Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

290 
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Pythagorean Neutrosophic Cubic Set 

 

1Berna Joyce L, 2Dr.Elvina Mary L 
1, 2PG and Research Department of Mathematics, Nirmala College for Women, Coimbatore, India. 

Email Id: bernajoyceslr@gmail.com1, corresponding author: elvinalawrence07@gmail.com 

 

Article History: 

Received: 19-10-2024 

Revised: 03-12-2024 

Accepted: 11-12-2024 

Abstract:  

The purpose of this work is to broaden the definition of Neutrosophic Cubic Set 

(NCS) and Pythagorean Cubic Set (PCS) to Pythagorean Neutrosophic Cubic Set 

(PNCS). Related qualities are examined and the concepts of T-external, I-external, and 

F-external Pythagorean Neutrosophic Cubic Set (PNCS) and T-internal, I-internal, and 

F-internal PNCS are conveyed. 

Keywords: Pythaogrean Neutrosophic Cubic Set Pythagorean Neutrosophic Internal 

Cubic Set, Pythagorean Neutrosophic External Cubic Set.  

 

1. Introduction 

Zadeh[8] established a foundation for fuzzy mathematics in 1965. Soon after, in 1975, Zadeh[8] 

proposed a reconfiguration of fuzzy sets with interval valued function membership. Neutrosophic 

Sets were initially defined in 1995 by Florentin Smarandache[5]. This enables ambiguity and 

uncertainty to be handled more thoroughly. In the year 2012, the introduction of the important theory 

of Cubic Sets was defined by Jun et al[2]. In 2017, Chang Su Kim, Florentin Smarandache, and 

Young Bae Jun[3] presented the an idea of Neutrosophic Cubic Sets. Yagar[7] first presented the 

Pythagorean Fuzzy set's evolution in 2013. In 2019, F. Khana, M. S. Ali Khana, M. Shahzada, and S. 

Abdullah[4] gave a concept of  the Pythagorean Cubic Fuzzy Set. R. Jhansi & K. Mohana[1] 

proposed the Neutrosophic Pythagorean Sets. The interval-valued Neutrosophic Pythagorean Sets 

were presented by Stephy et al[6]. This article tries to establish an innovative concept known as 

Pythagorean Neutrosophic Cubic Sets (PNCS) which is a combination of PNS and PNIVS. 

2. Preliminaries 

Definition 2.1[2] If Ι̂ ≠ 𝜙. Then a cubic set, �̂�𝐶𝑆 = {〈𝑖̂, �̂�𝐹𝐼𝑆(𝑖̂), �̂�𝐹𝑆(𝑖̂)〉: 𝑖̂ ∈ Ι̂} where �̂�𝐹𝐼𝑆(𝑖̂) is 

IVFS,  �̂�𝐹𝑆(𝑖̂) is a fuzzy set.  

Definition 2.2[2]  Let Ι̂ ≠ 𝜙. A cubic set �̂�𝐶𝑆 = 〈�̂�𝐹𝐼𝑆(𝑖̂), �̂�𝐹𝑆(𝑖̂)〉 in Ι̂ is called an ICS if �̂�𝐹𝐼𝑆
− (𝑖̂) ≤

�̂�𝐹𝑆(𝑖̂) ≤ �̂�𝐹𝐼𝑆
+ (𝑖̂) for all 𝑖̂ ∈ Ι̂. 

Definition 2.3 [2] If Ι̂ ≠ 𝜙 set. A cubic set �̂�𝐶𝑆 = 〈�̂�𝐹𝐼𝑆(𝑖̂), ℓ̂𝐹𝑆(𝑖̂)〉 in X is called an ECS if �̂�𝐹𝑆(𝑖̂) ∉

(�̂�𝐹𝐼𝑆
− (𝑖̂), �̂�𝐹𝐼𝑆

+ (𝑖̂)) . 

Definition 2.4[1] Let Ι̂ ≠ 𝜙. A PNS in Ι̂ 𝑖𝑠 �̂�𝑃𝑁𝑆 = {〈𝑖̂, 𝒯�̂�(𝑖̂), ℐ�̂�(𝑖)̂, ℱ�̂�(𝑖̂)〉: 𝑖̂ ∈ Ι̂} where 

𝒯𝐴(𝑖)̂, ℐ�̂�(𝑖̂), ℱ�̂�(𝑖̂): Ι̂ → [0,1] & 0 ≤ (𝒯�̂�(𝑖̂))
2

+ (ℐ�̂�(𝑖̂))
2

+ (ℱ�̂�(𝑖̂))
2

≤ 2 where 𝒯�̂�(𝑖̂) denotes the 

degree of membership, , ℐ�̂�(𝑖̂) denotes the degree of indeterminacy and ℱ�̂�(𝑖)̂ denotes the degree of 

non-membership. 



