Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2442 
https://internationalpubls.com 

Contra Continuous and Open Maps via Neutrosophic Soft Z – Open 

Sets 

 
1B. Vijayalakshmi* and 2S. Madhunika 

 

             1Assistant Professor, Department of Mathematics, Annamalai University,  Annamalai Nagar - 608002, 

Tamilnadu, India. 

             email: mathvijaya2006au@gmail.com 

          2Research Scholar, Department of Mathematics, Annamalai University, Annamalai Nagar - 608002, Tamilnadu, 

India. 

           email: madhunika2020@gmail.com 

Correspponding author: mathvijaya2006au@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: This paper investigates the concepts of contra Z-continuous, contra Z-

irresolute, contra Z-open and contra Z-closed maps in neutrosophic soft topological 

spaces. We also explore the notions of contra Z and Z-C homeomorphisms. Theoretical 

results are presented with examples and theorems, enhancing the understanding of 

these mappings within the framework of neutrosophic soft topology. 

Keywords: contra z-continuous maps, contra z-irresolute maps, contra z-open maps, 

contra z-closed maps, contra z homeomorphism and contra z-c homeomorphism. 

 

1. Introduction 

The foundational framework of fuzzy sets, introduced by Lofti A.Zadeh [20] in 1965, offers a 

powerful mathematical framework to handle the complexities that arise from ambiguity in practical, 

real world scenarios. This concept has been employed across various fields, including economics, 

sociology and medical science, where researchers frequently encounter vague, imprecise and 

occasionally incomplete information. These fields utilize fuzzy sets and fuzzy logic to model 

uncertain data for a range of specialized purposes. Standard fuzzy sets are defined by their 

membership value or degree of membership, although assigning this value can sometimes be 

challenging. Chang [6] in 1968 introduced fuzzy sets into topology under the framework known as 

fuzzy topological spaces. 

Building on this foundational idea, in the 1986, K T. Atanassov [2] introduced intuitionstic 

fuzzy sets, which build on fuzzy sets by including a non-membership degree in addition to degree of 

membership. Coker [7] in 1997 introduced intuitionistic fuzzy sets into the realm of topology, 

defining them as intuitionstic fuzzy topological spaces. Intuitionistic fuzzy sets are limited to 

managing incomplete information by considering both membership and non-membership values. 

However, they do not address uncertain and contradictory information often found in belief systems. 

To tackle these issues, Florentin Smarandache [17] introduced the concept of neutrosophic set in 

2005, which serves as a mathematical framework for dealing with imprecise, indeterminate, and 

inconsistent data. In 2012, Salama and Alblowi [14] proposed the concept of neutrosophic 

topological spaces. In 1999, Molodstov [12] initiated the soft set principle as a versatile 

mathematical approach that addresses parameterization issues and surpasses the limitations of other 

uncertainty theories. This theory is highly practical, efficient and widely applicable across different 

disciplines. Molodstov's implementations of soft set principle include domains such as function 

mailto:madhunika2020@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2443 
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analysis, decision making, operational research and integrative mathematics among others. As a 

result, soft set theory has gained significant traction and continuous to advance rapidly in diverse 

fields.     

In 2011, Shabir and Naz [16] initiated the notion of soft topological spaces. Subsequently, in 

2013 Maji [10] introduced the neutrosophic soft set concept, which inspired numerous 

mathematicians to explore its applications in various mathematical frameworks. Modification by 

Deli and Broumi [8] further refined this framework, while Bera and Mahapatra [3] explored its 

algebric structures. In 2011, A. I. EI-Magharabi and A. M. Mubarki [9] introduced Z-open sets in 

topological spaces. In 2020, A. Vadivel et al [18] proposed Z-open sets in neutrosophic topological 

spaces. 

This paper primarily aims to introduce and explore the concepts of contra Z-continuous maps, 

contra Z-irresolute maps, contra Z-open maps, and contra Z-closed maps in neutrosophic soft 

topological spaces, using neutrosophic soft Z-open sets. We analyze and discuss their fundamental 

properties, along with the notions of contra Z homeomorphisms and Z-C homeomorphisms, 

providing examples and theorems that contribute to further research in neutrosophic soft topology. 

 

2. Preliminaries 

This section offers a summary of essential definitions refers to neutrosophic sets, soft sets and 

neutrosophic soft sets to ensure thorough understanding. 

 

Definition 2.1 [15] Let 𝕎 be an underlying universe. A neutrosophic set (in short, NS) D is an 

object having the form D = {〈𝑤, 𝜇𝐷(𝑤), 𝜎𝐷(𝑤), 𝜈𝐷(𝑤)〉  ∶  𝑤 ∈  𝕎} where 𝜇𝐷  → [0, 1] denote the 

degree of membership function, 𝜎𝐷  → [0, 1] denote the degree of inderterminacy function and 𝜈𝐷  →
[0, 1] denote the degree of non-membership function respectively of each element 𝑤 ∈  𝕎 to the set 

D and 0 ≤   𝜇𝐷(𝑤) + 𝜎𝐷(𝑤) + 𝜈𝐷(𝑤) ≤ 3 for each 𝑤 ∈  𝕎. 

 

Definition 2.2 [12] Assume that 𝕎 is the underlying universe & let 𝜚 is a parameter set. Let 𝒫(𝕎) 

represent the collection of all neutrosophic sets within 𝕎. A pair (D, 𝜚) is known as the soft 

set(shortly, SS) over 𝕎, where D is a mapping D : 𝜚 → 𝒫(𝕎). In other terms, a soft set can be 

viewed as a collection of subsets of the set 𝕎, each associated with a specific parameter. 

 

Definition 2.3 [8] Assume that 𝕎 is the underlying universe & let 𝜚 is a parameter set. Let 𝒫(𝕎) 

represent the collection of all neutrosophic sets within  𝕎. Then a neutrosophic soft set 

(𝑆, 𝜚)  over  𝕎 (shortly, NSS) is characterized by (𝑆, 𝜚) =  {(𝜑, 〈휀, 𝜇𝑆(𝜑)(휀), 𝜎𝑆(𝜑)(휀), 𝜈𝑆(𝜑)(휀)〉  ∶

 휀 ∈  𝕎)  ∶  𝜑 ∈ 𝜚}, where  𝜇𝑆(𝜑)(휀), 𝜎𝑆(𝜑)(휀), 𝜈𝑆(𝜑)(휀)  ∈ [0, 1] are respectively called the degree 

of membership function, the degree of indeterminacy function and the degree of non-membership 

function of  𝑆(𝜑). As the maximum value for each of  𝜇, 𝜎, 𝜈 is 1. 

      The inequality 0 ≤   𝜇𝑆(𝜑)(휀) + 𝜎𝑆(𝜑)(휀) + 𝜈𝑆(𝜑)(휀) ≤ 3 naturally holds. 

 

Definition 2.4 [[10], [4]] Assume that 𝕎 is an underlying universe & NS sets (𝑆, 𝜚) & (𝐷, 𝜚) are in 

the form  

      (𝑆, 𝜚) = {(𝜑, 〈휀, 𝜇𝑆(𝜑)(휀), 𝜎𝑆(𝜑)(휀), 𝜈𝑆(𝜑)(휀)〉  ∶  휀 ∈  𝕎)  ∶  𝜑 ∈ 𝜚} & 

      (𝐷, 𝜚) = {(𝜑, 〈휀, 𝜇𝐷(𝜑)(휀), 𝜎𝐷(𝜑)(휀), 𝜈𝐷(𝜑)(휀)〉  ∶  휀 ∈  𝕎)  ∶  𝜑 ∈ 𝜚}, then 



Communications on Applied Nonlinear Analysis 

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1. 0(𝕎,𝜚) = {(𝜑, 〈휀, 0, 0, 1〉: 휀 ∈ 𝕎): 𝜑 ∈ 𝜚} and 1(𝕎,𝜚) = {(𝜑, 〈휀, 1, 1, 0〉 ∶ 휀 ∈ 𝕎): 𝜑 ∈ 𝜚}. 

2. (𝑆, 𝜚)  ⊆  (𝐷, 𝜚) iff 𝜇𝑆(𝜑)(휀) ≤ 𝜇𝐷(𝜑)(휀), 𝜎𝑆(𝜑)(휀)  ≤  𝜎𝐷(𝜑)(휀) and 𝜈𝑆(𝜑)(휀)  ≥  𝜈𝐷(𝜑)(휀) ∶

 휀 ∈  𝕎 ∶   𝜑 ∈ 𝜚. 

3. (𝑆, 𝜚) = (𝐷, 𝜚) iff  (𝑆, 𝜚) ⊆ (𝐷, 𝜚) and (𝐷, 𝜚) ⊆  (𝑆, 𝜚). 

4. (𝑆, 𝜚)𝑐 =  {(𝜑, 〈 휀,  𝜈𝑆(𝜑)(휀), 1 − 𝜎𝑆(𝜑)(휀),  𝜇𝑆(𝜑)(휀)〉 ∶ 휀 ∈ 𝕎)  ∶  𝜑 ∈ 𝜚}. 

5. (𝑆, 𝜚) ∪ (𝐷, 𝜚) = {(𝜑, 〈 휀, max(𝜇𝑆(𝜑)(휀), 𝜇𝐷(𝜑)(휀)),  max(𝜎𝑆(𝜑)(휀), 𝜎𝐷(𝜑)(휀)),

min( 𝜈𝑆(𝜑)(휀), 𝜈𝐷(𝜑)(휀))〉 ∶ 휀 ∈ 𝕎) ∶  𝜑 ∈ 𝜚}. 

6. (𝑆, 𝜚) ∩ (𝐷, 𝜚) = {(𝜑, 〈 휀, min(𝜇𝑆(𝜑)(휀), 𝜇𝐷(𝜑)(휀)),  min(𝜎𝑆(𝜑)(휀), 𝜎𝐷(𝜑)(휀)),

max( 𝜈𝑆(𝜑)(휀), 𝜈𝐷(𝜑)(휀))〉 ∶ 휀 ∈ 𝕎) ∶  𝜑 ∈ 𝜚}. 

 

Definition 2.5 [4] A neutrosophic soft topology (in short, NSt) on an underlying universe 𝕎 is a 

collection of  𝜏 of NS subsets (𝑆, 𝜚) of  𝕎 where 𝜚 be the parameters set, satisfying 

1. 0(𝕎 ,𝜚) ,   1(𝕎 ,𝜚) ∈ 𝜏. 

2. [(𝑆, 𝜚) ∩ (𝐷, 𝜚)] ∈ 𝜏 for any (𝑆, 𝜚), (𝐷, 𝜚) ∈ 𝜏. 

3. ⋃  (𝑆, 𝜚)𝑘 ∈ 𝜏k∈K  for all (𝑆, 𝜚𝑘) ∶ k ∈ K ⊆ 𝜏. 

          Then (𝕎, 𝜏, 𝜚) is known as a neutrosophic soft topological space (shortly, NSts) and the 

elements of  𝜏  elements are known as neutrosophic soft open sets (shortly, NSOS) in  𝕎. A NSS 
(𝑆, 𝜚) is called the neutrosophic soft closed set (in short, NSCS) if its complement (𝑆, 𝜚)𝑐 is NSOS. 

