5
⟩} is not 𝒫ℱ𝒩𝑍𝑜 set. 

Theorem 3.2   Let 𝐾 be a 𝒫ℱ𝔑𝑍𝑐 set in (𝑈, 𝜏ℛ(𝐴)) then 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 does not contain any 

non-empty 𝒫ℱ𝔑𝑐 set in (𝑈, 𝜏ℛ(𝐴)).  

  Proof. Let 𝐾  be a 𝒫ℱ𝔑𝑍𝑐  set in (𝑈, 𝜏ℛ(𝐴)) and 𝑆  be a 𝒫ℱ𝔑𝑐 subset of 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 . 

That is, 𝑆 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 implies 𝑆 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∩ (1𝒫 − 𝐾). That is 𝑆 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) and 

𝑆 ⊆ (1𝒫 − 𝐾) which implies 𝐾 ⊆ (1𝒫 − 𝑆) where 1𝒫 − 𝑆 is a 𝒫ℱ𝔑𝑜  set. Since 𝐾  is 𝒫ℱ𝔑𝑍𝑐 , 

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ (1𝒫 − 𝑆) . That is 𝑆 ⊆ (1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) . Thus 𝑆 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∩ (1𝒫 −

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) = 0𝑃. Hence 𝑆 = 0𝑃. Therefore 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 does not contain any non-empty 

𝒫ℱ𝔑𝑐 set in (𝑈, 𝜏ℛ(𝐴)).              

Remark 3.4  The converse of the theorem 3.2 need not be true as seen from the following example.  

Example 3.3  In Example 3.1, let 𝐾 = {𝑙2, 𝑙3, 𝑙4} be any subset of 𝑈 then 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 = 𝑈 −

{𝑙2, 𝑙3, 𝑙4} = {𝑙1} which contain any non-empty 𝒫ℱ𝔑𝑐 in 𝑈, but 𝐾 is not a 𝒫ℱ𝔑𝑍𝑐 in 𝑈.  

Theorem 3.3  Let 𝐾  be a 𝒫ℱ𝔑  set in (𝑈, 𝜏ℛ(𝐴))  then 𝐾  is 𝒫ℱ𝔑𝑍𝑐𝑠  if and only if 

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 is a 𝒫ℱ𝔑𝑐 set in (𝑈, 𝜏ℛ(𝐴)).  

  Proof. Let 𝐾  be a 𝒫ℱ𝔑𝑍𝑐  set. Assume that 𝐾  is 𝒫ℱ𝔑𝑍𝑐  then we have 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) = 𝐾 , 

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 = 0𝑃 which is 𝒫ℱ𝔑𝑐𝑠. 

Conversely, assume that 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾  be 𝒫ℱ𝔑𝑐𝑠  and 𝐾  is a 𝒫ℱ𝔑𝑍𝑐  set in 𝑈 . Now 

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 is a 𝒫ℱ𝔑𝑐 subset of itself. Therefore by theorem 3.2, 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 = 0𝑃 . 

That is 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) = 𝐾, implies 𝐾 is 𝒫ℱ𝔑𝑍𝑐𝑠.              

Theorem 3.4   If 𝐾  is 𝒫ℱ𝔑𝑍𝑐  set in (𝑈, 𝜏ℛ(𝐴))  and 𝐾 ⊆ 𝐿 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) , then 𝐿  is also 

𝒫ℱ𝔑𝑍𝑐 in (𝑈, 𝜏ℛ(𝐴)).  

  Proof. Let 𝐾 be a 𝒫ℱ𝔑𝑍𝑐 set in 𝑈 and 𝐾 ⊆ 𝐿 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾). Let 𝐿 ⊆ 𝑂 where 𝑂 be 𝒫ℱ𝔑𝑐 

set in 𝑈. Since 𝐾 ⊆ 𝐿, implies 𝐾 ⊂ 𝑂 and 𝐾 is 𝒫ℱ𝔑𝑍𝑐𝑠,  

implies 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑂 . By hypothesis 𝐿 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) , implies 𝒫ℱ𝔑𝑍𝑐𝑙(𝐿) ⊆

𝒫ℱ𝔑𝑍𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑂 , which implies 𝒫ℱ𝔑𝑍𝑐𝑙(𝐿) ⊆ 𝑂 . Therefore 𝐿  is 

𝒫ℱ𝔑𝑍𝑐𝑠 in 𝑈.              

