𝜼−𝐈𝐧𝐭𝐮𝐢𝐭𝐢𝐨𝐧𝐢𝐬𝐭𝐢𝐜 𝐅𝐮𝐳𝐳𝐲 𝐒𝐨𝐟𝐭 𝐆𝐫𝐨𝐮𝐩𝐬


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(3) (2022), Pages 354-361 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: December 23, 2021 Reviewed: May 25, 2022 Accepted: July 17, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i3.14555    

𝜼−𝐈𝐧𝐭𝐮𝐢𝐭𝐢𝐨𝐧𝐢𝐬𝐭𝐢𝐜 𝐅𝐮𝐳𝐳𝐲 𝐒𝐨𝐟𝐭 𝐆𝐫𝐨𝐮𝐩𝐬 

Mustika Ana Kurfia*, Noor Hidayat, Corina Karim 

Mathematics Department, Universitas Brawijaya, Malang, Indonesia 

Email: muzematika@gmail.com 

ABSTRACT 

The fuzzy set theory was introduced by Zadeh in 1965 and the soft set theory was introduced by 
Molodtsov in 1999. Recently, many researchers have developed these two theories and combined 
the theory of fuzzy set and soft set became the fuzzy soft set. In this research, we present the idea 
of the 𝜂 −intuitionistic fuzzy soft group defined on the 𝜂 −intuitionistic fuzzy soft set. The main 
purpose of this research is to create a new concept, which is an 𝜂 −intuitionistic fuzzy group. To 
achieve this, we combine the concept of 𝜂 −intuitionistic fuzzy group and intuitionistic fuzzy soft 
group. As the main result, we prove the correlation between intuitionistic fuzzy soft group and 
𝜂 −intuitionistic fuzzy soft group along with some properties of 𝜂 −intuitionistic fuzzy soft group. 
Also, we prove some properties of subgroup of an 𝜂 −intuitionistic fuzzy soft group. An 
𝜂 −intuitionistic fuzzy soft homomorphism is also proved. 

Keywords: intuitionistic fuzzy group; intuitionistic fuzzy soft group; 𝜂 −intuitionistic fuzzy 
group; 𝜂 −intuitionistic fuzzy soft group 

INTRODUCTION 

 The theory of fuzzy has been studied by many researchers in various fields. Zadeh 
introduced the fuzzy set theory in [1] by defining a membership function that maps each 
member of a set to a closed interval of 0 and 1. Then Atanassov formed the intuitionistic 
fuzzy set that consist of membership function and nonmembership function in [2]. The 
theory of fuzzy set and intuitionistic fuzzy set was developed into group theory became 
fuzzy subgroup in [3] and intuitionistic fuzzy subgroup in [4]. The intuitionistic fuzzy 
subgroup was studied in various types. For example, the intuitionistic L-fuzzy subgroups 
formed in [5], the (𝑠, 𝑡] −intuitionistic fuzzy subgroups defined in [6], definition of 
(𝛼, 𝛽)cut of intuitionistic fuzzy subgroups in [7], and t-intuitionistic fuzzy subgroups in 
[8]. Doda and Sharma studied the finite groups of different orders and gave the idea of 
recording the count of intuitionistic fuzzy subgroups in [9]. Zhou and Xu extended the 
intuitionistic fuzzy sets based on the hesitant fuzzy membership in [10]. The concept of 
the (𝜆, 𝜇) −intuitionistic fuzzy subgroups and normal subgroups were defined in [11]. 
Then the fundamental properties of t-intuitionistic fuzzy abelian subgroup along with the 
homomorphism of t-intuitionistic fuzzy abelian subgroup were studied in [12]. Latif, et al.  
studied the fundamental theorems of t-intuitionistic fuzzy isomorphism of t-intuitionistic 
fuzzy subgroup in [13]. Moreover, the concept of 𝜉 −intuitionistic fuzzy subgroup, 
𝜉 −intuitionistic fuzzy cosets, and 𝜉 −intuitionistic fuzzy normal subgroup were 
characterized in [14]. Based on those research, Shuaib, et al. in [15] formed a concept 

http://dx.doi.org/10.18860/ca.v7i3.14555
mailto:muzematika@gmail.com


𝜂 −Intuitionistic Fuzzy Soft Groups  

Mustika Ana Kurfia 355 

called an 𝜂 −intuitionistic fuzzy subgroup on an 𝜂 −intuitionistic fuzzy set along with 
𝜂 −intuitionistic fuzzy homomorphism.   