 

Definition 2.6 [4] Let (𝕎, τ, ϱ) act as a NSts on 𝕎 & let (S, ϱ) is a NSS on 𝕎. The neutrosophic soft 

interior of (S, ϱ) (in brief, NSint(S, ϱ)) and the neutrosophic soft closure of (S, ϱ) (in brief, 

NScl(S, ϱ)) are represented as 

(i) NSint(𝑆, 𝜚) =  ⋃{(𝐷, 𝜚) : (𝐷, 𝜚) ⊆ (𝑆, 𝜚) and (𝐷, 𝜚) is a NSOS in 𝕎}. 
(ii) NScl(𝑆, 𝜚) =  ⋂{(𝐷, 𝜚) : (𝐷, 𝜚) ⊇ (𝑆, 𝜚) and (𝐷, 𝜚) is a NSCS in 𝕎}. 

 

Definition 2.7 [4] Suppose (𝕎, τ, ϱ) act as a NSts on 𝕎 & let (S, ϱ) is a NSS on 𝕎. Then (S, ϱ) is 

called the NS 

 

     (i) regular-open set (in short, NSROS) if (S, 𝜚) = NSint(NScl(S, 𝜚)). 

     (ii) pre-open set (briefly, NSPOS) if (S, ϱ) ⊆  NSint(NScl(S, ϱ)). 

     (iii) semi-open set (briefly, NSSOS) if (S, ϱ) ⊆  NScl(NSint(S, ϱ)). 

     (iv) 𝛼-open set (shortly, NS𝛼OS) if (S, ϱ) ⊆ NSint(NScl(NSint(S, ϱ))). 

     (v) 𝛽 −open set (shortly, NS𝛽OS) if (S, ϱ) ⊆  NScl(NSint(NScl(S, ϱ))). 

     The complement of a NSROS(resp. NSPOS, NSSOS, NS𝛼OS, NS𝛽OS) is called a neutrosophic 

soft regular (resp. pre, semi, 𝛼, 𝛽) closed set (shortly, NSRCS(resp. NSPCS, NSSCS, NS𝛼CS, 

NS𝛽CS)) in 𝕎. 

    The family of all NSROS(resp. NSRCS, NSPOS, NSPCS, NSSOS, NSSCS, NS𝛼OS, NS𝛼CS, 

NS𝛽OS, NS𝛽CS) of 𝕎 is represented by NSROS(𝕎) (resp. NSRCS(𝕎), NSPOS(𝕎) NSPCS(𝕎), 

NSSOS(𝕎), NSSCS(𝕎), NS𝛼OS(𝕎), NS𝛼CS(𝕎), NS𝛽OS(𝕎), NS𝛽CS(𝕎)). 

 

 

 



Communications on Applied Nonlinear Analysis 

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Definition 2.8 [1]  Let (D, ϱ) be a NSts. Then 

(i) neutrosophic soft  δ-interior of (D, ϱ) (in short, NSδint(D, ϱ)) is defined by  

NSδint(D, ϱ) = ⋃{(D, ϱ) ∶ (S, ϱ)  ⊆ (D, ϱ)  and (S, ϱ)  is a NSROS in 𝕎 }  

(ii) neutrosophic soft δ-closure of (D, ϱ) (in short, NSδcl(D, ϱ)) is defined by           

NSδcl(D, ϱ)  =  ⋂{(S, ϱ) ∶ (S, ϱ) ⊇ (D, ϱ) & (S, ϱ)  is a NSRCS in 𝕎 } 

 

Definition 2.9 [1] A NSS (D, ϱ)  is referred as the neutrosophic soft δ-open set(shortly, NSδOS) if 

(D, ϱ)  = NSδint(D, ϱ). 

The complement of NSδOS is called NSδCS. 

 

Definition 2.10  [13] A NSS  (D, ϱ) is called the neutrosophic soft  

(i) δ-semiopen set (in short, NSδSOS) if (D, ϱ) ⊆ NScl(NSδint(D, ϱ)). 

(ii) e-open set (briefly, NSeOS) if (D, ϱ) ⊆ NScl(NSδint(D, ϱ)) ∪ NSint(NSδcl(D, ϱ)). 

  

The complement of NSδSOS and NSeOS is called NSδSCS and NSeCS. 

 

Throughout this paper, Let ( 𝕎, 𝜏, ϱ) be any NSts. Let (S, ϱ) & (D, ϱ)  be a neutrosophic soft sets in 

NSts. 

 

3.  Neutrosophic soft contra Z - continuous maps  

 

Definition 3.1 A mapping  𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is said to be a neutrosophic soft contra Z-

continuous (shortly, NSContraZCts) if the inverse image of each NSOS of (𝕋, σ, ϱ) is NSZCS in   

(𝕎, τ, ϱ). 

Example 3.1 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1,ϱ), (𝑆2,ϱ) and (𝑆3,ϱ) in 𝕎 and (𝑉1, ϱ)  in 𝕋  are defined as 

(𝑆1,e1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉 

(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.4
) , (

𝜇𝑤2

0.7
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.3
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.3
)〉  



Communications on Applied Nonlinear Analysis 

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(𝑉1, 𝑒1) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.4
) , (

𝜇𝑡2

0.7
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.3
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.3
)〉  

  Then, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑉1, ϱ)}.  Let 𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ)  be an identity mapping,   then 

1) 𝒢 is a NSContraZCts function. 

2) 𝒢 is a NSContraCts but not NSContraδCts, because the set 𝒢−1(𝑉1, ϱ) = (𝑆4, ϱ) is a NSCS but 

not NSδCS. 

Preposition 3.1  The statements hold true but not the converse. 

 

a)  Each NSContraδCts is a NSContraCts. 

b)  Each NSContraCts is a NSContraδSCts. 

c)  Each NSContraCts  is a NSContraPCts . 

d)  Each NSContraδSCts is a NSContraZCts. 

e)  Each NSContraPCts  is a NSContraZCts. 

f)  Each NSContraZCts  is a NSContraeCts. 

 

Proof. Consider the map 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ). 

(a) Let (𝑆, ϱ) be a NSOS in 𝕋. As 𝒢 is NSContraδCts, 𝒢−1(𝑆, ϱ) is a NSδCS in 𝕎. Since all NSδCS 

are NSCS,  𝒢−1(𝑆, ϱ) is NSCS in 𝕎. Thus, 𝒢 is a NSContraCts. 

 (b) Let (𝑆, ϱ) be a NSOS in 𝕋. As 𝒢 is NSContraCts, 𝒢−1(𝑆, ϱ) is a NSCS in 𝕎. Since all  NSCS are 

NSδCS, 𝒢−1(𝑆, ϱ) is a NSδSCS in 𝕎. Thus, 𝒢 is a NSContraδSCts. 

 (c) Let (𝑆, ϱ) be a NSOS in 𝕋. As 𝒢 is NSContraCts, 𝒢−1(𝑆, ϱ) is a NSCS in 𝕎. Since all NSCS is a 

NSPCS, 𝒢−1(𝑆, ϱ) is a NSPCS in 𝕎. Thus, 𝒢 is a NSContraPCts. 

(d) Let (𝑆, ϱ) be a NSOS in 𝕋. As 𝒢 is NSContraδSCts, 𝒢−1(𝑆, ϱ) is a NSδSCS in 𝕎. Since all 

NSδSCS is a NSZCS,  𝒢−1(𝑆, ϱ) is a NSZCS in 𝕎. Thus, 𝒢 is a NSContraZCts. 

 (e) Let (𝑆, ϱ) be a NSOS in 𝕋. As 𝒢 is NSContraPCts, 𝒢−1(𝑆, ϱ) is a NSPCS in 𝕎. Since all NSPCS 

is a NSZCS, 𝒢−1(𝑆, ϱ) is a NSZCS in 𝕎. Thus, 𝒢 is a NSContraZCts. 

 (f) Let (𝑆, ϱ) be a NSOS in 𝕋. As 𝒢 is NSContraZCts, 𝒢−1(𝑆, ϱ) is a NSZCS in 𝕎. Since all NSZCS 

is a NSeCS, 𝒢−1(𝑆, ϱ) is a NSeCS in 𝕎. Thus, 𝒢 is a NSContraeCts. 

Example 3.2 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1,ϱ), (𝑆2,ϱ) and (𝑆3,ϱ) in 𝕎 and (𝑉1, ϱ)  in 𝕋  are defined as 

(𝑆1,e1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉 

(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2447 
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(𝑆2, 𝑒2) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.2
) , (

𝜇𝑤2

0.8
,

𝜎𝑤2

0.6
,

𝜈𝑤2

0.4
) , (

𝜇𝑤3

0.7
,

𝜎𝑤3

0.8
,

𝜈𝑤3

0.3
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.2
) , (

𝜇𝑤2

0.7
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.2
) , (

𝜇𝑤3

0.9
,

𝜎𝑤3

0.6
,

𝜈𝑤3

0.2
)〉  

(𝑉1, 𝑒1) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.2
) , (

𝜇𝑡2

0.8
,

𝜎𝑡2

0.6
,

𝜈𝑡2

0.4
) , (

𝜇𝑡3

0.7
,

𝜎𝑡3

0.8
,

𝜈𝑡3

0.3
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.2
) , (

𝜇𝑡2

0.7
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.2
) , (

𝜇𝑡3

0.9
,

𝜎𝑡3

0.6
,

𝜈𝑡3

0.2
)〉  

Here, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑉1, ϱ)}.  Let  𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ)  be an identity mapping,   then 𝒢 is a NSContraPCts but not NSContraCts, 

because the set 𝒢−1(𝑉1, ϱ) = (𝑆4, ϱ) is a NSPCS but not NSCS. 

Remark 3.1 From the results discussed above, the following diagram is obtained. 

  

 

 

 

 

  

 

 

 

 

 

Diagram.1 Neutrosophic soft contra Z – continuous maps 

 

Example 3.3 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1,ϱ), (𝑆2,ϱ) and (𝑆3,ϱ) in 𝕎 and (𝑉1, ϱ)  in 𝕋  are defined as 

(𝑆1,e1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉 

 NSContra𝛿Cts  

NNNNnsNSC𝛿
OS 
 

 NSContraCts  

NS

Type equation here.

OS 
 

 NSContraPCts    NSContra𝛿SCts   

 NSContraZCts  

NS

Type equation here.

OS 
 

 NSContraeCts  

NS

Type equation here.

OS 
 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2448 
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(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.6
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.5
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.3
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.4
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.3
)〉  

(𝑉1, 𝑒1) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.6
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.5
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.3
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.4
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.3
)〉  

Here, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑉1, ϱ)}.  Let  𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) be an identity mapping,  then 

(i) 𝒢 is a NSContraZSCts but not NSContra𝛿Cts, because the set  𝒢−1(𝑉1, ϱ) = (𝑆4, ϱ) is a NS𝛿ZCS 

but not NSCS. 