Theorem 3.5   If a subset 𝐾  of 𝑈  is 𝒫ℱ𝔑𝑍𝑐  set, then 𝒫ℱ𝔑𝑐𝑙({𝑥𝑟}) ∩ 𝐾 ≠ 0𝑃  for each 𝑥𝑟 ∈

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾).  

  Proof. Suppose 𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set and 𝑥𝑟 ∈ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾). If possible 𝒫ℱ𝔑𝑐𝑙({𝑥𝑟}) ∩ 𝐾 = 0𝑃. 

Then 𝐾 ⊆ 1𝒫 − 𝒫ℱ𝔑𝑐𝑙({𝑥𝑟}) and 1𝒫 − 𝒫ℱ𝔑𝑐𝑙({𝑥𝑟}) is a 𝒫ℱ𝔑𝑜  set containing 𝐾 . Since 𝐾  is 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2012 
https://internationalpubls.com 

𝒫ℱ𝔑𝑍𝑐  set, implies 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 1𝒫 − 𝒫ℱ𝔑𝑐𝑙({𝑥𝑟})  which is a contradiction to 𝑥𝑟 ∈

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾). Therefore, 𝒫ℱ𝔑𝑐𝑙({𝑥𝑟}) ∩ 𝐾 ≠ 0𝑃.              

Theorem 3.6  If 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴) = 𝒫ℱ𝔑𝑍𝐶(𝑈, 𝐴), then 𝒫ℱ𝔑𝑍𝐶(𝑈, 𝐴) = 𝑃(𝑈) is the power set of 

𝑈.  

  Proof. Suppose 𝐾 ⊆ 𝑂 , where 𝑂  is 𝒫ℱ𝔑𝑜  in 𝑈 . Since every 𝒫ℱ𝔑𝑜  set is 𝒫ℱ𝔑𝑍𝑜 , 𝑂  is 

𝒫ℱ𝔑𝑍𝑜. By hypothesis, 𝑂 is 𝒫ℱ𝔑𝑍𝑐. Hence, 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑂. Therefore, 𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set. 

Since, 𝐾 is arbitrary, by Theorem 3.5, every subset of 𝑈 is 𝒫ℱ𝔑𝑍𝑐. Thus 𝒫ℱ𝔑𝑍𝐶(𝑈, 𝐴) = 𝑃(𝑈).              

Definition 3.6  The intersection of all 𝒫ℱ𝔑𝑜 subset of 𝑈 containing 𝐾 is called the Pythagorean 

fuzzy nano kernel of 𝐾  (briefly, 𝒫ℱ𝔑𝑘𝑒𝑟(𝐾) ), this means 𝒫ℱ𝔑𝑘𝑒𝑟(𝐾) =∩ {𝐺 ∈

𝒫ℱ𝔑𝑂(𝑈, 𝐴): 𝐾 ⊆ 𝐺}.  

Theorem 3.7  A subset 𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set iff 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝑘𝑒𝑟(𝐾).  

  Proof. Suppose 𝐾 is 𝒫ℱ𝔑𝑍𝑐 set, then 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑂 whenever 𝐾 ⊆ 𝑂 and 𝑂 is 𝒫ℱ𝔑𝑜. Let 

𝑥𝑟 ∈ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾). If 𝑥𝑟 ∉ 𝒫ℱ𝔑𝑘𝑒𝑟(𝐾), then there exist a 𝒫ℱ𝔑𝑜 set 𝑂 containing 𝐾  such that 

𝑥𝑟 ∉ 𝑂. Since 𝑂 is a 𝒫ℱ𝔑𝑜 set containing 𝐾, implies 𝑥𝑟 ∉ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾), which is a contradiction. 

Therefore 𝑥𝑟 ∈ 𝒫ℱ𝔑𝑘𝑒𝑟(𝐾). 

Conversely, let 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝑘𝑒𝑟(𝐾) . If 𝑂  is a 𝒫ℱ𝔑𝑜  set containing 𝐾 , then 

𝒫ℱ𝔑𝑘𝑒𝑟(𝐾) ⊆ 𝑂, which implies 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑂. Therefore, 𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set.              

Theorem 3.8  If 𝐾 ⊆ 𝑉 ⊆ 1𝑃  and suppose that 𝐾  is a 𝒫ℱ𝔑𝑍𝑐 set in 1𝑃 , then 𝐾  is 𝒫ℱ𝔑𝑍𝑐𝑠 

relative to 𝑉.  