While the theory of soft set is introduced in [16] which is an ordered pair of 
parameter and function that maps each member of parameter to the power set of an 
empty set. The soft sets constructed in the form of membership function became fuzzy 
soft sets defined in [17]. Maji, et al. introduced the concept of intuitionistic fuzzy soft set 
which is a generalization of intuitionistic fuzzy set and soft set in [18]. Furthermore, the 
operation properties and algebraic structure of intuitionistic fuzzy soft set were discussed 
in [19]. Soft group on the soft set is defined in [16]. Then Aygünoglu and Aygun in [20] 
developed the concept of soft group in the form of membership function became fuzzy 
soft group. The concept of intuitionistic fuzzy soft set to semigroup was applied in [21] 
and intuitionistic fuzzy soft ideals over ordered ternary semigroup was defined in [22]. 
The concept of soft group is developed into an intersection called soft int-group in [23]. 
Moreover, Karaaslan, et al. in [24] applied the concept of soft int-group into intuitionistic 
fuzzy soft set by forming the intuitionistic fuzzy soft group and investigate some 
properties of intuitionistic fuzzy soft group.    

Based on [15] and [24], we introduce the notion of the 𝜂 −intuitionistic fuzzy soft 
group on the 𝜂 −intuitionistic fuzzy soft set and give some basic properties. Moreover, we 
define the notion of the 𝜂 −intuitionistic fuzzy subgroup and investigate the properties of 
homomorphism of an 𝜂 −intuitionistic fuzzy soft group. 
 

METHODS  

The method of this research is literature review, data collecting techniques by 
conducting review studies of books, notes, and other scientific research results related to 
the object of the problem. In this paper, we begin by forming the definition of 
𝜂 −intuitionistic fuzzy soft set based on the definition of intuitionistic fuzzy soft set and 
𝜂 −intuitionistic fuzzy set. Then, we form the definition of 𝜂 −intuitionistic fuzzy soft 
group based on the definition of intuitionistic fuzzy soft group. We continue to prove some 
properties of the 𝜂 −intuitionistic fuzzy soft group along with subgroup of an 
𝜂 −intuitionistic fuzzy soft group. Moreover, we continue to define the definition of image 
and pre-image of an 𝜂 −intuitionistic fuzzy soft group and prove the homomorphism of 
an 𝜂 −intuitionistic fuzzy soft group. Here is shown the definition of intuitionistic fuzzy 
soft set, 𝜂 −intuitionistic fuzzy set, and intuitionistic fuzzy soft group. 

Definition 1[18]. Let 𝑋 be a non empty set and 𝐸 be a set of parameter with 𝐴 ⊆ 𝐸. Let 
ℐℱ(𝑋) be a set of all intuitionistic fuzzy set of 𝑋. An intuitionistic fuzzy soft set of 𝐴 over 𝑋 
is defined by 

Γ𝐴 = {(𝑎, 𝛾𝐴(𝑎)): 𝑎 ∈ 𝐴}, 

where 𝛾𝐴(𝑎): 𝐸 → ℐℱ(𝑋) such as 𝛾𝐴(𝑎) = ∅ if 𝑎 ∉ 𝐴. Therefore, for all 𝑎 ∈ 𝐸, 𝛾𝐴(𝑎) is called 
intuitionistic fuzzy value set of 𝑎. 𝛾𝐴(𝑎) can be written as 

𝛾𝐴(𝑎) = {(𝑥, 𝜇𝛾𝐴(𝑎)(𝑥), 𝜇𝛾𝐴(𝑎)(𝑥)) : 𝑥 ∈ 𝑋}, 

for all 𝑎 ∈ 𝐸. 

Definition 2 [15]. Suppose 𝐴 is an intuitionistic fuzzy set of a non empty set 𝑋 where 𝜇𝐴 
be a membership function and 𝜈𝐴 be a nonmembership function in 𝐴. Let 𝜂 ∈ [0, 1]. An 
𝜂 −intuitionistic fuzzy set is defined by 

𝐴𝜂 = {(𝑥, 𝜇𝐴𝜂 (𝑥), 𝜈𝐴𝜂 (𝑥)): 𝑥 ∈ 𝑋}, 

where 𝜇𝐴𝜂 (𝑥) = 𝜓[𝜇𝐴(𝑥), 𝜂] = √𝜇𝐴(𝑥) ⋅ 𝜂 and 𝜈𝐴𝜂 (𝑥) = 𝜓
′[𝜈𝐴(𝑥), 1 − 𝜂] = √𝜈𝐴(𝑥) ⋅ (1 − 𝜂). 



𝜂 −Intuitionistic Fuzzy Soft Groups  

Mustika Ana Kurfia 356 

Definition 3 [24]. Let 𝐺 be an arbitrary group and Γ𝐺  be an intuitionistic fuzzy soft set 
over universe 𝑋, then Γ𝐺  is called intuitionistic fuzzy soft group on 𝐺 over 𝑋 if  
1. 𝛾𝐺 (𝑥𝑦) ⊇ 𝛾𝐺 (𝑥) ∩ 𝛾𝐺 (𝑦), 
2. 𝛾𝐺 (𝑥

−1) = 𝛾𝐺 (𝑥), 
for all 𝑥 ∈ 𝐺. 