Communications on Applied Nonlinear Analysis 

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Definition 2.5[6] Let Ι̂ ≠ 𝜙, PNIVS is of the form, 

�̂�𝑃𝑁𝐼𝑉𝑆{〈𝑖̂, 𝒯�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆), ℐ�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆), ℱ�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆)〉: 𝑖̂𝑃𝑁𝐼𝑉𝑆Ι̂} 

where𝒯�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆)[𝒯�̂�
−(𝑖̂𝑃𝑁𝐼𝑉𝑆), 𝒯�̂�

+(𝑖̂𝑃𝑁𝐼𝑉𝑆)], ℐ�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆) = [ℐ�̂�
−(𝑖̂𝑃𝑁𝐼𝑉𝑆), ℐ�̂�

+(𝑖̂𝑃𝑁𝐼𝑉𝑆)] and ℱ�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆) =

[ℱ�̂�
−(𝑖̂𝑃𝑁𝐼𝑉𝑆), ℱ�̂�

+(𝑖�̂�𝑁𝐼𝑉𝑆)]. Consider the mapping 𝒯�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆), ℐ�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆), ℱ�̂�(𝑖̂𝑃𝑁𝐼𝑉𝑆): Ι̂ → [0,1] and  

0 ≤ [ 
𝒯

�̂�
−(�̂�𝑃𝑁𝐼𝑉𝑆)+𝒯

�̂�
+(�̂�𝑃𝑁𝐼𝑉𝑆)

2
]

2

+ [ 
ℐ

�̂�
−(�̂�𝑃𝑁𝐼𝑉𝑆)+ℐ

�̂�
+(�̂�𝑃𝑁𝐼𝑉𝑆)

2
]

2

+ [ 
ℱ

�̂�
−(�̂�𝑃𝑁𝐼𝑉𝑆)+ℱ

�̂�
+(�̂�𝑃𝑁𝐼𝑉𝑆)

2
]

2

≤ 2  

3. Pythagorean Neutrosophic Cubic Set (PNCS) 

Definition 3.1 Let �̂� ≠ 𝜙, a Pythagorean Neutrosophic Cubic Set (PNCS), having a form, �̂�𝑃𝑁𝐶𝑆 =

{〈𝜄,̂ �̂�𝑃𝑁𝐼𝑉𝑆(𝜄 ̂̂), 𝜛𝑃𝑁𝑆(𝜄)̂〉: 𝜄 ̂ ∈ Ι̂} where �̂�𝑃𝑁𝐼𝑉𝑆(𝜄)̂ represent the Pythagorean Neutrosophic Interval 

valued set in Ι̂. 𝜛𝑃𝑁𝑆(𝜄)̂ represent the Pythagorean Neutrosophic Set. PNCS can be denoted as a pair 

�̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛). 

Example 3.2 If �̂� = {ℎ, 𝑟, 𝑣} then �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛)with the tabular representation given below, 

�̂� �̂�𝑃𝑁𝐼𝑉𝑆(𝜄 ̂̂) 𝜛𝑃𝑁𝑆(𝜄)̂ 

   h ([0.2,0.3], [0.3,0.5], [0.4,0.6]) (0.1,0.2,0.3) 

r ([0.1,0.4], [0.3,0.7], [0.4,0.8]) (0.3,0.2,0.7) 

v ([0.2,0.3], [0.4,0.8], [0.3,0.5]) (0.5,0.2,0.3) 

     Table 1 – Example of PNCS 

Therefore, �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) �̂�𝑃𝑁𝐶𝑆 = {

〈ℎ, ([0.2,0.3], [0.3,0.5], [0.4,0.6]), (0.1,0.2,0.3)〉,
〈𝑟, ([0.1,0.4], [0.3,0.7], [0.4,0.8]), (0.3,0.2,0.7)〉,
〈𝑣, ([0.2,0.3], [0.4,0.8], [0.3,0.5]), (0.5,0.2,0.3)〉 

} The pair 

�̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is called PNCS. 

Definition 3.3 Let �̂� ≠ 𝜙. A PNCS  �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) in �̂�is said to be �̂�-internal, 

(𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕋𝒟

+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�.  