  Proof. Given that 𝐾 ⊆ 𝑉 ⊆ 1𝑃  and let 𝐾 ⊆ 𝑉 ∩ 𝑂  where 𝑂  is 𝒫ℱ𝔑𝑜𝑠  in 𝑈 . Since 𝐾  is a 

𝒫ℱ𝔑𝑍𝑐 set in 𝑈, 𝐾 ⊆ 𝑂 which implies that 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑂. That is, 𝑉 ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝑉 ∩ 𝑂, 

where 𝑉 ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) is 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) in 𝑉. Thus 𝐾 is 𝒫ℱ𝔑𝑍𝑐𝑠 relative to 𝑉.              

Lemma 3.2   Let 𝐾 and 𝐿 be two 𝑝𝑓𝑠’s of (𝑈, 𝜏ℛ(𝐴)). Then:  

    (i)  1𝒫 − 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) = 𝒫ℱ𝔑𝛿𝑐𝑙(1𝒫 − 𝐾) and 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(1𝒫 − 𝐾) = 1𝒫 − 𝒫ℱ𝔑𝛿𝑐𝑙(𝐾),  

    (ii)  𝒫ℱ𝔑𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝛿𝑐𝑙(𝐾) (resp. 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾)), for any subset 𝐾 of 𝑈,  

    (iii)  𝒫ℱ𝔑𝛿𝑐𝑙(𝐾 ∪ 𝐿) = 𝒫ℱ𝔑𝛿𝑐𝑙(𝐾) ∪ 𝒫ℱ𝔑𝛿𝑐𝑙(𝐿) , 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾 ∩ 𝐿) = 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩

𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿). 

Proposition 3.1   Let 𝐾 be a 𝑝𝑓𝑠 in a 𝒫ℱ𝒩𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then:   

    (i) 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) = 𝐾 ∪ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾)), 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾) = 𝐾 ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾)),  

    (ii)  𝒫ℱ𝔑𝛿𝒮𝑐𝑙(1𝒫 − 𝐾) = 1𝒫 − 𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾) , 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾 ∩ 𝐿) ⊆ 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) ∪

𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐿),  

Lemma 3.3   The following hold for a 𝑝𝑓𝑠 𝐻 in a 𝒫ℱ𝒩𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)).  

    1.  𝒫ℱ𝔑𝒫𝑐𝑙(𝐻) = 𝐻 ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐻))  and 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐻) = 𝐻 ∩

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐻)),  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2013 
https://internationalpubls.com 

    2.  𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐻) = 𝐻 ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐻))  and 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐻) = 𝐻 ∪

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻)). 

Lemma 3.4   The following hold for a 𝑝𝑓𝑠 𝐻 in a 𝒫ℱ𝒩𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐻)) =

𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑 𝛿𝑖𝑛𝑡(𝐻)) and 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻)) = 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻)).  

Theorem 3.9   Let (𝑈, 𝜏ℛ(𝐴)) be a 𝒫ℱ𝔑𝑡𝑠. Then:  

    (i)  If 𝐾 ∈ 𝒫ℱ𝔑𝛿𝑂(𝑈, 𝐴) and 𝐿 ∈ 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴), then 𝐾 ∩ 𝐿 is 𝒫ℱ𝔑𝑍𝑜𝑠,  

    (ii)  If 𝐾 ∈ 𝒫ℱ𝔑𝑎𝑂(𝑈, 𝐴) and 𝐿 ∈ 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴), then 𝐾 ∩ 𝐿 ∈ 𝒫ℱ𝔑𝑍𝑜𝑠.  

 

 Proof. (i) Suppose that 𝐾 ∈ 𝒫ℱ𝔑𝛿𝑂(𝑈, 𝐴). Then 𝐾 = 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾). Since 𝐿 ∈ 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴), 

then 𝐿 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑 𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿)) and hence  

𝐾 ∩ 𝐿 ⊆ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿))) 

    = (𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿))) ∪ (𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿))

⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ (𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿))) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝑐𝑙(𝐿))

⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾 ∩ 𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾 ∩ 𝐿)). 

 Thus 𝐾 ∩ 𝐿 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾 ∩ 𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾 ∩ 𝐿)).  Therefore, 𝐾 ∩ 𝐿  is 

𝒫ℱ𝔑𝑍𝑜𝑠. 

(ii) Suppose that 𝐾 ∈ 𝒫ℱ𝔑𝑎𝑂(𝑈, 𝐴)  and 𝐿 ∈ 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴), The 𝐾 ∩ 𝐿 ⊆

𝒫ℱ𝔑𝑖𝑛𝑡 (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾))) ∩ (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿))) 

= (𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾))) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿))) ∪ (𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙 

(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾))) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿))) 

   ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾))

∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿))) 

        ⊆ 𝒫ℱ𝔑𝑐𝑙 (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿))) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡 (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ∩

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿))) 

and hence 

        𝐾 ∩ 𝐿 ⊆ (𝐾 ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿))) ∪ (𝐾 ∩

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿)))) 

         ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿)))) 

        ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝑐𝑙(𝐿))) 

    ⊆ 𝒫ℱ𝔑𝑐𝑙(𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝐿)). 