RESULTS AND DISCUSSION  

In this section, we introduce the notion of 𝜂 −intuitionistic fuzzy soft group on 
𝜂 −intuitionistic fuzzy soft set. Inspiring from Definition 1 and Definition 2, we define the 
notion of 𝜂 −intuitionistic fuzzy soft set.  
Definition 4. Suppose Γ𝐴 be an intuitionistic fuzzy soft set of parameter 𝐴 over universe 
𝑋 and 𝜂 ∈ [0, 1]. An 𝜂 −intuitionistic fuzzy soft set of 𝐴 over 𝑋 is defined by 

Γ𝐴
𝜂

= {(𝑎, 𝛾𝐴
𝜂 (𝑎)): 𝑎 ∈ 𝐴}, 

where 𝛾𝐴
𝜂(𝑎) is an 𝜂 −intuitionistic fuzzy set of 𝑋 defined as 

𝛾𝐴
𝜂 (𝑎) = Ψ[𝛾𝐴(𝑎), 𝜂], 

for all 𝑎 ∈ 𝐴, where  
Ψ[𝛾𝐴(𝑎), 𝜂] = {(𝑥, 𝜓[𝜇𝛾𝐴(𝑎)(𝑥), 𝜂], 𝜓[𝜈𝛾𝐴(𝑎)(𝑥), 1 − 𝜂]): 𝑥 ∈ 𝑋}. 

The value of 𝜓[𝜇𝛾𝐴(𝑎)(𝑥), 𝜂] = √𝜇𝛾𝐴(𝑎)(𝑥) ⋅  𝜂 and 𝜓[𝜈𝛾𝐴(𝑎)(𝑥), 1 − 𝜂] = √𝜇𝛾𝐴(𝑎)(𝑥) ⋅ (1 − 𝜂). 

Some basic properties such as intersection and union of 𝜂 −intuitionistic fuzzy soft set is 
proved by the following proposition. 

Proposition 1. The intersection of any two 𝜂 −intuitionistic fuzzy soft sets is an 
𝜂 −intuitionistic fuzzy soft set. 
Proof. Let Γ𝐴

𝜂 = {(𝑎, 𝛾𝐴
𝜂 (𝑎)): 𝑎 ∈ 𝐴} and Γ𝐵

𝜂 = {(𝑏, 𝛾𝐵
𝜂(𝑏)): 𝑏 ∈ 𝐵}, respectively be two 

𝜂 −intuitionistic fuzzy soft sets of 𝐴 and 𝐵 over 𝑋. For any 𝑐 ∈ 𝐴 ∩ 𝐵, then 
𝛾(𝐴∩̌𝐵)𝜂 (𝑐) = Ψ[𝛾𝐴∩̌𝐵 (𝑐), 𝜂]  

= {(
𝑥, 𝜓[min[𝜇𝛾𝐴(𝑐)(𝑥), 𝜇𝛾𝐵(𝑐)(𝑥)] , 𝜂],   

𝜓[max[𝜈𝛾𝐴(𝑐)(𝑥), 𝜈𝛾𝐵(𝑐)(𝑥)] , 1 − 𝜂]
) : 𝑥 ∈ 𝑋}  

= {(
𝑥, 𝜓[min[𝜇𝛾𝐴(𝑐)(𝑥), 𝜂] , min[ 𝜇𝛾𝐵(𝑐)(𝑥), 𝜂]],           

𝜓[max[𝜈𝛾𝐴(𝑐)(𝑥), 1 − 𝜂] , max[𝜈𝛾𝐵(𝑐)(𝑥), 1 − 𝜂]]
) : 𝑥 ∈ 𝑋}  

= {(
𝑥, min[𝜓[𝜇𝛾𝐴(𝑐)(𝑥), 𝜂] , 𝜓[ 𝜇𝛾𝐵(𝑐)(𝑥), 𝜂]],            

max[𝜓[𝜈𝛾𝐴(𝑐)(𝑥), 1 − 𝜂] , 𝜓[ 𝜈𝛾𝐵(𝑐)(𝑥), 1 − 𝜂]]
) : 𝑥 ∈ 𝑋}  

= Ψ[𝛾𝐴(𝑐), 𝜂] ∩ Ψ[𝛾𝐵 (𝑐), 𝜂]  
= 𝛾𝐴𝜂∩̃𝐵𝜂 (𝑐).  

Hence Γ𝐴
𝜂

∩̃ Γ𝐵
𝜂

 is an 𝜂 −intuitionistic fuzzy soft set.  ∎ 

Remark 1. The union of any two 𝜂 −intuitionistic fuzzy soft set is an 𝜂 −intuitionistic 
fuzzy soft set. 