𝐼-internal if (𝕀𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕀𝒟

+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�. �̂�-internal if (𝔽𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤

𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ≤ 𝔽𝒟
+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�. If a PNCS �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛)in �̂� adheres to the 

aforementioned shortcomings then, �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is an IPNCS. 

Example 3.4 

Let Ι̂ = {ℎ, 𝑟, 𝑣}. Then the pair �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) with the tabular representation given below, 

�̂� �̂�𝑃𝑁𝐼𝑉𝑆(�̂�𝑃𝑁𝐶𝑆) 𝜛𝑃𝑁𝑆(�̂�𝑃𝑁𝐶𝑆) 

h ([0.1,0.3], [0.4,0.6], [0.5,0.8]) (0.25,0.48,0.62) 

r ([0.1,0.4], [0.3,0.7], [0.4,0.8]) (0.30,0.58,0.71) 

v ([0.2,0.3], [0.4,0.8], [0.3,0.5]) (0.27,0.65,0.47) 

     Table 2 – Example of IPNCS 



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Therefore, �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛),  �̂�𝑃𝑁𝐶𝑆 = {

〈ℎ, ([0.1,0.3], [0.4,0.6], [0.5,0.8]), (0.25,0.48,0.62)〉,
〈𝑟, ([0.1,0.4], [0.3,0.7], [0.4,0.8]), (0.30,0.58,0.71)〉,
〈𝑣, ([0.2,0.3], [0.4,0.8], [0.3,0.5]), (0.27,0.65,0.47)〉 

} 

The pair �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) satisfies, (𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕋𝒟

+(�̂�𝑃𝑁𝐶𝑆)), (𝕀𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤

𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕀𝒟
+(�̂�𝑃𝑁𝐶𝑆)),(𝔽𝒟

−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ≤ 𝔽𝒟
+(�̂�𝑃𝑁𝐶𝑆)). Then the pair �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) 

is called IPNCS. 

Definition 3.5 Let �̂� ≠ 𝜙. A PNCS �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) in �̂� is said to be �̂�-external, 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∉

(𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆), 𝕋𝒟

+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�, 𝐼-external if   

𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ∉ (𝕀𝒟
−(�̂�𝑃𝑁𝐶𝑆), 𝕀𝒟

+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂� &  �̂�-external if  

𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ∉ (𝔽𝒟
−(�̂�𝑃𝑁𝐶𝑆), 𝔽𝒟

+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�. If a PNCS satisfies the above inequalities then, 

�̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is an EPNCS. 

Example 3.6 Let Ι̂ = {ℎ, 𝑟, 𝑣}. Then the pair �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) with the tabular representation given 

below, 

�̂� �̂�𝑃𝑁𝐼𝑉𝑆(�̂�𝑃𝑁𝐶𝑆) 𝜛𝑃𝑁𝑆(�̂�𝑃𝑁𝐶𝑆) 

h ([0.1,0.3], [0.4,0.6], [0.5,0.8]) (0.4,0.7,0.3) 

    r ([0.1,0.4], [0.3,0.7], [0.4,0.8]) (0.5, 0.2, 0.9) 

   v ([0.2,0.3], [0.6,0.8], [0.3,0.5]) (0.1, 0.5, 0.6) 

     Table 2 – Example of EPNCS 

Therefore,�̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) �̂�𝑃𝑁𝐶𝑆 = {

〈ℎ, ([0.1,0.3], [0.4,0.6], [0.5,0.8]), (0.4,0.7,0.3)〉,
〈𝑟, ([0.1,0.4], [0.3,0.7], [0.4,0.8]), (0.5, 0.2, 0.9)〉,
〈𝑣, ([0.2,0.3], [0.6,0.8], [0.3,0.5]), (0.1, 0.5, 0.6)〉 

}. Then the 

pair �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛)is called EPNCS. 

Theorem 3.7 

Let  �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be a PNCS in �̂� ≠ 𝜙 which is not EPNCS. Then there exists �̂�𝑃𝑁𝐶𝑆 ∈ Ι̂ such 

that 𝜛𝑃𝑁𝑆(�̂�𝑃𝑁𝐶𝑆) ∈ �̂�𝑃𝑁𝐼𝑉𝑆(�̂�𝑃𝑁𝐶𝑆).  