 Since 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿) ⊆ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ⊆ 𝐾 which is 𝒫ℱ𝔑𝛿𝑜 in 𝐾,  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2014 
https://internationalpubls.com 

Then 

𝐾 ∩ 𝐿 ⊆ 𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾 ∪ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)

∩ 𝐿)) 

 ⊆ 𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾 ∩ 𝐿) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) ∩ 𝐿) 

 ⊆ 𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾 ∩ 𝐿) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡𝒫ℱ𝔑𝑐𝑙(𝐾 ∩ 𝐿). 

 Therefore 𝐾 ∩ 𝐿 ∈ 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴).             

Proposition 3.2   Let (𝑈, 𝜏ℛ(𝐴))  be a 𝒫ℱ𝔑𝑡𝑠 . Then the closure of a 𝒫ℱ𝔑𝑍𝑜  set of 𝐴  is 

𝒫ℱ𝔑𝒮𝑜𝑠.  

 Proof. Let 𝐾 ∈ 𝒫ℱ𝔑𝑍𝑂(𝑈, 𝐴). 

 Then 𝒫ℱ𝔑𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))) ⊆

𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ∪ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))) = 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))). 

Therefore, 𝒫ℱ𝔑𝑐𝑙(𝐾) is 𝒫ℱ𝔑𝒮𝑜𝑠.              

Proposition 3.3   Let 𝐾 be a 𝒫ℱ𝔑𝑍𝑜 set of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)) and 𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾) = 0𝑃. Then 

𝐾 is 𝒫ℱ𝔑𝒫𝑜𝑠.  

 Proof. Obvious.              

Theorem 3.10   Let 𝐾 and 𝐿 be two subsets of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then the following are hold:   

    (i)  𝒫ℱ𝔑𝑍𝑐𝑙(1𝒫 − 𝐾) = 1𝒫 − 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾),  

    (ii)  𝒫ℱ𝔑𝑍𝑖𝑛𝑡(1𝒫 − 𝐾) = 1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾),  

    (iii)  If 𝐾 ⊆ 𝐿, then 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐿) and 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) ⊆ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿),  

    (iV)  𝑙 ∈ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) iff for each 𝒫ℱ𝔑𝑍𝑜 set 𝐴 contains 𝑙, 𝐴 ∩ 𝐾 ≠ 0𝑃,  

    (V)  𝑙 ∈ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) iff there exist a 𝒫ℱ𝔑𝑍𝑜 set 𝑊 such that 𝑙 ∈ 𝑊 ⊆ 𝐾,  

    (Vi)  𝒫ℱ𝔑𝑍𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) and 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾)) = 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾),  

    (Vii)  𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∪ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐿) ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾 ∪ 𝐿)  and 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) ∪ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿) ⊆

𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾 ∪ 𝐿),  

    (Viii)  𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾 ∩ 𝐿) ⊆ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) ∩ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿)  and 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾 ∩ 𝐿) ⊆

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐿).  

 Proof. (i) It follows from Definition 3.5.              

Remark 3.5   By the following example we show that the inclusion relation in parts (vii) and (viii) of 

the above theorem cannot be replaced by equality.  

Example 3.4  In Example 3.1, the sets  

    1.  𝐴 = {⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.4
⟩}  and 𝐵 = {⟨

𝑠1,𝑠4

0.2
⟩ , ⟨

𝑠2

0.4
⟩ , ⟨

𝑠3

0.6
⟩} , then 𝐴 ∨ 𝐵 = {⟨

𝑠1,𝑠4

0.2
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.6
⟩} . 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2015 
https://internationalpubls.com 

𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴) = {⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.3
⟩ , ⟨

𝑠3

0.4
⟩} , 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐵) = {⟨

𝑠1,𝑠4

0.2
⟩ , ⟨

𝑠2

0.4
⟩ , ⟨

𝑠3

0.6
⟩}  and 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴 ∨ 𝐵) =

{⟨
𝑠1,𝑠4

0.2
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.6
⟩}. Thus 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴 ∨ 𝐵) < 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴) ∨ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐵).  