The notion of 𝜂 −intuitionistic fuzzy soft group is defined based on Definition 3. An 
𝜂 −intuitionistic fuzzy soft group must satisfy 2 axioms to be an 𝜂 −intuitionistic fuzzy 
soft group. 

Definition 5. Let 𝐺 be a group and Γ𝐺
𝜂  be an 𝜂 −intuitionistic fuzzy soft set over universe 

𝑈, then Γ𝐺
𝜂  is called an 𝜂 −intuitionistic fuzzy soft group over 𝑈 if  

1. 𝛾𝐺
𝜂 (𝑥𝑦) ⊇ 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑦), 

2. 𝛾𝐺
𝜂 (𝑥−1) = 𝛾𝐺

𝜂 (𝑥), 
for all 𝑥, 𝑦 ∈ 𝐺. 



𝜂 −Intuitionistic Fuzzy Soft Groups  

Mustika Ana Kurfia 357 

The property of the identity element of the group on an 𝜂 −intuitionistic fuzzy soft group 
is  proved by the following proposition. 

Proposition 2. Let 𝐺 be a group and Γ𝐺
𝜂

 be an 𝜂 −intuitionistic fuzzy soft group over 
universe 𝑈, then 𝛾𝐺

𝜂 (𝑒) ⊇ 𝛾𝐺
𝜂 (𝑥) for all 𝑥 ∈ 𝐺. 

Proof. Let Γ𝐺
𝜂 = {(𝑥, 𝛾𝐺

𝜂 (𝑥)): 𝑥 ∈ 𝐺} be an 𝜂 −intuitionistic fuzzy soft group and 𝑒 be an 

identity element of 𝐺. For 𝑒 ∈ 𝐺, then 𝛾𝐺
𝜂 (𝑒) = 𝛾𝐺

𝜂 (𝑥𝑥 −1). Since Γ𝐺
𝜂

 is an 𝜂 −intuitionistic 
fuzzy soft group, then 
𝛾𝐺

𝜂 (𝑒) ⊇ 𝛾𝐺
𝜂 (𝑥) ∩ 𝛾𝐺

𝜂 (𝑥−1)  
 = 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑥)  

 = 𝛾𝐺
𝜂 (𝑥).  

So, 𝛾𝐺
𝜂 (𝑒) ⊇ 𝛾𝐺

𝜂 (𝑥), for all 𝑥 ∈ 𝐺.  ∎ 

An 𝜂 −intuitionistic fuzzy soft set is called an 𝜂 −intuitionistic fuzzy soft group if it 
satisfies 2 axioms in Definition 5. Here we prove the alternative way of 𝜂 −intuitionistic 
fuzzy soft group. 

Theorem 1. An 𝜂 −intuitionistic fuzzy soft set is called 𝜂 −intuitionistic fuzzy soft group 
if and only if 𝛾𝐺

𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺
𝜂 (𝑥) ∩ 𝛾𝐺

𝜂 (𝑦) for all 𝑥, 𝑦 ∈ 𝐺. 
Proof.  
(⇒) Let Γ𝐺

𝜂 = {(𝑥, 𝛾𝐺
𝜂 (𝑥)): 𝑥 ∈ 𝐺} be an 𝜂 −intuitionistic fuzzy soft group. From 

Definition 5, then for all 𝑥, 𝑦 ∈ 𝐺 we have 
𝛾𝐺

𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺
𝜂 (𝑥) ∩ 𝛾𝐺

𝜂 (𝑦−1) = 𝛾𝐺
𝜂 (𝑥) ∩ 𝛾𝐺

𝜂 (𝑦). 
(⇐) Since 𝛾𝐺

𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺
𝜂 (𝑥) ∩ 𝛾𝐺

𝜂 (𝑦) for all 𝑥, 𝑦 ∈ 𝐺, then 
𝛾𝐺

𝜂 (𝑥−1) = 𝛾𝐺
𝜂 (𝑒𝑥) ⊇ 𝛾𝐺

𝜂 (𝑒) ∩ 𝛾𝐺
𝜂 (𝑥) ⊇ 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑥) = 𝛾𝐺

𝜂 (𝑥), 
and 

𝛾𝐺
𝜂 (𝑥) = 𝛾𝐺

𝜂 (𝑒(𝑥−1)−1) ⊇ 𝛾𝐺
𝜂 (𝑒) ∩ 𝛾𝐺

𝜂 (𝑥−1) ⊇ 𝛾𝐺
𝜂 (𝑥)−1 ∩ 𝛾𝐺

𝜂 (𝑥−1) = 𝛾𝐺
𝜂 (𝑥−1). 