(ii) 𝒢 is a NSContraZCts but not NSContrPCts, because the set  𝒢−1(𝑉1, ϱ) = (𝑆4, ϱ) is a NSZCS but 

not NS𝑃CS. 

Example 3.4 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1,ϱ), (𝑆2,ϱ) and (𝑆3,ϱ) in 𝕎 and (𝑉1, ϱ)  in 𝕋  are defined as 

 

(𝑆1, 𝑒1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉  

 

(𝑆1, 𝑒2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉  

(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.7
,

𝜈𝑤1

0.1
) , (

𝜇𝑤2

0.8
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.1
) , (

𝜇𝑤3

0.8
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.2
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.7
,

𝜈𝑤1

0.1
) , (

𝜇𝑤2

0.8
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.11
) , (

𝜇𝑤3

0.9
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.1
)〉  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2449 
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(𝑉1, 𝑒1) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.3
) , (

𝜇𝑡2

0.8
,

𝜎𝑡2

0.7
,

𝜈𝑡2

0.1
) , (

𝜇𝑡3

0.8
,

𝜎𝑡3

0.7
,

𝜈𝑡3

0.2
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.7
,

𝜈𝑡1

0.1
) , (

𝜇𝑡2

0.8
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.1
) , (

𝜇𝑡3

0.9
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.1
)〉  

Here, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑉1, ϱ)}.  Let  𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) be an identity mapping,  the 𝒢 is a NSContraZCts but not NSContra𝛿SCts, 

because the set  𝒢−1(𝑉1, ϱ) = (𝑆4, ϱ) is a NSZCS but not NS𝛿SCS. 

Example 3.5 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑃1,ϱ), (𝑃2,ϱ) and (𝑃3,ϱ) in 𝕎 and (𝑄1, ϱ)  in 𝕋  are defined as 

 

(𝑃1, 𝑒1) =  〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.4
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.9
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.4
,

𝜈𝑤3

0.8
)〉  

 

(𝑃1, 𝑒2) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑃2, 𝑒1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.6
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.5
)〉  

(𝑃2, 𝑒2) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.4
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.4
,

𝜈𝑤3

0.6
)〉  

(𝑃3, 𝑒1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.4
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉  

(𝑃3, 𝑒2) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.4
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.7
)〉  

(𝑄1, 𝑒1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.4
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉  

(𝑄1, 𝑒2) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.4
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.7
)〉  

     Here, we have  τ = {0(𝕎 ,𝜚) , 1(𝕎 ,𝜚), (𝑃1, ϱ), (𝑃2, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑄1, ϱ)}.  Let  𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ)  be an identity mapping,   then 𝒢  is a NSContraeCts but not 

NSContraZCts, because the set 𝒢−1(𝑄1, ϱ) = (𝑃3, ϱ) is a NSeCS but not NS𝑍CS. 

Theorem 3.1 A map 𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ,   ϱ)  is NSContraZCts iff the inverse image of every 

NSCS in 𝕋 is NSZOS in 𝕎. 

Proof. Consider  a  NSCS (𝑆, ϱ) in 𝕋. Then  (𝑆, ϱ)c is NSOS in 𝕋.  As 𝒢 is NSContraZCts,  𝒢−1 

((𝑆, ϱ)c)  is  NSZCS in 𝕎. As 𝒢−1((𝑆, ϱ)c) = (𝒢−1 (𝑆, ϱ))𝑐,  𝒢−1(𝑆, ϱ) is a NSZOS in 𝕎.                   

       Conversely, consider a NSCS (𝑆, ϱ) in 𝕋. So (𝑆, ϱ)c is a NSOS in 𝕋.  By hypothesis, 

𝒢−1((𝑆, ϱ)c)  is NSZCS  in  𝕎.   As  𝒢−1 ((𝑆, ϱ)c) = (𝒢−1 (𝑆, ϱ))𝑐 , (𝒢−1 (𝑆, ϱ))𝑐 is a NSZCS in 𝕎. 

Hence, 𝒢−1(𝑆, ϱ) is a NSZOS in 𝕎. Hence 𝒢 is NSContraZCts. 

Theorem 3.2 Let 𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ)  be a NSContraZCts where every NSZOS in 𝕎 is a 

NSOS in 𝕎, then 𝒢 is a NSContraCts. 

Proof. Let (𝑆, ϱ) be a NSOS in 𝕋. Then 𝒢−1 (𝑆, ϱ) is a NSZCS in 𝕎. By hypothesis,  𝒢−1(𝑆, ϱ) is a 

NSOS in 𝕎, then 𝒢 is a NSContraZCts. 



Communications on Applied Nonlinear Analysis 

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Theorem 3.3 Let  𝒢 ∶  (𝕎, τ, ϱ) → (𝕋 , σ, ϱ)  be a NSContraZCts map and ℋ : (𝕋, σ, ϱ) → (𝕌, 𝜌, 

ϱ) be an NSContraCts, then ℋ ∘ 𝒢: (𝕎, τ, ϱ) → (𝕌, 𝜌, ϱ) is a NSContraZCts map. 

Proof. Let (𝑆, ϱ) be a NSOS in 𝕌. Then ℋ −1(𝑆, ϱ) is a NSZCS in 𝕋, by hypothesis, Since 𝒢 is a 

NSContraZCts maps, 𝒢−1 (ℋ −1(𝑆, ϱ)) is a NSZCS in 𝕎. Hence ℋ ∘ 𝒢 is a NSContraZCts map. 

Theorem 3.4 Let  𝒢: (𝕎, τ, ϱ) → (𝕋 , σ, ϱ)  be a NSContraZCts map. Then the following 

conditions are hold. 

(i) 𝒢(NSZcl(𝑆, ϱ)) ⊇ NSint(𝒢(𝑆, ϱ)), for all NSS (𝑆, ϱ) in 𝕎. 

(ii) NSZcl(𝒢−1 (𝐷, ϱ)) ⊇ 𝒢−1(NSint(𝐷, ϱ)), for all NSS in 𝕋. 

Proof. (i) As NSZcl(𝒢(𝑆, ϱ)) is a NSZCS in 𝕋 and 𝒢 is NSContraZCts, then 𝒢−1(NSZcl(𝒢(𝑆, ϱ)) is 

NSZOS in 𝕎. Now, as (𝑆, ϱ) ⊇ 𝒢−1(NSint(𝒢(𝑆, ϱ))). NSZcl(𝑆, ϱ)  ⊇ 𝒢−1(NSint(𝒢(𝑆, ϱ))). 

Therefore, 𝒢(NSZcl(𝑆, ϱ)) ⊇ NSint(𝒢(𝑆, ϱ)). 

(ii) By replacing (𝑆, ϱ) by (𝐷, ϱ) in (i), we obtain 𝒢(NSZcl(𝒢−1(𝐷, ϱ))) ⊇  NSint(𝒢 (𝒢−1(𝐷, ϱ))) ⊇ 

NSint(𝐷, ϱ). Hence NSZcl(𝒢−1(𝐷, ϱ)) ⊇ 𝒢−1 (NSint(𝐷, ϱ)). 

4.  Neutrosophic soft contra Z – irresolute maps  

  
Definition 4.1 A map  𝒢: (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is known as a neutrosophic soft contra Z-irresolute 

(briefly, NSContraZ-irr)    map   if    𝒢−1(𝑆, ϱ) is a NSZCS in (𝕎, τ, ϱ) for each NSZOS (𝑆, ϱ) in 

(𝕋, σ, ϱ). 

Theorem 4.1 Let  𝒢: (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a NSContraZ-irr. Then 𝒢 is a NSContraZCts map. 

But not conversely. 

Proof. Assume 𝒢 is a NSContraZ-irr map. Consider a NSOS (𝑆, ϱ) in 𝕋. As each NSOS is a NSZOS, 

(𝑆, ϱ) is a NSZOS in 𝕋. By hypothesis,  𝒢−1(𝑆, ϱ) is a NSZCS in 𝕎. Hence 𝒢 is a NSContraZCts 

map. 

Example 4.1 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } = {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1,ϱ), (𝑆2,ϱ), (𝑆3,ϱ), and (𝑆4, ϱ) in 𝕎 and (𝑉1,ϱ) and (𝑉2,ϱ) in 𝕋 are defined as 

(𝑆1,e1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉 

(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  



Communications on Applied Nonlinear Analysis 

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2451 
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(𝑆4, 𝑒1) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.4
) , (

𝜇𝑤2

0.7
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.3
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.3
)〉  

(𝑉1, 𝑒1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.2
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.8
)〉  

(𝑉2, 𝑒1) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑉2, 𝑒2) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.4
) , (

𝜇𝑡2

0.7
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.3
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.3
)〉  

Here, we have  τ = {0(𝕎 ,𝜚) ,1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)} and  𝜎 = {0(𝕋,𝜚),    1(𝕋,𝜚),   (𝑉1, ϱ)}.  Let  𝒢 ∶

 (𝕎, τ, ϱ) → (𝕋, σ, ϱ)  be an identity mapping,   then 𝒢 is a NSContraZCts but not  NSContraZ-irr, 

because  the set (V2, ϱ) is a NSZCS in 𝕋  but 𝒢−1(V2, ϱ) = (𝑆4, ϱ) is not NSZOS in 𝕎. 

Theorem 4.2 Let  𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ)  be a NSContraZ-irr. If  𝕎 is a NSZ𝑈1

2

 – space, then 𝒢 is 

a NSContraCts map. 

Proof. Consider a  NSOS (𝑆, ϱ) in 𝕋. Then (𝑆, ϱ) is a NSZOS in 𝕋. Hence 𝒢−1(𝑆, ϱ) is a NSZCS in 

𝕎. As 𝕎 is a  NSZ𝑈1

2

 – space,  𝒢−1(𝑆, ϱ) is a NSCS in 𝕎. Thus 𝒢 is a NSContraCts map. 

Theorem 4. 3 Let  𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a NSContraZ-irr and ℋ : (𝕋, σ, ϱ) → (𝕌, 𝜌, ϱ) be a 

NSZCts maps. Then ℋ ∘ 𝒢: (𝕎, τ, ϱ)  → (𝕌, 𝜌, ϱ) is a NSContraZCts map. 

Proof. Consider a NSOS (𝑆, ϱ) in 𝕌. Then ℋ −1(𝑆, ϱ) is a NSZOS in 𝕋. As 𝒢 is a NSContraZ-irr, 

𝒢−1 (ℋ −1(𝑆, ϱ)) is a NSZCS in 𝕎. Hence ℋ ∘ 𝒢 is a NSContraZCts map. 

Theorem 4.4 Let 𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ) and  ℋ : (𝕋, σ, ϱ) → (𝕌, 𝜌, ϱ) be mappings. Then ℋ ∘ 𝒢: 

(𝕎, τ, ϱ) → (𝕌, 𝜌, ϱ) is  

(i) NSContraZCts if  𝒢 is NSZirr and  ℋ is NSContraZCts. 

(ii) NSContraZ-irr if 𝒢 is NSContraZirr (resp. NSZirr) and ℋ is NSZ-irr (resp NSContraZ-irr). 