    2.  𝐶 = {⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.7
⟩}  and 𝐷 = {⟨

𝑠1,𝑠4

0.2
⟩ , ⟨

𝑠2

0.4
⟩ , ⟨

𝑠3

0.6
⟩} , then 𝐶 ∧ 𝐷 = {⟨

𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.4
⟩ , ⟨

𝑠3

0.6
⟩} . 

𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶) = {⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.7
⟩} , 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐷) = {⟨

𝑠1,𝑠4

0.2
⟩ , ⟨

𝑠2

0.4
⟩ , ⟨

𝑠3

0.6
⟩}  and 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶 ∧ 𝐷) =

{⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.3
⟩ , ⟨

𝑠3

0.4
⟩}.  

Thus 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶) ∧ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐷) < 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶 ∧ 𝐷). 

Theorem 3.11   Let (𝑈, 𝜏ℛ(𝐴))  be a 𝒫ℱ𝔑𝑡𝑠  and 𝐾 ⊆ 𝑆.  Then 𝐾  is a 𝒫ℱ𝔑𝑍𝑜  set iff 𝐾 =

𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾) ∪ 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾).  

 Proof. Let 𝐾 be a 𝒫ℱ𝔑𝑍𝑜 set. 

 Then 𝐾 ⊆ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))  and hence by Proposition 3.1 and 

Lemma 3.3, 

𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾) ∪ 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾) = (𝑆 ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾))) ∪ (𝑆 ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))) =

𝐾 ∩ (𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿  𝑖𝑛𝑡(𝐾)) ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾))) = 𝐾.  The Converse, it follows from 

Proposition 3.1 and Lemma 3.3.              

Proposition 3.4   Let (𝑈, 𝜏ℛ(𝐴))  be a 𝒫ℱ𝔑𝑡𝑠  and 𝐾 ⊆ 𝑆.  Then 𝐾  is a 𝒫ℱ𝔑𝑍𝑐  set iff 𝐾 =

𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾).  

 Proof. It follows from Theorem 3.11.              

Theorem 3.12   Let 𝐾 be a subset of a space (𝑈, 𝜏ℛ(𝐴)). Then:   

    (i)  𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) = 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾),  

    (ii)  𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) = 𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾) ∪ 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾).  

  Proof. (i) It is easy to see that 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) . Also, 

𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) = (𝐾 ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)) ∩ (𝐾 ∪ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾)) = 𝐾 ∪

(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾))) . Since 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)  is 𝒫ℱ𝔑𝑍𝑐𝑠 , then 

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))) ⊇

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾)). 

Thus 𝐾 ∪ (𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)) ∩ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾))) ⊆ 𝐾 ∪ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) 

and hence, 𝒫ℱ𝔑𝛿𝒮  𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾).  So, 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) = 𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) ∩

𝒫ℱ𝔑𝒫𝑐𝑙(𝐾). 

(ii) It follows from (i).              

Theorem 3.13   Let 𝐾 be a Pythagorean fuzzy subset of a space (𝑈, 𝜏ℛ(𝐴)) Then  

    (i)  𝐾 is a 𝒫ℱ𝔑𝑍𝑜 set iff 𝐾 = 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾),  

    (ii)  𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set iff 𝐾 = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾).  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2016 
https://internationalpubls.com 

 Proof. (i) It follows from Theorems 3.11 & 3.12.              

Lemma 3.5   Let 𝐾 be a 𝑝𝑓𝑠 of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then the following statement are hold:   

    (i)  𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)) = 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)),  

    (ii)  𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)) = 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾) ∪ 𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾)).  

 Proof. (i) By Lemma 3.4, 

 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)) = 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾))) = 𝒫ℱ𝔑𝒫 

𝑐𝑙(𝐾) ∩ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾 ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑  𝑐𝑙(𝐾)))) = 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) ∩

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾)). 

(ii) It follows from (i).              

Proposition 3.5   Let 𝐾 be a 𝑝𝑓𝑠 of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then:   

    (i)  𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) = 𝐾 ∪ 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)),  

    (ii)  𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) = 𝐾 ∩ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)).  

  Proof. (i) By Lemma 3.5, 𝐾 ∪ 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)) = 𝐾 ∪ (𝒫ℱ𝔑𝒫𝑐𝑙(𝐾) ∩

𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾))) = (𝐾 ∪ 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)) ∩ (𝐾 ∪ 𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾))) = 𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)  ∩

𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾). 

(ii) It follows from (i).              