Proof:  Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS. Given that �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is not external. Then �̂�𝑃𝑁𝐶𝑆 

does not satisfies the following conditions, i.e., 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∉ (𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆),

𝕋𝒟
+(�̂�𝑃𝑁𝐶𝑆)) , 𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ∉ (𝕀𝒟

−(�̂�𝑃𝑁𝐶𝑆), 𝕀𝒟
+(�̂�𝑃𝑁𝐶𝑆)) , 𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ∉

(𝔽𝒟
−(�̂�𝑃𝑁𝐶𝑆), 𝔽𝒟

+(�̂�𝑃𝑁𝐶𝑆)) ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂� . Then �̂�𝑃𝑁𝐶𝑆 must be IPNCS.∴ 𝜛𝑃𝑁𝑆(�̂�𝑃𝑁𝐶𝑆) ∈

�̂�𝑃𝑁𝐼𝑉𝑆(�̂�𝑃𝑁𝐶𝑆). 

Theorem 3.8 



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Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS in �̂� ≠ 𝜙 which is not T-IPNCS then exists �̂�𝑃𝑁𝐶𝑆 ∈ �̂�, 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∉

𝕋𝒟(�̂�𝑃𝑁𝐶𝑆). 

Proof 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS. Given that �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is not T-IPNCS. Then �̂�𝑃𝑁𝐶𝑆 does not 

satisfies, (𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕋𝒟

+(�̂�𝑃𝑁𝐶𝑆)). Therefore,𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∉ 𝕋𝒟(�̂�𝑃𝑁𝐶𝑆) 

Theorem 3.10 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS in �̂� ≠ 𝜙 which is not I-IPNCS then exists �̂�𝑃𝑁𝐶𝑆 ∈ �̂�, 𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ∉

𝕀𝒟(�̂�𝑃𝑁𝐶𝑆) 

Proof 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS. Given that �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is not I-IPNCS. Then �̂�𝑃𝑁𝐶𝑆 does not 

satisfies, (𝕀𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕀𝒟

+(�̂�𝑃𝑁𝐶𝑆)). Therefore,𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ∉ 𝕀𝒟(�̂�𝑃𝑁𝐶𝑆) 

Theorem 3.11 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS in �̂� ≠ 𝜙 which is not F-IPNCS then exists �̂�𝑃𝑁𝐶𝑆 ∈ �̂�, 𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ∉

𝔽𝒟(�̂�𝑃𝑁𝐶𝑆). 

Proof 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS. Given that �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is not F-IPNCS. Then �̂�𝑃𝑁𝐶𝑆 does not 

satisfies, (𝔽𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤ 𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ≤ 𝔽𝒟

+(�̂�𝑃𝑁𝐶𝑆)). Therefore,𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ∉ 𝔽𝒟(�̂�𝑃𝑁𝐶𝑆). 

Theorem 3.12 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS �̂� ≠ 𝜙. If �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is both T-IPNCS and T-EPNCS then 

(∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�) (𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∈ {𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�} ∪ {𝕋𝒟

+(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�}) 

Proof 

Let�̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS. If �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is T-IPNCS then ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�, (𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆) ≤

𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ≤ 𝕋𝒟
+(�̂�𝑃𝑁𝐶𝑆)). If �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is T-EPNC then ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�  𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∉

(𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆), 𝕋𝒟

+(�̂�𝑃𝑁𝐶𝑆))Then it indicates ∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�,  𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) = 𝕋𝒟
−(�̂�𝑃𝑁𝐶𝑆) or 𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) =

𝕋𝒟
+(�̂�𝑃𝑁𝐶𝑆) and so that  (𝜛𝕋(�̂�𝑃𝑁𝐶𝑆) ∈ {𝕋𝒟

−(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�} ∪ {𝕋𝒟
+(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�})  

Theorem 3.13 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS �̂� ≠ 𝜙. If �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is both I-IPNCS and I-EPNCS then 

(∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�) (𝜛𝕀(�̂�𝑃𝑁𝐶𝑆) ∈ {𝕀𝒟
−(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�} ∪ {𝕀𝒟

+(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�}). 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

294 
https://internationalpubls.com 

Proof: Forthright. 

Theorem 3.14 

Let �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) be PNCS �̂� ≠ 𝜙. If �̂�𝑃𝑁𝐶𝑆 = (�̂�, 𝜛) is both F-IPNCS and F-EPNCS then 

(∀�̂�𝑃𝑁𝐶𝑆 ∈ �̂�) (𝜛𝔽(�̂�𝑃𝑁𝐶𝑆) ∈ {𝔽𝒟
−(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�} ∪ {𝔽𝒟

+(�̂�𝑃𝑁𝐶𝑆)|�̂�𝑃𝑁𝐶𝑆 ∈ �̂�}). 

Proof: Forthright. 

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