Proof. (i) Let (𝑆, ϱ) be a NSOS in 𝕌. Then ℋ −1(𝑆, ϱ) is a NSZCS in 𝕋. As ℋ is a NSZ-irr map, 

𝒢−1(ℋ −1(𝑆, ϱ)) is a NSZCS in 𝕎. Hence ℋ ∘ 𝒢 is a NSContraZCts map. 

The other cases are similar. 

Theorem 4.5 Let  𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a mapping. 

(i) If  (𝕎, τ, ϱ) is NSZ𝑈1

2

 – space, then the concepts of NSContraCts and NSContraZCts are 

equivalent. 

(ii) If  (𝕋, σ, ϱ) is NSZ𝑈1

2

 – space, then the concepts of NSContraZCts and NSContraZ-irr are 

equivalent. 



Communications on Applied Nonlinear Analysis 

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2452 
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(iii) If  (𝕎, τ, ϱ) and (𝕋, σ, ϱ) are NSZ𝑈1

2

 – space, then the concepts of NSContraCts, NSContraZCts 

and NSContraZ-irr are equivalent. 

Proof. (i) Let (𝑆, ϱ) be a NSCS in 𝕋. Then ℋ −1(𝑆, ϱ) is a NSZOS in 𝕎 if  𝒢 is NSContraZCts. As  

(𝕎, τ, ϱ) is a NSZ𝑈1

2

 – space, ℋ−1(𝑆, ϱ) is a  NSOS in  𝕎. Hence 𝒢 is also NSContraCts map. 

The other cases are similar. 

Theorem 4.6 Let 𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ) and ℋ : (𝕋, σ, ϱ) → (𝕌, 𝜌, ϱ) be  NSContraZCts 

mappings and (𝕋, σ, ϱ) be a  NSZ𝑈1

2

 – space.  Then ℋ ∘ 𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ)  is a NSZCts map. 

Proof. Let (𝑆, ϱ) be a NSCS in 𝕌. Then ℋ −1(𝑆, ϱ) is a NSZOS in 𝕋. Since ℋ  is NSContraZCts. As 

(𝕋, σ, ϱ) is a NSZ𝑈1

2

 – space,  ℋ −1(𝑆, ϱ) is a NSOS in 𝕋. Then, 𝒢(ℋ −1(𝑆, ϱ)) is a NSZCS in  𝕎 

because 𝒢 is NSContraZCts. Hence, ℋ ∘ 𝒢 is a NSZCts map. 

Theorem 4.7 Let 𝒢 ∶  (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a map from a NSts 𝕎 into a NSts 𝕋. If  𝕎 and 𝕋 are 

NSZ𝑈1

2

 – space, then the following are equivalent. 

(i) 𝒢 is a NSContraZ-irr map. 

(ii) 𝒢−1(𝑆, ϱ) is a NSZOS in 𝕎 for every NSZCS (𝑆, ϱ) in 𝕋. 

(iii) NScl(𝒢−1(𝑆, ϱ)) ⊇ 𝒢−1 (NSint(𝑆, ϱ)) for each (𝑆, ϱ) of 𝕋. 

Proof. (i) → (ii): Consider a NSZCS (𝑆, ϱ) in 𝕋. Then (𝑆, 𝜚)𝑐 is a NSZOS in 𝕋. As 𝒢 is NSContraZ-

irr, 𝒢−1 ((𝑆, 𝜚)𝑐) is a NSZCS in 𝕎. We know that,  𝒢−1((𝑆, 𝜚)𝑐) = (𝒢−1 (𝑆, 𝜚))𝑐. Thus 𝒢−1(𝑆, ϱ) is a 

NSZOS in 𝕎. 

(ii) → (iii) : Consider a NSS (𝑆, ϱ) in 𝕋 and NSint(𝑆, ϱ) ⊆ (𝑆, ϱ). Then 𝒢−1(𝑁𝑆𝑖𝑛𝑡(𝑆, ϱ))  ⊆  

𝒢−1(𝑆, ϱ). As NSint(𝑆, ϱ) is a NSOS in 𝕋, NSint(𝑆, ϱ) is a NSZOS in 𝕋. Therefore (𝑁𝑆𝑖𝑛𝑡(𝑆, 𝜚))𝑐 is 

a NSZCS in 𝕋. By presumption,  𝒢−1(NSint(𝑆, 𝜚))𝑐 is a NSZOS in 𝕎. As 𝒢−1 ((𝑁𝑆𝑖𝑛𝑡(𝑆, 𝜚))𝑐) = 

(𝒢−1(NSint(𝑆, 𝜚)))𝑐, 𝒢−1(NSint(𝑆, ϱ)) is a NSZOS in 𝕎. As 𝕎 is a NSZ𝑈1

2

 – space, 𝒢−1 

(NSint(𝑆, ϱ)) is a NSOS in 𝕎. Thus, NScl(𝒢−1(𝑆, ϱ)) ⊇ NScl(𝒢−1(NSint(𝑆, ϱ))) = 𝒢−1 

(NSint(𝑆, ϱ)). That is, NScl(𝒢−1(𝑆, ϱ)) ⊇  𝒢−1(NSint(𝑆, ϱ)). 

(iii) → (i): Consider a NSZCS (𝑆, ϱ) in 𝕋. As 𝕋 is NSZ𝑈1

2

 – space, (𝑆, ϱ) is a NSCS in 𝕋 and 

NScl(𝑆, ϱ) = (𝑆, ϱ). Hence 𝒢−1(𝑆, ϱ) = 𝒢−1(NScl(𝑆, ϱ)) ⊇ NSint(𝒢−1(𝑆, ϱ)). But clearly 𝒢−1(𝑆, ϱ) ⊇ 

NSint(𝒢−1(𝑆, ϱ)).  

Therefore NSint(𝒢−1(𝑆, ϱ)) = 𝒢−1(𝑆, ϱ). So,  𝒢−1(𝑆, ϱ) is a NSOS and hence it is a NSZOS in 𝕎. 

Thus 𝒢 is a NSContraZ-irr map.  

5.  Neutrosophic soft contra Z - open mapping  

 

Definition 5.1 A mapping  𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is neutrosophic soft contra Z – open (in short, 

NSContraZO) if the  image of each NSOS of (𝕎, τ, ϱ) is a NSZCS in (𝕋, σ, ϱ). 

Theorem 5.1  The statements are hold  but  the converse does not true. 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 10s (2025) 

  

2453 
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a)  Each NSContraδO is a NSContraO. 

b)  Each NSContraO is a NSContraδSO. 

c)  Each NSContraO  is a NSContraPO. 

d)  Each NSContraδSO is a NSContraZO. 

e)  Each NSContraPO  is a NSContraZO. 

f)  Each NSContraZO  is a NSContraeO. 

 

Proof.  

(a) Let (𝑆, ϱ) be a NSOS in 𝕎. As 𝒢 is NSContraδO, 𝒢(𝑆, ϱ) is a NSδCS in 𝕋. Since all NSδCS are 

NSCS,  𝒢(𝑆, ϱ) is NSCS in 𝕋. Thus, 𝒢 is a NSContraO. 

(b) Let (𝑆, ϱ) be a NSOS in 𝕎. As 𝒢 is NSContraO, 𝒢(𝑆, ϱ) is a NSCS in 𝕋. Since all  NSCS are 

NSδSCS, 𝒢(𝑆, ϱ) is a NSδSCS in 𝕋. Thus, 𝒢 is a NSContraδSO. 

(c) Let (𝑆, ϱ) be a NSOS in 𝕎. As 𝒢 is NSContraO, 𝒢(𝑆, ϱ) is a NSCS in 𝕋. Since all NSCS are 

NSPCS, 𝒢(𝑆, ϱ) is a NSPCS in 𝕋. Hence, 𝒢 is a NSContraPO. 

(d) Let (𝑆, ϱ) be a NSOS in 𝕎. As 𝒢 is NSContraδSO, 𝒢(𝑆, ϱ) is a NSδSCS in 𝕋. Since all NSδSCS 

is a NSZCS,  𝒢(𝑆, ϱ) is a NSZCS in 𝕋. Hence, 𝒢 is a NSContraZO. 

(e) Let (𝑆, ϱ) be a NSOS in 𝕎. As 𝒢 is NSContraPO, 𝒢(𝑆, ϱ) is a NSPCS in 𝕋. Since all NSPCS are 

NSZCS, 𝒢(𝑆, ϱ) is a NSZCS in 𝕋. Hence, 𝒢 is a NSContraZO. 

 (f) Let (𝑆, ϱ) be a NSOS in 𝕎. As 𝒢 is NSContraZO, 𝒢(𝑆, ϱ) is a NSZCS in 𝕋. Since all NSZCS is a 

NSeCS, 𝒢(𝑆, ϱ) is a NSeCS in 𝕋. Hence, 𝒢 is a NSContraeO. 

Example 5.1 Let 𝕎 = { 𝑤1 , 𝑤2   , 𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets  (𝑉1, ϱ)  in 𝕎 

and (𝑆1,ϱ), (𝑆2,ϱ), (𝑆3,ϱ) and (𝑆4,ϱ) in 𝕋  are defined as 

(𝑉1, 𝑒1) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.4
) , (

𝜇𝑤2

0.7
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.3
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.3
)〉  

(𝑆1,e1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.2
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.8
)〉 

(𝑆2, 𝑒1) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.3
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.7
,

𝜈𝑡3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑡1,

0.3
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.3
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑡1,

0.1
,

𝜎𝑡1

0.3
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.1
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2454 
https://internationalpubls.com 

(𝑆4, 𝑒2) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.4
) , (

𝜇𝑡2

0.7
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.3
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.3
)〉                     

Then, we have τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚), (𝑉1, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)}.  Let 𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ)  be an identity mapping,   then 𝒢 is a NSContraO but not NSContraδO, 

because the set 𝒢 (𝑉1, ϱ) = (𝑆4, ϱ) is a NSCS but not NSδCS. 

Example 5.2 Let 𝕎 = { 𝑤1 , 𝑤2   , 𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets  (𝑉1, ϱ)  in 𝕎 

and (𝑆1,ϱ), (𝑆2,ϱ), (𝑆3,ϱ) and (𝑆4,ϱ) in 𝕋  are defined as 

(𝑉1, 𝑒1) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.2
) , (

𝜇𝑤2

0.8
,

𝜎𝑤2

0.6
,

𝜈𝑤2

0.4
) , (

𝜇𝑤3

0.7
,

𝜎𝑤3

0.8
,

𝜈𝑤3

0.3
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.2
) , (

𝜇𝑤2

0.7
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.2
) , (

𝜇𝑤3

0.9
,

𝜎𝑤3

0.6
,

𝜈𝑤3

0.2
)〉  

(𝑆1,e1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.2
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.8
)〉 

(𝑆2, 𝑒1) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.3
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.7
,

𝜈𝑡3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑡1,

0.3
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.3
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑡1,

0.1
,

𝜎𝑡1

0.3
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.1
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.2
) , (

𝜇𝑡2

0.8
,

𝜎𝑡2

0.6
,

𝜈𝑡2

0.4
) , (

𝜇𝑡3

0.7
,

𝜎𝑡3

0.8
,

𝜈𝑡3

0.3
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.2
) , (

𝜇𝑡2

0.7
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.2
) , (

𝜇𝑡3

0.7
,

𝜎𝑡3

0.6
,

𝜈𝑡3

0.2
)〉  

                          Then, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚),  (𝑉1, ϱ)} and 𝜎 =

{0(𝕋,𝜚),   1(𝕋,𝜚),    (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)}.  Let 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ)  be  an identity mapping,   

then 𝒢 is a NSContraPO but not NSContraO, because the set 𝒢 (𝑉1, ϱ) = (𝑆4, ϱ) is a NSPCS but not 

NSCS. 