Theorem 3.14   Let 𝐾 be a fuzzy subset of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then the following are equivalent:  

    (i)  𝐾 is a 𝒫ℱ𝔑𝑍𝑜 set,  

    (ii)  𝐾 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)),  

    (iii)  there exists 𝑂 ∈ 𝒫ℱ𝔑𝒫𝑂(𝐴) such that 𝑂 ⊆ 𝐾 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂),  

    (iV)  𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝐾) = 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)).  

  Proof. (i) ⇒ (ii): Let 𝐾 be a 𝒫ℱ𝔑𝑍𝑜 set. Then by Theorem 3.13, 𝐾 = 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾) and by 

Proposition 3.5, 𝐾 = 𝐾 ∩ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)) and hence, 𝐾 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)). 

(ii) ⇒  (i): Let 𝐾 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)).  Then by Proposition 3.5, 𝐾 ⊆ 𝐾 ∩

𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)) = 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾), and hence 𝐾 = 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾). Thus 𝐾 is 𝒫ℱ𝔑𝑍𝑜𝑠. 

(ii) ⇒ (iii): It follows from putting 𝑂 = 𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾), 

(iii) ⇒ (ii). Let there exists 𝑂 ∈ 𝒫ℱ𝔑𝒫𝑂(𝐴) such that 𝑂 ⊆ 𝐾 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂). Since 𝑂 ⊆ 𝐾, then 

𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂) ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)) , therefore 𝐾 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂) ⊆

𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾)). 

(ii) ⇔ (iv): It is clear.              

Theorem 3.15   Let 𝐾 be a 𝑝𝑓𝑠 of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then the following are equivalent:  

    (i)  𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set,  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2017 
https://internationalpubls.com 

    (ii)  𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)) ⊆ 𝐾,  

    (iii)  there exists 𝑂 ∈ 𝒫ℱ𝔑𝑃𝐶(𝐴) such that 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑂) ⊆ 𝐾 ⊆ 𝑂,  

    (iV)  𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝐾) = 𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾)).  

  Proof. It follows from Theorem 3.14.              

Proposition 3.6  If 𝐾 is a 𝒫ℱ𝔑𝑍𝑜 set of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)) such that 𝐾 ⊆ 𝐿 ⊆ 𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝐾), 

then 𝐿 is 𝒫ℱ𝔑𝑍𝑜.  

 Proof. It is clear.              

Definition 3.7   A set 𝐾  of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)) is said to be locally 𝒫ℱ𝔑𝑍𝑐𝑠 if 𝐾 = 𝑂 ∩ 𝑆, 

where 𝑂 ∈ 𝑂(𝐴) and 𝑆 ∈ 𝒫ℱ𝔑𝑍𝐶(𝐴).  

Theorem 3.16   Let 𝐻 be a 𝑝𝑓𝑠 of a 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then 𝐻 is locally 𝒫ℱ𝔑𝑍𝑐 iff 𝐻 = 𝑂 ∩

𝒫ℱ𝔑𝑍𝑐𝑙(𝐻).  

 Proof. Since 𝐻 is a locally 𝒫ℱ𝔑𝑍𝑐 set, then 𝐻 = 𝑂 ∩ 𝑆, where 𝑂 ∈ 𝑂(𝐴) and 𝑆 ∈ 𝒫ℱ𝔑𝑍𝐶(𝐴) 

and hence 𝐻 ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐻) ⊆ 𝒫ℱ𝔑𝑍𝑐𝑙(𝑆) = 𝑆.  Thus 𝐻 ⊆ 𝑂 ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐻) ⊆ 𝑂 ∩

𝒫ℱ𝔑𝑍𝑐𝑙(𝑆) = 𝐻. 

Therefore 𝐻 = 𝑂 ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐻). The Converse is clear.              

Theorem 3.17   Let 𝐾  be a locally 𝒫ℱ𝔑𝑍𝑐  set of a space (𝑈, 𝜏ℛ(𝐴)) . Then the following 

statements are hold: 

    (i) 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 is a 𝒫ℱ𝔑𝑍𝑐 set,  

    (ii)  (𝐾 ∪ (1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))) is a 𝒫ℱ𝔑𝑍𝑜,  

    (iii)  𝐾 ⊆ 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾 ∪ (1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))).  

 Proof. (i) If 𝐾  is a locally 𝒫ℱ𝔑𝑍𝑐  set, then there exists an 𝒫ℱ𝔑𝑜  set 𝑂  such that 𝐾 = 𝑂 ∩

𝒫ℱ𝔑𝑍𝑐𝑙(𝐾).  Hence, 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾 = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − (𝑂 ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∩

(1𝒫 − (𝑂 ∩ 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∩ (1𝒫 − 𝑂) ∪ (1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))) = 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) ∩

(1𝒫 − 𝑂) which is 𝒫ℱ𝔑𝑍𝑐. 