Hence 𝛾𝐺
𝜂 (𝑥−1) = 𝛾𝐺

𝜂 (𝑥) for all 𝑥 ∈ 𝐺. Then for all 𝑥, 𝑦 ∈ 𝐺 we have 
𝛾𝐺

𝜂 (𝑥𝑦) = 𝛾𝐺
𝜂 (𝑥(𝑦−1)−1)  

 ⊇ 𝛾𝐺
𝜂 (𝑥) ∩ 𝛾𝐺

𝜂 (𝑦−1)  
 = 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑦).  

Hence 𝛾𝐺
𝜂 (𝑥𝑦) ⊇ 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑦) for all 𝑥, 𝑦 ∈ 𝐺. Therefore, Γ𝐺

𝜂  is an 𝜂 −intuitionistic 
fuzzy soft group.  ∎ 

Since 𝜂 −intuitionistic fuzzy soft group is defined based on intuitionistic fuzzy soft group, 
so there is a correlation between those concepts. The following theorem, we prove that 
every intuitionistic fuzzy soft group is an 𝜂 −intuitionistic fuzzy soft group. 

Theorem 2. If Γ𝐺  is an intuitionistic fuzzy soft group, then Γ𝐺
𝜂

 is an 𝜂 −intuitionistic fuzzy 
soft group for all 𝜂 ∈ [0, 1]. 

Proof. Let Γ𝐺 = {(𝑥, 𝛾𝐺 (𝑥)): 𝑥 ∈ 𝐺} be an intuitionistic fuzzy soft group over the universe 

𝑈 where 𝛾𝐺 (𝑥) = {(𝑢, 𝜇𝛾𝐺(𝑥)(𝑢), 𝜈𝛾𝐺(𝑥)(𝑢)) : 𝑢 ∈ 𝑈}. For any 𝑥, 𝑦 ∈ 𝐺 and 𝜂 ∈ [0, 1], then 

𝛾𝐺
𝜂 (𝑥𝑦−1) = Ψ[𝛾𝐺 (𝑥𝑦

−1), 𝜂]. 
Since Γ𝐺  is an intuitionistic fuzzy soft group, then 
𝛾𝐺

𝜂 (𝑥𝑦−1) ⊇ Ψ[(𝛾𝐺 (𝑥) ∩ 𝛾𝐺 (𝑦)), 𝜂]  

= {(
𝑢, 𝜓[min[𝜇𝛾𝐺(𝑥)(𝑢), 𝜇𝛾𝐺(𝑦)(𝑢)], 𝜂],

𝜓[max[𝜈𝛾𝐺(𝑥)(𝑢), 𝜈𝛾𝐺(𝑦)(𝑢)], 1 − 𝜂]
) : 𝑢 ∈ 𝑈}  

= {(
𝑥, 𝜓[min[𝜇𝛾𝐺(𝑥)(𝑢), 𝜂] , min[ 𝜇𝛾𝐺(𝑦)(𝑢), 𝜂]],           

𝜓[max[𝜈𝛾𝐺(𝑥)(𝑢), 1 − 𝜂] , max[𝜈𝛾𝐺(𝑦)(𝑢), 1 − 𝜂]]
) : 𝑢 ∈ 𝑈}  



𝜂 −Intuitionistic Fuzzy Soft Groups  

Mustika Ana Kurfia 358 

= {(
𝑥, min[𝜓[𝜇𝛾𝐺(𝑥)(𝑢), 𝜂] , 𝜓[ 𝜇𝛾𝐺(𝑦)(𝑢), 𝜂]],            

max[𝜓[𝜈𝛾𝐺(𝑥)(𝑢), 1 − 𝜂] , 𝜓[ 𝜈𝛾𝐺(𝑦)(𝑢), 1 − 𝜂]]
) : 𝑢 ∈ 𝑈}  

= Ψ[𝛾𝐺 (𝑥), 𝜂] ∩ Ψ[𝛾𝐺 (𝑦), 𝜂]  
= 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑦). 

Therefore, Γ𝐺
𝜂

 is an 𝜂 −intuitionistic fuzzy soft group. ∎  
Here we define the 𝜂 −intuitionistic fuzzy subgroup of an 𝜂 −intuitionistic fuzzy group. 
The properties of 𝜂 −intuitionistic fuzzy subgroup are proved in the following theorem. 

Definition 6. Suppose 𝐺 be a group and 𝐻 be a subgroup of 𝐺. Let Γ𝐺
𝜂

 be an 
𝜂 −intuitionistic fuzzy soft group over universe 𝑈, then Γ𝐻

𝜂  is called an 𝜂 −intuitionistic 
fuzzy soft subgroup of Γ𝐺

𝜂
 if Γ𝐻

𝜂
 is an 𝜂 −intuitionistic fuzzy soft group over 𝑈.  

Theorem 3. Let Γ𝐺
𝜂

 be an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈. Suppose Γ𝐻
𝜂

 
and Γ𝑁

𝜂
 be two 𝜂 −intuitionistic fuzzy soft subgroups of Γ𝐺

𝜂
, then Γ𝐻

𝜂
∩̃ Γ𝑁

𝜂
 is an 

𝜂 −intuitionistic fuzzy soft subgroup of Γ𝐺
𝜂 . 