Example 5.3 Let 𝕎 = { 𝑤1 , 𝑤2   , 𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets  (𝑉1, ϱ)  in 𝕎 

and (𝑆1,ϱ), (𝑆2,ϱ), (𝑆3,ϱ) and (𝑆4,ϱ) in 𝕋  are defined as 

(𝑉1, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.6
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.5
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.3
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.9
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.3
)〉  

(𝑆1,e1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.2
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.8
)〉 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2455 
https://internationalpubls.com 

(𝑆2, 𝑒1) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.3
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.7
,

𝜈𝑡3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑡1,

0.3
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.3
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑡1,

0.1
,

𝜎𝑡1

0.3
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.1
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.6
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.5
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.3
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.9
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.3
)〉  

                          Then, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚),  (𝑉1, ϱ)} and 𝜎 =

{0(𝕋,𝜚),   1(𝕋,𝜚),    (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)}.  Let 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) be  an identity mapping. 

Then  

(i) 𝒢 is a NSContraδSO but not NSContraO, because the set 𝒢 (𝑉1, ϱ) = (𝑆4, ϱ) is a NSδSCS but not 

NSCS. 

(ii) 𝒢 is a NSContraZO but not NSContraPO, because the set 𝒢 (𝑉1, ϱ) = (𝑆4, ϱ) is a NSZCS but not 

NSPCS. 

Example 5.4 Let 𝕎 = { 𝑤1 , 𝑤2   , 𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets  (𝑉1, ϱ)  in 𝕎 

and (𝑆1,ϱ), (𝑆2,ϱ), (𝑆3,ϱ) and (𝑆4,ϱ) in 𝕋  are defined as 

(𝑉1, 𝑒1) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.3
) , (

𝜇𝑤2

0.8
,

𝜎𝑤2

0.7
,

𝜈𝑤2

0.1
) , (

𝜇𝑤3

0.8
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.2
)〉  

(𝑉1, 𝑒2) =  〈(
𝜇𝑤1,

0.7
,

𝜎𝑤1

0.7
,

𝜈𝑤1

0.1
) , (

𝜇𝑤2

0.8
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.1
) , (

𝜇𝑤3

0.9
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.1
)〉  

(𝑆1,e1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉 

(𝑆1,e2) =  〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.2
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.8
)〉 

(𝑆2, 𝑒1) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑆2, 𝑒2) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.3
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.7
,

𝜈𝑡3

0.4
)〉  

(𝑆3, 𝑒1) =  〈(
𝜇𝑡1,

0.3
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.3
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.8
)〉  

(𝑆3, 𝑒2) =  〈(
𝜇𝑡1,

0.1
,

𝜎𝑡1

0.3
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.1
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.1
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.9
)〉  

(𝑆4, 𝑒1) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.6
,

𝜈𝑡1

0.3
) , (

𝜇𝑡2

0.8
,

𝜎𝑡2

0.7
,

𝜈𝑡2

0.1
) , (

𝜇𝑡3

0.8
,

𝜎𝑡3

0.7
,

𝜈𝑡3

0.2
)〉  

(𝑆4, 𝑒2) =  〈(
𝜇𝑡1,

0.7
,

𝜎𝑡1

0.7
,

𝜈𝑡1

0.1
) , (

𝜇𝑡2

0.8
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.1
) , (

𝜇𝑡3

0.9
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.1
)〉  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2456 
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Then, we have  τ = {0(𝕎,𝜚) , 1(𝕎 ,𝜚),  (𝑉1, ϱ)} and 𝜎 =

{0(𝕋,𝜚),   1(𝕋,𝜚),    (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ)}.  Let 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) 

 be  an identity mapping, then 𝒢 is a NSContraZO but not NSContraδSO, because the set 𝒢 (𝑉1, ϱ) = 

(𝑆4, ϱ) is a NSZCS but not NSδSCS. 

Remark  5.1  From the results discussed above, the following diagram is obtained. 

 

 

 

 

 

 

 

 

 

 

Diagram.2  Neutrosophic soft contra Z – open maps 

 

Example  5.5  Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets (𝑄1, ϱ) in 𝕎 and 

(𝑃1,ϱ), (𝑃2,ϱ) and (𝑃3,ϱ) in 𝕋  are defined as 

 

(𝑄1, 𝑒1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.4
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉  

(𝑄1, 𝑒2) =  〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.4
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.7
)〉  

(𝑃1, 𝑒1) =  〈(
𝜇𝑡1,

0.3
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.7
) , (

𝜇𝑡2

0.4
,

𝜎𝑡2

0.3
,

𝜈𝑡2

0.9
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.4
,

𝜈𝑡3

0.8
)〉  

 

(𝑃1, 𝑒2) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑤1

0.7
) , (

𝜇𝑡2

0.3
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.1
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.8
)〉  

(𝑃2, 𝑒1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.3
,

𝜈𝑡2

0.6
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.5
)〉  

(𝑃2, 𝑒2) =  〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑡2

0.4
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.4
,

𝜈𝑡3

0.6
)〉  

(𝑃3, 𝑒1) =  〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑤1

0.4
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.3
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉  

 NSContra𝛿O  

NNNNnsNSC𝛿
OS 
 
 NSContraO  

NS

Type equation here.

OS 
 

 NSContraPO 
 NSContra𝛿SO   

 NSContraZO  

NS

Type equation here.

OS 
 

 NSContraeO  

NS

Type equation here.

OS 
 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2457 
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(𝑃3, 𝑒2) =  〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.4
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.7
)〉  

Then, we have  τ = {0(𝕎 ,𝜚) , 1(𝕎 ,𝜚), (𝑄1, ϱ)} and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑃1, ϱ), (𝑃2, ϱ)}.  Let  𝒢 ∶

 (𝕎, τ, ϱ) →  (𝕋, σ, ϱ)  be an identity mapping,   then 𝒢 is a NSContraeO but not NSContraZO, 

because the set 𝒢(𝑄1, ϱ) = (𝑃3, ϱ) is a NSeCS but not NS𝑍CS. 

Theorem 5.2 A mapping 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) is NSContraZO iff for every NSS (𝑆, ϱ) of 

(𝕎, τ, ϱ),  𝒢(NSint(S, ϱ)) ⊇  NSZcl(𝒢(S, ϱ)). 

Proof.  Necessity: Assume 𝒢 is a NSContraZO mapping and (𝑆, ϱ) is a NSOS (𝕎, τ, ϱ). Now,  

𝒢(NSint(S, ϱ)) ⊆ (𝑆, ϱ) implies 𝒢(NSint(S, ϱ))  ⊆ 𝒢(𝑆, ϱ). Since 𝒢 is a NSContraZO mapping, 

𝒢(NSint(S, ϱ)) is a NSZCS in (𝕋, σ, ϱ) such that 𝒢(NSint(S, ϱ)) ⊇ 𝒢(𝑆, ϱ). Therefore, 

𝒢(NSint(S, ϱ)) ⊇  NSZcl(𝒢(S, ϱ)).  

Sufficiency: Assume  (𝑆, ϱ) is a NSOS (𝕎, τ, ϱ). Then 𝒢(𝑆, ϱ) = 𝒢(NSint(S, ϱ)) ⊇

 NSZcl(𝒢(S, ϱ)).  But NSZcl(𝒢(S, ϱ)) ⊇ 𝒢(𝑆, ϱ). So, 𝒢(𝑆, ϱ) = NSZcl(𝒢(S, ϱ)) which implies 𝒢(𝑆, ϱ) 

is a NSZCS of  (𝕋, σ, ϱ) and hence 𝒢 is a NSContraZO. 

Theorem 5.3 If 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) is NSContraZO mapping, then  NSint (𝒢 −1 (S, ϱ) ⊆

 𝒢 −1(NSZcl(S, ϱ))  for every NSS (𝑆, ϱ) of (𝕋, σ, ϱ). 

Proof. Consider a NSS (𝑆, ϱ) in (𝕋, σ, ϱ).  Then,  NSint (𝒢 −1(S, ϱ) is a NSOS in (𝕎, τ,

ϱ).  Since 𝒢  is NSContraZO, 𝒢(NSint (𝒢 −1 (S, ϱ)) is a NSZCS in (𝕋, σ, ϱ) and hence 

𝒢(NSint (𝒢 −1 (S, ϱ)) ⊆  NSZcl (𝒢 (𝒢 −1(S, ϱ)))⊆  NSZcl(S, ϱ).  Thus, NSint (𝒢 −1 (S, ϱ) ⊆

 𝒢 −1(NSZcl(S, ϱ)). 

Theorem 5.4 A mapping 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) is NSContraZO iff for every NSS (𝑆, ϱ) of 

(𝕋, σ, ϱ), and for each NSOS (𝐵, ϱ) of (𝕎, τ, ϱ) containing 𝒢 −1 (S, ϱ), there is a NSZOS  

(𝐾, ϱ) of (𝕋, σ, ϱ)  such that (S, ϱ)  ⊆  (B, ϱ) and 𝒢 −1 (K, ϱ) ⊆ (B, ϱ). 

Proof.  Necessity: Assume 𝒢 be a NSContraZO mapping. Let a NSCS (𝑆, ϱ) in (𝕋, σ, ϱ) and a 

NSOS (𝐵, ϱ) in (𝕎, τ, ϱ) such that 𝒢 −1 (S, ϱ) ⊆ (B, ϱ). Then, (𝐾, ϱ) = (𝒢(B, ϱ)c)c is NSZOS of 

(𝕋, σ, ϱ) ∋ 𝒢 −1 (K, ϱ) ⊆ (B, ϱ). 

Sufficiency: Assume  (B, ϱ) is a NSOS (𝕎, τ, ϱ).  So,  𝒢 −1(𝒢(B, ϱ)c) ⊆ (B, ϱ)c and (B, ϱ)c is 

NSCS in (𝕎, τ, ϱ). By presumption, there is a NSZOS (𝐾, ϱ) of (𝕋, σ, ϱ) such that (𝒢(B, ϱ))c  ⊆ 

(𝐾, ϱ) and 𝒢 −1 (K, ϱ)  ⊆ (B, ϱ)c. Therefore, (B, ϱ) ⊆  (𝒢 −1 (K, ϱ))
c
.  Hence (K, ϱ)c ⊆  𝒢(B, ϱ) ⊆

 𝒢((𝒢 −1 (K, ϱ))
c
) ⊆ (K, ϱ)c which implies  𝒢(B, ϱ) =  (K, ϱ)c.  As  (K, ϱ)c is NSZCS of (𝕋, σ, ϱ).  