(ii) Since 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾  is 𝒫ℱ𝔑𝑍𝑐 , then 1𝒫 − (𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾)  is a 𝒫ℱ𝔑𝑍𝑜  set. Since 

1𝒫 − (𝒫ℱ𝔑𝑍𝑐𝑙(𝐾) − 𝐾) = ((1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) ∪ (1𝒫 ∩ 𝐾)) = (𝐾 ∪ (1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))),  then 

𝐾 ∪ (1𝒫 − 𝒫ℱ𝔑𝑍𝑐𝑙(𝐾)) is 𝒫ℱ𝔑𝑍𝑜𝑠. 

(iii) It follows from (ii).              

Definition 3.8   A Pythagorean fuzzy set 𝐾 of a space (𝑈, 𝜏ℛ(𝐴)) is said to be Pythagorean fuzzy 

nano 𝐷(𝑐, 𝑧) (briefly, 𝒫ℱ𝔑𝐷(𝑐, 𝑧)) iff 𝒫ℱ𝔑𝑖𝑛𝑡(𝐾) = 𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾).  

Remark 3.6   One may notice that the concepts of 𝒫ℱ𝔑𝑍𝑜 and 𝒫ℱ𝔑𝐷(𝑐, 𝑧) are independent and 

by we show this the following example.  

Example 3.5  In Example 3.1, the nano sets   



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2018 
https://internationalpubls.com 

    1.  {⟨
𝑠1,𝑠4

0.3
⟩ , ⟨

𝑠2

0.5
⟩ , ⟨

𝑠3

0.7
⟩} is a 𝒫ℱ𝔑𝑍𝑜𝑠 but not 𝒫ℱ𝔑𝐷(𝑐, 𝑧).  

    2.  {⟨
𝑠1,𝑠4

0.1
⟩ , ⟨

𝑠2

0.3
⟩ , ⟨

𝑠3

0.5
⟩} is a 𝒫ℱ𝔑𝐷(𝑐, 𝑧) but not 𝒫ℱ𝔑𝑍𝑜𝑠.  

Theorem 3.18   Let 𝐾 be a 𝑝𝑓𝑠 of 𝒫ℱ𝔑𝑡𝑠 (𝑈, 𝜏ℛ(𝐴)). Then the following are equivalent:   

    1.  𝐾 is an 𝒫ℱ𝔑𝑜 set,  

    2.  𝐾 is 𝒫ℱ𝔑𝑍𝑜 and 𝒫ℱ𝔑𝐷(𝑐, 𝑧).  

  Proof. Obvious.            

4  Application 

Entropy as a measure of fuzziness was first proposed by Zadeh [21]. 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 decision making. 

Definition 4.1  Let 𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜆𝐴(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 (𝜇𝐴 − 𝜆𝐴)2;  𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦`𝑥𝑖 ∈ 𝐴, where 𝜏𝑝𝑓𝑠(𝑋) denote the family of all 𝑝𝑓𝑠’s on 𝑋.  

Example 4.1  The association of the tourism wants to announce that best “Hotel of the year", for 

each year. The actual problem is, they want to select the best hotel based on reviews and ratings. There 

are four nominees namely Hotel 1, Hotel 2, Hotel 3 and Hotel 4 for this award and they have to 

reviewed based on the four criteria namely Ambiance, Good food, Clean and Tidy, Cyber security 

facility. Here we use entropy measure to find the best hotel by the overall entropy measure with the 

Pythagorean fuzzy sets. 

Table 1. Reviews of the Hotels based on the Criteria 

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

Hotel 1 (𝐻1) < 𝐻1, 𝐶1; 0.9,0.3 > < 𝐻1, 𝐶2; 0.7,0.6 > < 𝐻1, 𝐶3; 0.5,0.8 > < 𝐻1, 𝐶4; 0.6,0.4 > 

Hotel 2 (𝐻2) < 𝐻2, 𝐶1; 0.7,0.1 > < 𝐻2, 𝐶2; 0.9,0.2 > < 𝐻2, 𝐶3; 0.8,0.1 > < 𝐻2, 𝐶4; 0.6,0.3 > 

Hotel 3 (𝐻3) < 𝐻3, 𝐶1; 0.8,0.4 > < 𝐻3, 𝐶2; 0.7,0.5 > < 𝐻3, 𝐶3; 0.6,0.2 > < 𝐻3, 𝐶4; 0.7,0.5 > 

Hotel 4 (𝐻4) < 𝐻4, 𝐶1; 0.7,0.2 > < 𝐻4, 𝐶2; 0.8,0.2 > < 𝐻4, 𝐶3; 0.8,0.4 > < 𝐻4, 𝐶4; 0.6,0.6 > 

 

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

Table 2. Entropy measure of each Hotel. 