Proof. Defined Γ𝐻
𝜂 ∩̃ Γ𝑁

𝜂 = {(𝑥, 𝛾𝐻∩̃𝑁
𝜂 (𝑥)): 𝑥 ∈ 𝐻 ∩ 𝑁}. For any 𝑥, 𝑦 ∈ 𝐺, then 

𝛾𝐻∩̃𝑁
𝜂 (𝑥𝑦−1) =  𝛾𝐻

𝜂 (𝑥𝑦−1) ∩  𝛾𝑁
𝜂 (𝑥𝑦−1)  

⊇ (𝛾𝐻
𝜂 (𝑥) ∩ 𝛾𝐻

𝜂 (𝑦)) ∩  (𝛾𝑁
𝜂 (𝑥) ∩ 𝛾𝑁

𝜂 (𝑦))  

= (𝛾𝐻
𝜂 (𝑥) ∩ 𝛾𝑁

𝜂 (𝑥)) ∩  (𝛾𝐻
𝜂 (𝑦) ∩ 𝛾𝑁

𝜂 (𝑦))  

= 𝛾𝐻∩̃𝑁
𝜂 (𝑥) ∩ 𝛾𝐻∩̃𝑁

𝜂 (𝑦).  
Hence Γ𝐻

𝜂 ∩̃ Γ𝑁
𝜂 is an 𝜂 −intuitionistic fuzzy soft subgroup of Γ𝐺

𝜂 .  ∎  

Theorem 4. Let Γ𝐺
𝜂

 be an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈 and 𝑒 be an 
identity elemen of 𝐺. Then Γ𝐺

𝜂 |𝑒 = {(𝑥, 𝛾𝐺
𝜂 (𝑥)): 𝛾𝐺

𝜂 (𝑥) =  𝛾𝐺
𝜂 (𝑒), 𝑥 ∈ 𝐺} is an 

𝜂 −intuitionistic fuzzy subgroup of 𝐺.  
Proof. Since (𝑒, 𝛾𝐺

𝜂 (𝑒)) ∈ Γ𝐺
𝜂 |𝑒, then Γ𝐺

𝜂 |𝑒 ≠ ∅. Let (𝑥, 𝛾𝐺
𝜂 (𝑥)), (𝑦, 𝛾𝐺

𝜂 (𝑦)) ∈ Γ𝐺
𝜂 |𝑒 , we 

have 𝛾𝐺
𝜂 (𝑥) = 𝛾𝐺

𝜂 (𝑦) =  𝛾𝐺
𝜂 (𝑒) for 𝑥, 𝑦 ∈ 𝐺. Then 

𝛾𝐺
𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺

𝜂 (𝑥) ∩ 𝛾𝐺
𝜂 (𝑦)  

= 𝛾𝐺
𝜂 (𝑒) ∩ 𝛾𝐺

𝜂 (𝑒)  
= 𝛾𝐺

𝜂 (𝑒). 
Since 𝛾𝐺

𝜂 (𝑒) ⊇ 𝛾𝐺
𝜂 (𝑥𝑦−1), then 𝛾𝐺

𝜂 (𝑥𝑦−1) = 𝛾𝐺
𝜂 (𝑒). Thus (𝑥𝑦−1, 𝛾𝐺

𝜂 (𝑥𝑦−1)) ∈ Γ𝐺
𝜂 |𝑒 . 

Therefore Γ𝐺
𝜂 |𝑒  is an 𝜂 −intuitionistic fuzzy subgroup of 𝐺.   ∎  

Here notioned the definition of image and pre-image of an 𝜂 −intuitionistic fuzzy soft 
group. 

Definition 7. Let Γ𝐴
𝜂

 and Γ𝐵
𝜂

 be two 𝜂 −intuitionistic fuzzy soft sets over universe 𝑈, and 
let 𝜙: 𝐴 → 𝐵, then 
1. Image of Γ𝐴

𝜂  under the map 𝜙, denoted by 𝜙(Γ𝐴
𝜂 ) defined by 

𝜙(Γ𝐴
𝜂 ) = {(𝑏, 𝜙(𝛾𝐴

𝜂 )(𝑏)): 𝑏 ∈ 𝐵}, 

where for all 𝑏 ∈ 𝐵, then 

𝜙(𝛾𝐴
𝜂 )(𝑏) = { 

∩ {𝛾𝐴
𝜂 (𝑎): 𝑎 ∈ 𝐴, 𝑓(𝑎) = 𝑏},   if 𝜙(𝑎) ∈ 𝜙(𝐴),   

𝛾∅
𝜂                                             ,    others.                   