Hence 𝒢(B, ϱ) is NSZCS in (𝕋, σ, ϱ) and thus 𝒢 is NSContraZO mapping. 

Theorem 5.5 A mapping 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) is NSContraZO iff 𝒢 −1(NScl(S, ϱ)) ⊇ 

NSint(𝒢 −1(S, ϱ)) for every NSS (S, ϱ) of  (𝕋, σ, ϱ). 

Proof. Necessity: Let 𝒢 be a NSContraZO mapping. For any NSS (S, ϱ) of (𝕋, σ, ϱ), 𝒢 −1(S, ϱ) ⊆ 

NScl(𝒢 −1(S, ϱ)). Therefore, by Theorem 5.4 there exists a NSZOS (B, ϱ) in (𝕋, σ, ϱ) ∋ (S, ϱ)  ⊇

 (B, ϱ) &𝒢 −1(B, ϱ) ⊇ 𝑁𝑆𝑖𝑛𝑡(𝒢 −1(S, ϱ)).  Hence 𝒢 −1(NSZcl(S, ϱ))   ⊇  𝒢 −1(B, ϱ) ⊇ 

NSint(𝒢 −1(S, ϱ)). 



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Sufficiency: Let (S, ϱ) be a NSS in (𝕋, σ, ϱ) and (B, ϱ) be a NSCS of  (𝕎, τ, ϱ) containing 

𝒢 −1(S, ϱ). Put (K, ϱ) = NScl(S, ϱ), then (S, ϱ) ⊆ (K, ϱ) and (K, ϱ) is NSZC and 𝒢 −1(S, ϱ) ⊆ 

NSint(𝒢 −1(S, ϱ)) ⊆ (B, ϱ). Thus by Theorem 5.4, 𝒢 is NSZO mapping. 

Theorem 5.6 If 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) and ℋ : (𝕋, σ, ϱ) → (𝕌, 𝜌, ϱ) be two neutrosophic 

soft mappings and ℋ ∘ 𝒢: (𝕎, τ, ϱ) →  (𝕌, 𝜌, ϱ) is NSContraZO. If  ℋ: (𝕋, σ, ϱ) → (𝕌, 𝜌, ϱ) is 

NSContraZ-irr then 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) is NSZO mapping. 

Proof. Let (S, ϱ) be a NSOS in (𝕎, τ, ϱ). Then  ℋ ∘ 𝒢(S, ϱ) is NSZCS of  (𝕌, 𝜌, ϱ) because ℋ ∘ 𝒢 

is NSContraZO  mapping. As ℋ is NSContraZ-irr and  ℋ ∘ 𝒢(S, ϱ) is NSZCS of (𝕌, 𝜌, ϱ) therefore 

ℋ −1(ℋ ∘ 𝒢(S, ϱ)) =  𝒢(S, ϱ) is NSZOS in (𝕋, σ, ϱ). Hence 𝒢 is NSZO mapping. 

Theorem 5.7 If 𝒢 ∶  (𝕎, τ, ϱ) →  (𝕋, σ, ϱ) is NSO and ℋ : (𝕋, σ, ϱ) → (𝕌, 𝜌, ϱ) is NSContraZO 

mappings, then ℋ ∘ 𝒢: (𝕎, τ, ϱ) →  (𝕌, 𝜌, ϱ) is NSContraZO. 

Proof.  Let (S, ϱ) be a NSOS in (𝕎, τ, ϱ).  Then 𝒢(S, ϱ) is a NSOS of  (𝕋, σ,

ϱ) because 𝒢 is a NSO mapping. Since ℋ is NSContraZO, ℋ(𝒢(S, ϱ)) = (ℋ ∘ 𝒢)(S, ϱ) is NSZCS of 

(𝕌, 𝜌, ϱ). Hence ℋ ∘ 𝒢 is NSContraZO mapping. 

6.  Neutrosophic soft contra Z - closed mapping  

 

Definition 6.1 A mapping  𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is neutrosophic soft contra Z – closed (briefly, 

NSContraZC) if  image of every NSCS of (𝕎, τ, ϱ) is a NSZOS in (𝕋, σ, ϱ). 

Theorem 6.1  The statements are hold  but  the converse does not true. Every 

a)  NSContraδC is a NSContraC. 

b)  NSContraC is a NSContraδSC. 

c)  NSContraC  is a NSContraPC. 

d)  NSContraδSC is a NSContraZC. 

e)  NSContraPC  is a NSContraZC. 

f)  NSContraZC  is a NSContraeC. 

 

Proof. Consider the map 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) 

(a) Let (𝑆, ϱ) be a NSCS in 𝕎. As 𝒢 is NSContraδC, 𝒢(𝑆, ϱ) is a NSδOS in 𝕋. Since all NSδOS are 

NSOS,  𝒢(𝑆, ϱ) is NSOS in 𝕋. Then, 𝒢 is a NSContraC. 

(b) Let (𝑆, ϱ) be a NSCS in 𝕎. As 𝒢 is NSContraC, 𝒢(𝑆, ϱ) is a NSOS in 𝕋. Since all  NSOS are 

NSδSOS, 𝒢(𝑆, ϱ) is a NSδSOS in 𝕋. Then, 𝒢 is a NSContraδS. 

(c) Let (𝑆, ϱ) be a NSCS in 𝕎. As 𝒢 is NSContraC, 𝒢(𝑆, ϱ) is a NSOS in 𝕋. Since all NSOS are 

NSPOS, 𝒢(𝑆, ϱ) is a NSPOS in 𝕋. Hence, 𝒢 is a NSContraPC. 

(d) Let (𝑆, ϱ) be a NSCS in 𝕎. As 𝒢 is NSContraδSC, 𝒢(𝑆, ϱ) is a NSδSOS in 𝕋. Since all NSδSOS 

is a NSZOS,  𝒢(𝑆, ϱ) is a NSZOS in 𝕋. Hence, 𝒢 is a NSContraZC. 

 (e) Let (𝑆, ϱ) be a NSCS in 𝕎. As 𝒢 is NSContraPC, 𝒢(𝑆, ϱ) is a NSPOS in 𝕋. Since all NSPOS are 

NSZOS, 𝒢(𝑆, ϱ) is a NSZOS in 𝕋. Hence, 𝒢 is a NSContraZC. 



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 (f) Let (𝑆, ϱ) be a NSCS in 𝕎. As 𝒢 is NSContraZC, 𝒢(𝑆, ϱ) is a NSZOS in 𝕋. Since all NSZOS is a 

NSeOS, 𝒢(𝑆, ϱ) is a NSeOS in 𝕋. Hence, 𝒢 is a NSContraeC 

Example 6.1 In example 5.1, 𝒢 is a NSContraC but not NSContraδC mapping because the set 

(𝑉1, ϱ)𝑐 is NSCS in 𝕎 and  𝒢(𝑉1, ϱ)𝑐  =  (𝑆4, ϱ)𝑐 is NSOS but not NSδOS in 𝕋. 

Example 6.2 In example 5.2, 𝒢 is a NSContraPC but not NSContraC mapping because the set 

(𝑉1, ϱ)𝑐 is NSCS in 𝕎 and  𝒢(𝑉1, ϱ)𝑐  =  (𝑆4, ϱ)𝑐 is NSPOS but not NSOS in 𝕋. 

Example 6.3 In example 5.3, 

(i) 𝒢 is a NSContraδSC but not NSContraC mapping because the set (𝑉1, ϱ)𝑐 is NSCS in 

𝕎 and  𝒢(𝑉1, ϱ)𝑐  =  (𝑆4, ϱ)𝑐 is NSδSOS but not NSOS in 𝕋. 

(ii) 𝒢 is a NSContraZC but not NSContraPC mapping because the set (𝑉1, ϱ)𝑐 is NSCS in 

𝕎 and  𝒢(𝑉1, ϱ)𝑐  =  (𝑆4, ϱ)𝑐 is NSZOS but not NSPOS in 𝕋. 

Example 6.4 In example 5.4, 𝒢 is a NSContraZC but not NSContraδSC mapping because the set 

(𝑉1, ϱ)𝑐 is NSCS in 𝕎 and  𝒢(𝑉1, ϱ)𝑐  =  (𝑆4, ϱ)𝑐 is NSZOS but not NSδSOS in 𝕋. 

Example 6.5 In example 5.5, 𝒢 is a NSContraeC but not NSContraZC mapping because the set 

(𝑄1, ϱ)𝑐 is NSCS in 𝕎 and  𝒢(𝑄1, ϱ)𝑐  =  (𝑃3, ϱ)𝑐 is NSeOS but not NSZOS in 𝕋. 

Remark 6.1  The diagram shows NSConraZO mappings in NSts. 

 

 

 

 

 

 

 

 

Diagram.3  Neutrosophic soft contra Z – closed maps 

 

Theorem 6.2 A mapping  𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is NSContraZC iff for each NSS (S, ϱ) of 
(𝕋, σ, ϱ) and for each NSCS (B, ϱ) of (𝕎, τ, ϱ)  containing  𝒢 −1(S, ϱ) there is a NSZCS (K, ϱ)  of 
(𝕋, σ, ϱ)  such that (S, ϱ) ⊆  (K, ϱ)  and  𝒢 −1 (K, ϱ) ⊆  (B, ϱ). 

 

Proof. Necessity: Assume 𝒢 be a NSContraZC mapping. Let a NSOS (S, ϱ) in (𝕋, σ, ϱ) and a NSCS 

(B, ϱ) in (𝕎, τ, ϱ) such that 𝒢 −1(S, ϱ) ⊆  (B, ϱ). Then (K, ϱ) = 𝕋 – 𝒢 ((𝐵, 𝜚) 𝑐)𝑐 is NSZCS of 
(𝕋, σ, ϱ) such that 𝒢 −1(K, ϱ) ⊆  (B, ϱ).  

 Suffciency: Assume (B, ϱ) is a NSCS of (𝕎, τ, ϱ). Then, ((𝒢(𝐵, 𝜚))𝑐 is a NSS of (𝕋, σ, ϱ)  and 

(𝐵, 𝜚) 𝑐 is NSOS in (𝕎, τ, ϱ) such that 𝒢 −1((𝒢(𝐵, 𝜚))𝑐 ⊆ (𝐵, 𝜚) 𝑐. By presumption, there is a 

NSZCS (K, ϱ) of (𝕋, σ, ϱ) such that (𝒢(𝐵, 𝜚))𝑐  ⊆  (K, ϱ) and 𝒢 −1 (K, ϱ) ⊆ (𝐵, 𝜚) 𝑐.  Therefore,   

 NSContra𝛿C  

NNNNnsNS

C𝛿OS 
 
 NSContraC  

NS

Type equation here.

OS 
 

 NSContraPC 
 NSContra𝛿SC   

 NSContraZC 

NS

Type equation here.

OS 
 

 NSContraeC  

NS

Type equation here.