 휀𝑝𝑓𝑠(𝐻𝑖) 

𝐻1 0.87 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2019 
https://internationalpubls.com 

𝐻2 0.64 

𝐻3 0.9 

𝐻4 0.81 

 

From Table 2, Clearly that 휀𝑝𝑓𝑠(𝐻2) < 휀𝑝𝑓𝑠(𝐻4) < 휀𝑝𝑓𝑠(𝐻1) < 휀𝑝𝑓𝑠(𝐻3). 

Hence we conclude that 𝐻2 is the best Hotel of the year with less fuzziness. 

5  Conclusion 

 In this paper, we have studied a new class of sets called Pythagorean fuzzy nano 𝑍-open sets in 

Pythagorean fuzzy nano topological spaces and their properties, and also discussed about Pythagorean 

fuzzy nano 𝑍-closure, Pythagorean fuzzy nano 𝑍-interior and their relations with already existing 

well known fuzzy sets. In future, this can be extended to Pythagorean fuzzy nano 𝑍 continuous 

function, Pythagorean fuzzy nano 𝑍 open mapping, Pythagorean fuzzy nano 𝑍 closed mapping and 

Pythagorean fuzzy nano 𝑍 homeomorphic functions. 

References  

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

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

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

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

[3]  X. Arul Selvaraj and U. Balakrishna,  𝑍-open sets in nano topological spaces, AIP Conference 

Proceedings, 2364 (2021), 020037. 

[4]  K. T. Atanassov,  Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central 

Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian), (1983). 

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

[6]  D. Coker,  An introduction to intuitionistic fuzzy topological spaces, Fuzzy sets and systems,  

88 (1997), 81-89. 

[7]  R. Lowen,  Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical analysis 

and applications,  56 (3) (1976), 621–633. 

[8]  A. I. EL-Magharabi and A. M. Mubarki,  𝑍-open sets and 𝑍-continuity in topological spaces, 

International Journal of Mathematical Archive, 2 (10) (2011), 1819-1827. 

[9]  M. Lellis Thivagar and C. Richard,  On nano forms of weekly open sets, International journal of 

mathematics and statistics invention,  1 (1) (2013), 31-37. 

[10]  M. Lellis Thivagar, S. Jafari, V. Sutha Devi and V. Antonysamy,  A novel approach to nano 

topology via neutrosophic sets, Neutrosophic Sets and Systems,  20 (2018), 86-94. 

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



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2020 
https://internationalpubls.com 

spaces, Complex & Intelligent Systems, 177-183. https://doi.org/10.1007/s40747-019-0095-2. 

[12]  A. Padma, M. Saraswathi, A. Vadivel and G. Saravanakumar,  New Notions of Nano 𝑀-open 

Sets, Malaya Journal of Matematik, S (1) (2019), 656-660. 

[13]  V. Pankajam and K. Kavitha,  𝛿 -open sets and 𝛿 -nano continuity in 𝛿 -nano topological 

spaces, International Journal of Innovative Science and Research Technology,  2 (12) (2017), 

110-118. 

[14]  Shiventhiradevi Sathaananthan, S. Tamilselvan, A. Vadivel and G. Saravanakumar,  Fuzzy 𝒵 

closed sets in double fuzzy topological spaces, AIP Conf Proc., 2277 (2020), 090001. 

[15]  Shiventhiradevi Sathaananthan, A. Vadivel, S. Tamilselvan and G. Saravanakumar,  

Generalized fuzzy 𝒵 closed sets in double fuzzy topological spaces, Adv. Math: Sci. J., 9 (4) 

(2020), 2107-2112. 

[16]  R. Thangammal, M. Saraswathi, A. Vadivel and C. John Sundar, Fuzzy Nano 𝑍-open Sets in 

Fuzzy Nano Topological Spaces, Journal of Linear and Topological Algebra Vol. 11, No. 01, 

2022, 27- 38. 

[17]  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. 

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

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

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

congress 𝑁𝐴𝐹𝐼𝑃𝑆 annual meeting, 57-61. 

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

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

[21]  L. A. Zadeh,  Fuzzy sets, Information and control,  8 (1965), 338-353.