 

2. Pre-image of Γ𝐵
𝜂  under 𝜙, denoted by 𝜙−1(Γ𝐵

𝜂 ) defined by 
𝜙−1(Γ𝐵

𝜂 ) = {(𝑎, 𝜙−1(𝛾𝐵
𝜂)(𝑎)): 𝑎 ∈ 𝐴}, 

where for all 𝑎 ∈ 𝐴, then 
𝜙−1(𝛾𝐵

𝜂)(𝑎) = 𝛾𝐵
𝜂 (𝜙(𝑎)). 

Lemma 1. Let Γ𝐴
𝜂  and Γ𝐵

𝜂  be two 𝜂 −intuitionistic fuzzy soft sets over 𝑈, then 𝜙(Γ𝐴
𝜂 ) and 

𝜙−1(Γ𝐵
𝜂 ) are 𝜂 −intuitionistic fuzzy soft sets over universe 𝑈. 



𝜂 −Intuitionistic Fuzzy Soft Groups  

Mustika Ana Kurfia 359 

Proof. Let Γ𝐴
𝜂

= {(𝑎, 𝛾𝐴
𝜂 (𝑎)): 𝑎 ∈ 𝐴} and Γ𝐵

𝜂
= {(𝑏, 𝛾𝐵

𝜂(𝑏)): 𝑏 ∈ 𝐵} respectively be two 

𝜂 −intuitionistic fuzzy soft sets of 𝐴 and 𝐵 over 𝑈. Let 𝜙: 𝐴 → 𝐵. From Definition 7, there 
is 𝑏 ∈ 𝜙(𝐴) such that 𝜙(𝑎) = 𝑏, then 

𝜙(𝛾𝐴
𝜂 )(𝑏) =∩ {𝛾𝐴

𝜂 (𝑎): 𝑎 ∈ 𝐴, 𝜙(𝑎) = 𝑏}. 
From proposition 1, we have 𝜙(𝛾𝐴

𝜂)(𝑏) is an 𝜂 −intuitionistic fuzzy set.  
If 𝜙(𝑎) ∉ 𝜙(𝐴), then 𝜙(𝛾𝐴

𝜂 )(𝑏) = 𝛾∅
𝜂  is an 𝜂 −intuitionistic fuzzy set. Then ∀𝑎 ∈ 𝐴, we 

have 𝜙−1(𝛾𝐵
𝜂 )(𝑎) = 𝛾𝐵

𝜂 (𝜙(𝑎)) is an 𝜂 −intuitionistic fuzzy set. Therefore, 𝜙(Γ𝐴
𝜂) and 

𝜙−1(Γ𝐵
𝜂 ) are 𝜂 −intuitionistic fuzzy soft sets over 𝑈.   ∎  

The map 𝑓: 𝐺1 → 𝐺2 of a group 𝐺1 to a group 𝐺2 is called homomorphism if for all 𝑥, 𝑦 ∈ 𝐺1 
then 𝑓(𝑥𝑦) = 𝑓(𝑥)𝑓(𝑦). The following theorems, we prove that every image and pre-
image of the 𝜂 −intuitionistic fuzzy group under the homomorphism function are the 
𝜂 −intuitionistic fuzzy soft group. 

Theorem 5. Let Γ𝐺1
𝜂

 is an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈 and 

𝜙hom: 𝐺1 − 𝐺2, then 𝜙(Γ𝐺1
𝜂

) is an 𝜂 −intuitionistic fuzzy soft group. 

Proof. Let 𝐺1 and 𝐺2 are two groups, 𝜙hom: 𝐺1 − 𝐺2, and Γ𝐺1
𝜂

 is an 𝜂 −intuitionistic fuzzy 

soft group over 𝑈. Let 𝑢, 𝑣 ∈ 𝐺2, 𝑢 ∉ 𝜙(𝐺1) or 𝑣 ∉ 𝜙(𝐺1), then  

𝜙(𝛾𝐺1
𝜂 )(𝑢) ∩ 𝜙(𝛾𝐺1

𝜂 )(𝑣) = 𝛾∅
𝜂, 

means 
𝜙(𝛾𝐺1

𝜂 )(𝑢𝑣) ⊇ 𝜙(𝛾𝐺1
𝜂 )(𝑢) ∩ 𝜙(𝛾𝐺1

𝜂 )(𝑣). 

Since 𝑢 ∉ 𝜙(𝐺1), then 𝑢
−1 ∉ 𝜙(𝐺1), so 

𝜙(𝛾𝐺1
𝜂 )(𝑢−1) = 𝜙(𝛾𝐺1

𝜂 )(𝑢) = 𝛾∅
𝜂 . 