OS 
 



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(B, ϱ) ⊆  (𝒢−1(𝐾, 𝜚))𝑐. Hence (𝐾, 𝜚) 𝑐 ⊆ 𝒢(𝐵, ϱ) ⊆ 𝒢 ((𝒢 −1(K, ϱ) )𝑐) which implies 𝒢(B, ϱ)  = 

(𝐾, 𝜚) 𝑐.  As (𝐾, 𝜚) 𝑐 is NSZOS of (𝕋, σ, ϱ), 𝒢 (B, ϱ) is NSZOS in (𝕋, σ, ϱ) and hence 𝒢 is 

NSContraZC mapping. 

 

Theorem 6.3 If 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is NSC and ℋ : (𝕋, σ, ϱ)  → (𝕌, ρ, ϱ)  is NSContraZC. 

Then ℋ ∘  𝒢: (𝕎, τ, ϱ) → (𝕌, ρ, ϱ)  is NSContraZC. 

 

 Proof. Let (S, ϱ) be a NSCS in (𝕎, τ, ϱ). As 𝒢 is NSC mapping, 𝒢 (S, ϱ) is NSCS in (𝕋, σ, ϱ). As ℋ 

is NSContraZC mapping (ℋ ∘ 𝒢) (S, ϱ) = ℋ(𝒢 S, ϱ)) is NSZOS in (𝕌, ρ, ϱ). Hence ℋ ∘ 𝒢 is 

NSContraZC mapping. 

 

Theorem 6.4 If 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is NSContraZCmap, then NSZint(𝒢 (S, ϱ) ) ⊇ 𝒢 

(NSint(S, ϱ)).  

Proof. The proof is obvious from Defnition 6.1 and Defnition neutrosophic soft Z-interior.  

Theorem 6.5 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) and ℋ : (𝕋, σ, ϱ)  → (𝕌, ρ, ϱ) be NSContraZC mappings. 

If every NSZOS of (𝕋, σ, ϱ)  is NSOS, then  ℋ ∘  𝒢: (𝕎, τ, ϱ) → (𝕌, ρ, ϱ) is NSZC.  

Proof. Let (S, ϱ) be a NSCS in (𝕎, τ, ϱ). As, 𝒢 is NSContraZC mapping, 𝒢(S, ϱ) is NSZOS in 

(𝕋, σ, ϱ). By presumption, 𝒢(S, ϱ) is NSOS of (𝕋, σ, ϱ). As ℋ is NSContraZC mapping, ℋ (𝒢 (S, ϱ)) 

= (ℋ ∘  𝒢) (S, ϱ) is NSZCS in (𝕌, ρ, ϱ). Hence ℋ ∘  𝒢 is NSZC mapping. 

Theorem 6.6 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a bijective mapping. Then, the following statements 

are equivalent.  

(i) 𝒢 is a NSContraZO mapping.  

(ii) 𝒢 is a NSContraZC mapping.  

(iii) 𝒢 −1 is NSZCts mapping. 

Proof. (i) ⟹ (ii) : Assume 𝒢 is a NSContraZO mapping. If NSOS (S, ϱ) in (𝕎, τ, ϱ), by 

presumption 𝒢(S, ϱ) is a NSZCS in (𝕋, σ, ϱ). But now, (S, ϱ) is NSCS in (𝕎, τ, ϱ). So, 1(𝕎,𝜚) − (S, 

ϱ) is a NSOS in (𝕎, τ, ϱ). By assumption, 𝒢(1(𝕋,𝜚) − (S, ϱ)) is a NSZCS in (𝕋, σ, ϱ). Hence 1(𝕎,𝜚) − 

𝒢(1(𝕋,𝜚)) − (S, ϱ) is a NSZOS in (𝕋, σ, ϱ). Thus, 𝒢 is a NSContraZC mapping.  

(ii) ⟹ (iii) : Consider a NSCS in (𝕎, τ, ϱ). By assumption, 𝒢(S, ϱ) is a NSZOS in (𝕋, σ, ϱ). Hence, 

𝒢(S, ϱ)  = (𝒢 −1) −1 (S, ϱ). So, 𝒢 −1  is a NSZOS in (𝕋, σ, ϱ). Thus, 𝒢 −1  is NSZCts.  

(iii) ⟹ (i) : Consider a NSOS (S, ϱ) in (𝕎, τ, ϱ). By assumption, (𝒢 −1) −1 (S, ϱ) = 𝒢 (S, ϱ)  is a 

NSContraZO mapping. 

7 Neutrosophic soft contra Z-homeomorphism 

Defnition7.1 A bijection 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is called a neutrosophic soft contra Z-

homeomorphism (briefly, NSContraZHom) if 𝒢 and 𝒢 −1 are NSContraZCts mappings.  

 

 



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Theorem7.1 Each NSContraHom is a NSContraZHom. But the converse not true.  

Proof.: Assume 𝒢 is NSContraHom. Then 𝒢 and 𝒢 −1 are NSContraCts. We know that each 

NSContraCts function is NSContraZCts. So, 𝒢 and 𝒢 −1 are NSContraZCts. Thus, 𝒢 is a 

NSContraZHom. 

Example7.1 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1, ϱ), (𝑆2, ϱ) (𝑆3, ϱ) and (𝑆4, ϱ)  in 𝕎 and (𝑉1,ϱ) in 𝕋  are defined as  

(𝑆1, 𝑒1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉  

(𝑆1, 𝑒2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉  

(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

 

(𝑆2, 𝑒2) = 〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) = 〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) = 〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  

(𝑆4, 𝑒1) = 〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.6
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.5
)〉  

(𝑆4, 𝑒2) = 〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.3
) , (

𝜇𝑤2

0.6
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.4
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.3
)〉  

(𝑉1, 𝑒1) = 〈(
𝜇𝑡1,

0.5
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.6
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.5
)〉  

(𝑉2, 𝑒2) = 〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.3
) , (

𝜇𝑡2

0.6
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.4
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.3
)〉  

Here, we have τ = {0(𝕎 ,𝜚) , 1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ) }  and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑉1, ϱ)}.   Let 𝒢 

: (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be an identity mapping. Then 𝒢 is a NSContraZHom because 

(𝑆1, ϱ), (𝑆2, ϱ) and (𝑆3, ϱ)  are NSOS in 𝕎 and 𝒢 (𝑆1, ϱ), 𝒢(𝑆2, ϱ) and 𝒢(𝑆3, ϱ) are NSZCS  in 𝕋 and 

𝒢 −1 (𝑉1, ϱ)  = (𝑆4, ϱ) is NSZCS in 𝕎. But 𝒢 is not NSContraHom because 𝒢(𝑆1, ϱ), 𝒢(𝑆2, ϱ) and 

𝒢(𝑆3, ϱ) are not NSCS in 𝕋 and 𝒢 −1(𝑉1, ϱ) = (𝑆4, ϱ) is a NSCS in 𝕎. 

Theorem7.2 Consider a bijective mapping 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ). The following statements are 

equivalent if 𝒢 is NSContraZCts.  

(i) 𝒢 is a NSContraZC mapping.  

(ii) 𝒢 is a NSContraZO mapping. 

(iii) 𝒢 −1 is a NSContraZHom. 

Proof.: (i) ⟹ (ii): Let 𝒢 be a bijective mapping and a NSContraZC mapping. Therefore, 𝒢 −1  is a 

NSContraZCts mapping. As each NSOS in (𝕎, τ, ϱ) is a NSZCS in (𝕋, σ, ϱ), 𝒢 is a NSContraZO 

mapping.  



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(ii) ⟹  (iii): Assume 𝒢 is a bijective and NSConraO  mapping. Also, 𝒢 −1  is a NSContraZCts 

mapping. Therefore, 𝒢 and  𝒢 −1  are NSContraZCts. Thus, 𝒢 is a NSContraZHom.  

(iii) ⟹  (i): Assume 𝒢 is a NSContraZHom. So, 𝒢 and 𝒢 −1  are NSContraZCts. As every NSCS in 
(𝕎, τ, ϱ)  is a NSZOS in (𝕋, σ, ϱ), 𝒢 is a NSContraZC mapping.  

Theorem 7.3 Let  𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a NSContraZHom. If (𝕎, τ, ϱ)  and (𝕋, σ, ϱ) are 

NSZT1

2

 - spaces, then 𝒢 is a NSContraHom. 

Proof. Consider a NSCS (S, ϱ) in (𝕋, σ, ϱ). So, 𝒢 −1 (S, ϱ) is a NSZOS in (𝕎, τ, ϱ). As, (𝕎, τ, ϱ)  is 

a NSZT1

2

 -space, 𝒢 −1 (S, ϱ) is a NSOS in (𝕎, τ, ϱ). Therefore, 𝒢 is NSContraCts. By hypothesis, 

𝒢 −1   is NSContraZCts. Let (B, ϱ) be a NSCS in (𝕎, τ, ϱ). Then 𝒢(B, ϱ)  is a NSZOS in (𝕋, σ, ϱ), by 

presumption. Since (𝕋, σ, ϱ) is a NSZT1

2

 - space, 𝒢(B, ϱ) is a NSOS in (𝕋, σ, ϱ). Therefore, 𝒢 −1  is 

NSContraCts. Thus  𝒢 is a NSContraHom. 

Theorem 7.4 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be a NSCts. If (𝕋, σ, ϱ) are NSZT1

2

 - space, then the 

following are equivalent.  

1. 𝒢 is NSContraZC mapping.  

2. If (B, ϱ)  is a NSOS in (𝕎, τ, ϱ), then 𝒢 (B, ϱ)  is NSZCS in (𝕋, σ, ϱ).  

3. 𝒢 (NSint(B, ϱ)) ⊆ NScl(NSint(𝒢 (B, ϱ))) for every NSS (B, ϱ) in (𝕎, τ, ϱ). 

Proof. (i) ⟹  (ii) : Obvious.  

(ii) ⟹  (iii) Consider a NSS (B, ϱ)  in (𝕎, τ, ϱ). We know that, NSint(B, ϱ)  is a NSOS in (𝕎, τ, ϱ). 

Then, 𝒢(NSint((B, ϱ)) is a NSZCS in (𝕋, σ, ϱ). Since (𝕋, σ, ϱ) is a NSZT1

2

 -space 𝒢(NSint(B, ϱ)) is a 

NSCS in (𝕋, σ, ϱ). Therefore, 𝒢(NSint(B, ϱ)) = NScl(𝒢(𝑁𝑆𝑖𝑛𝑡((B, ϱ))) ⊆ NScl(NSint(𝒢((B, ϱ))). 

NScl(NSint(G(B ))).  

(iii) ⟹  (i) Let (B, ϱ) be a NSCS in (𝕎, τ, ϱ). Then, (𝐵, 𝜚) 𝑐 is a NSOS in (𝕎, τ, ϱ). As, 𝒢 

(NSint((𝐵, 𝜚) 𝑐) ⊆ NScl(NSint(𝒢(𝐵, 𝜚) 𝑐)), we get 𝒢((𝐵, 𝜚) 𝑐) ⊆ NScl(NSint(𝒢(𝐵, 𝜚) 𝑐)). Therefore, 

𝒢((𝐵, 𝜚) 𝑐) is NSZCS in (𝕋, σ, ϱ). Thus, 𝒢(B, ϱ) is a NSZOS in (𝕎, τ, ϱ). Hence, 𝒢 is a NSContraZC 

mapping. 

Theorem 7.5 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) and ℋ : (𝕋, σ, ϱ)  → (𝕌, ρ, ϱ) be a NSContraZC, where 

(𝕎, τ, ϱ) and (𝕌, ρ, ϱ)  are two NSts’s and (𝕋, σ, ϱ) a NSZT1

2

 - space, then the composition ℋ ∘  𝒢  is 

NSZC.  

Proof. Consider a NSCS(B, ϱ)  in (𝕎, τ, ϱ). As 𝒢 is NSContraZC and 𝒢(B, ϱ)  is a NSZOS in 

(𝕋, σ, ϱ), by assumption, 𝒢(B, ϱ)  is a NSOS in (𝕋, σ, ϱ). Since ℋ is NSContraZC, then ℋ (𝒢(B, ϱ)) 

is NSZCS in (𝕌, ρ, ϱ) and ℋ (𝒢(B, ϱ)) = ( ℋ ∘  𝒢 ) (B, ϱ). Thus, ℋ ∘  𝒢 is NSZC.  

Theorem 7.6 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) and ℋ : (𝕋, σ, ϱ)  → (𝕌, ρ, ϱ) be two NSts’s, then the 

following hold.  

1. If ℋ ∘  𝒢 is NSContraZO and 𝒢 is NSCts, then ℋ is NSContraZO.  



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2. If ℋ ∘  𝒢 is NSO and ℋ is NSContraZCts, then ℋ is NSContraZO.  

Proof. The proof is obvious from Definition 5.1, Definitions of neutrosophic soft Z continuous 

function, Definition of neutrosophic soft Z open mapping and Definitions 3.1. 

8 NeutrosophicsoftcontraZ-Chomeomorphism 

Definition 8.1 A bijection 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is called a neutrosophic soft contraZ-

Chomeomorphism (briefly, NSContraZCHom) if 𝒢 and 𝒢 −1 are NSContraZ-irr mappings.  

Theorem 8.1 Each NSContraZCHom is a NSContraZHom. But the converse not true. 

 Proof.: Consider a NSOS(S, ϱ)  in (𝕋, σ, ϱ). Then, (S, ϱ) is a NSZOS in (𝕋, σ, ϱ). By presumption, 

𝒢 −1 (S, ϱ) is a NSZCS in (𝕎, τ, ϱ). Therefore, 𝒢 is a NSContraZCts mapping. So 𝒢 and 𝒢 −1 are 

NSContraZCts  mapping. Thus, 𝒢 is a NSContraZHom. 

Example 8.1 Let 𝕎 = { 𝑤1, 𝑤2,   𝑤3 } =  {𝑡1 , 𝑡2,  𝑡3} = 𝕋, ϱ = { 𝑒1, 𝑒2} and NS sets 

(𝑆1, ϱ), (𝑆2, ϱ) (𝑆3, ϱ) and (𝑆4, ϱ)  in 𝕎 and (𝑉1,ϱ) in 𝕋  are defined as  

(𝑆1, 𝑒1) =  〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.4
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.7
)〉  

(𝑆1, 𝑒2) =  〈(
𝜇𝑤1,

0.2
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.2
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.8
)〉  

(𝑆2, 𝑒1) =  〈(
𝜇𝑤1,

0.5
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

 

(𝑆2, 𝑒2) = 〈(
𝜇𝑤1,

0.4
,

𝜎𝑤1

0.6
,

𝜈𝑤1

0.6
) , (

𝜇𝑤2

0.3
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.7
) , (

𝜇𝑤3

0.3
,

𝜎𝑤3

0.7
,

𝜈𝑤3

0.4
)〉  

(𝑆3, 𝑒1) = 〈(
𝜇𝑤1,

0.3
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.3
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.2
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.8
)〉  

(𝑆3, 𝑒2) = 〈(
𝜇𝑤1,

0.1
,

𝜎𝑤1

0.3
,

𝜈𝑤1

0.7
) , (

𝜇𝑤2

0.1
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.8
) , (

𝜇𝑤3

0.1
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.9
)〉  

(𝑆4, 𝑒1) = 〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.5
,

𝜈𝑤1

0.5
) , (

𝜇𝑤2

0.5
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.5
) , (

𝜇𝑤3

0.6
,

𝜎𝑤3

0.5
,

𝜈𝑤3

0.6
)〉  

(𝑆4, 𝑒2) = 〈(
𝜇𝑤1,

0.6
,

𝜎𝑤1

0.4
,

𝜈𝑤1

0.4
) , (

𝜇𝑤2

0.7
,

𝜎𝑤2

0.5
,

𝜈𝑤2

0.3
) , (

𝜇𝑤3

0.4
,

𝜎𝑤3

0.3
,

𝜈𝑤3

0.3
)〉  

(𝑉1, 𝑒1) = 〈(
𝜇𝑡1,

0.4
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.4
,

𝜈𝑡2

0.8
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.7
)〉  

(𝑉1, 𝑒2) = 〈(
𝜇𝑡1,

0.2
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.6
) , (

𝜇𝑡2

0.2
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.7
) , (

𝜇𝑡3

0.2
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.8
)〉  

(𝑉2, 𝑒1) = 〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.5
,

𝜈𝑡1

0.5
) , (

𝜇𝑡2

0.5
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.5
) , (

𝜇𝑡3

0.6
,

𝜎𝑡3

0.5
,

𝜈𝑡3

0.6
)〉  

(𝑉2, 𝑒2) = 〈(
𝜇𝑡1,

0.6
,

𝜎𝑡1

0.4
,

𝜈𝑡1

0.4
) , (

𝜇𝑡2

0.7
,

𝜎𝑡2

0.5
,

𝜈𝑡2

0.3
) , (

𝜇𝑡3

0.4
,

𝜎𝑡3

0.3
,

𝜈𝑡3

0.3
)〉  

Here, we have τ = {0(𝕎 ,𝜚) , 1(𝕎 ,𝜚), (𝑆1, ϱ), (𝑆2, ϱ), (𝑆3, ϱ) } and 𝜎 = {0(𝕋,𝜚),   1(𝕋,𝜚),   (𝑉1, ϱ)}.   Let 𝒢 

: (𝕎, τ, ϱ) → (𝕋, σ, ϱ) be an identity mapping. Then 𝒢 is a NSContraZHom because 

(𝑆1, ϱ), (𝑆2, ϱ) and (𝑆3, ϱ)  are NSOS in 𝕎 and 𝒢 (𝑆1, ϱ), 𝒢(𝑆2, ϱ) and 𝒢(𝑆3, ϱ) are NSZCS  in 𝕋. 



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Also(𝑉1, ϱ) is NSOS in 𝕋 and 𝒢 −1 (𝑉1, ϱ)  = (𝑆1, ϱ) is NSZCS in 𝕎. But 𝒢 is not NSContraHom 

because 𝒢(𝑉2, ϱ)  is NSZCS in 𝕋 but (𝑉2, ϱ) is not NSZOS in 𝕎. 

Theorem 8.2 If  𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) is a NSContraZCHom, then NSint(𝒢 −1 (S, ϱ)) ⊆ 𝒢 −1 

(NScl(B, ϱ)) for every NSS(S, ϱ) in (𝕋, σ, ϱ). 

Proof.: Consider a NSS(S, ϱ) in (𝕋, σ, ϱ). Since NScl(S, ϱ) is a NSCS in (𝕋, σ, ϱ)  and every NSCS  

is a NSZCS in (𝕋, σ, ϱ).  As 𝒢 is NSContraZ-irr, 𝒢 −1(NScl(S, ϱ)) is a NSZOS in (𝕎, τ, ϱ). Then, 

NSint(𝒢 −1 (NScl(S, ϱ))) = 𝒢 −1 (NScl(S, ϱ)). Here, NSZint(𝒢 −1 (S, ϱ)) ⊆ NSZint(𝒢 −1 (NScl(S, ϱ))) 

= 𝒢 −1 (NScl(S, ϱ)). Therefore NSZint(𝒢 −1(S, ϱ)) ⊆ 𝒢 −1 (NScl(S, ϱ)) for every NSS(S, ϱ)  in 
(𝕋, σ, ϱ). 

Theorem 8.3 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ)  be a NSContraZCHom. Then NSZint(𝒢 −1 (S, ϱ)) ⊆ 𝒢 −1 

(NSZcl(S, ϱ)) for every NSS(S, ϱ) in (𝕋, σ, ϱ).  

Proof.: As 𝒢 is a NSContraZCHom, 𝒢 is a NSContraZ-irr mapping. Consider a NSS(S, ϱ). It is 

obvious that, NSZcl(S, ϱ) is a NSZCS in (𝕋, σ, ϱ). As 𝒢 −1(S, ϱ) ⊆ 𝒢 −1(NSZcl(S, ϱ)), we have 

NSZint(𝒢 −1(S, ϱ)) ⊆ NSZint(𝒢 −1 (NSZcl(S, ϱ)) ⊆ 𝒢 −1 (NSZcl(S, ϱ)). Thus, NSZint(𝒢 −1 (S, ϱ)) ⊆ 

𝒢 −1 (NSZcl(S, ϱ)).  

Theorem 8.4 Let 𝒢 : (𝕎, τ, ϱ) → (𝕋, σ, ϱ) and ℋ : (𝕋, σ, ϱ)  → (𝕌, ρ, ϱ)  be a NSContraZCHom’s.  

Then  ℋ ∘  𝒢  is a NSZCHom.  

Proof.: Assume that 𝒢 and ℋ are two NSContraZCHom’s. Let (S, ϱ) be a NSZCS in (𝕌, ρ, ϱ). Then, 

ℋ−1(S, ϱ) is a NSZOS in (𝕋, σ, ϱ). By presumption, 𝒢−1((ℋ−1((S, ϱ)) is a NSZCS in  (𝕎, τ, ϱ). 

Therefore,(ℋ ∘  𝒢  )−1  is a NSZ-irr mapping. Assume (B, ϱ) is NSZCS in (𝕎, τ, ϱ). Then, by 

hypothesis, 𝒢 (ℋ) is a NSZOS in (𝕋, σ, ϱ). Hence, ℋ(𝒢 (B, ϱ)) is a NSZCS in (𝕌, ρ, ϱ). This implies 

that ℋ ∘  𝒢  is NSZ-irr mapping. Thus, ℋ ∘  𝒢 is a NSZCHom. 

9.  Conclusion 

In this paper, we have introduced and explored contra Z-continuous, contra Z-irresolute, contra Z-

open, and contra Z-closed maps in neutrosophic soft topological spaces. Additionally, we have 

investigated contra Z and Z-C homeomorphisms, with relevant theorems and examples, thereby 

contributing to the expansion of neutrosophic soft topology. These results pave the way for future 

research and potential applications in this emerging field. 

 

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