Suppose 𝜙(𝑥) = 𝑢 and 𝜙(𝑦) = 𝑣 for 𝑥, 𝑦 ∈ 𝐺1. Let 𝑧 = 𝑥𝑦, then  

1. 𝜙(𝛾𝐺1
𝜂 )(𝑢𝑣)  =∩ {𝛾𝐺1

𝜂 (𝑧): 𝑧 ∈ 𝐺1, 𝜙(𝑧) = 𝑢𝑣} 

=∩ {𝛾𝐺1
𝜂 (𝑥𝑦): 𝑥, 𝑦 ∈ 𝐺1, 𝜙(𝑥𝑦) = 𝑢𝑣}  

⊇∩ {𝛾𝐺1
𝜂 (𝑥) ∩ 𝛾𝐺1

𝜂 (𝑦): 𝑥, 𝑦 ∈ 𝐺1, 𝜙(𝑥) = 𝑢, 𝜙(𝑦) = 𝑣}  

= (∩ {𝛾𝐺1
𝜂 (𝑥): 𝑥 ∈ 𝐺1, 𝜙(𝑥) = 𝑢}) ∩ (∩ {𝛾𝐺1

𝜂 (𝑦): 𝑦 ∈ 𝐺1, 𝜙(𝑦) = 𝑣})  

= 𝜙(𝛾𝐺1
𝜂 )(𝑢) ∩ 𝜙(𝛾𝐺1

𝜂 )(𝑣). 

2. 𝜙(𝛾𝐺1
𝜂 )(𝑢)  =∩ {𝛾𝐺1

𝜂 (𝑥): 𝑥 ∈ 𝐺1, 𝜙(𝑥) = 𝑢} 

=∩ {𝛾𝐺1
𝜂 (𝑥−1): 𝑥 ∈ 𝐺1, 𝜙(𝑥

−1) = 𝑢−1}  

= 𝜙(𝛾𝐺1
𝜂 )(𝑢−1). 

Therefore, 𝜙(Γ𝐺1
𝜂 ) is an 𝜂 −intuitionistic fuzzy soft group over 𝑈.  ∎  

Theorem 6. Let Γ𝐺2
𝜂  is an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈 and 

𝜙hom: 𝐺1 − 𝐺2, then 𝜙
−1(Γ𝐺2

𝜂
) is an 𝜂 −intuitionistic fuzzy soft group. 

Proof. Let 𝐺1 and 𝐺2 are two groups, 𝜙hom: 𝐺1 − 𝐺2, and Γ𝐺2
𝜂  is an 𝜂 −intuitionistic fuzzy 

soft group over 𝑈. For all 𝑥, 𝑦 ∈ 𝐺1 we have 

1. 𝜙−1(𝛾𝐺2
𝜂)(𝑥𝑦) = 𝛾𝐺2

𝜂 (𝜙(𝑥𝑦)) 

 = 𝛾𝐺2
𝜂 (𝜙(𝑥)𝜙(𝑦))  

 ⊇ 𝛾𝐺2
𝜂 (𝜙(𝑥)) ∩ 𝛾𝐺2

𝜂 (𝜙(𝑥))  

 = 𝜙−1(𝛾𝐺2
𝜂 )(𝑥) ∩ 𝜙−1(𝛾𝐺2

𝜂 )(𝑦).  

2. 𝜙−1(𝛾𝐺2
𝜂)(𝑥−1) = 𝛾𝐺2

𝜂 (𝜙(𝑥−1)) 

= 𝛾𝐺2
𝜂 ((𝜙(𝑥))

−1
)  

= 𝛾𝐺2
𝜂 (𝜙(𝑥))  

= 𝜙−1(𝛾𝐺2
𝜂 )(𝑥). 



𝜂 −Intuitionistic Fuzzy Soft Groups  

Mustika Ana Kurfia 360 

Therefore, 𝜙−1(Γ𝐺2
𝜂

) is an 𝜂 −intuitionistic fuzzy soft group over 𝑈.  ∎  

 
Based on Theorem 5 and Theorem 6, we know that image and pre-image of an 
𝜂 −intuitionistic fuzzy soft group under 𝜂 −intuitionistic fuzzy soft homomorphism are 
also an 𝜂 −intuitionistic fuzzy group. 

CONCLUSIONS 

Based on the result, it is concluded that 𝜂 −intuitionistic fuzzy soft group depends on 
intuitionistic fuzzy soft group. It is proved that every intuitionistic fuzzy soft group is an 
𝜂 −intuitionistic fuzzy soft group. The definition of 𝜂 −intuitionistic fuzzy soft subgroup 
and it’s properties are presented. The 𝜂 −intuitionistic fuzzy soft homomorphism show 
that image and pre-image of an 𝜂 −intuitionistic fuzzy soft group are also 𝜂 −intuitionistic 
fuzzy soft groups. For the future research, it is suggested to notion the 𝜂 −intuitionistic 
fuzzy soft ring along with the properties. 

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Mustika Ana Kurfia 361 

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