EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6514
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Generalization of Similarity Measure in Collection of
Intuitionistic Fuzzy Sets

Dwi Nur Yunianti1,2,∗, Noor Hidayat1, Raden Sulaiman2, Abdul Rouf Alghofari1

1 Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Brawijaya University, Malang, East Java, Indonesia
2 Department of Mathematics, Faculty of Mathematics and Science,
State University of Surabaya, Surabaya, East Java, Indonesia

Abstract. A collection of intuitionistic fuzzy sets is a new approach to intuitionistic fuzzy set
theory. In collections of intuitionistic fuzzy sets, a similarity measure can determine the degree of
similarity based on the information carried by the collections. However, an existing similarity mea-
sure is limited to evaluating similarity between two collections defined over the same universal set.
To overcome this limitation, thus, in this paper, we propose a generalized similarity measure that
can be applied to collections defined over different universal sets. To construct the generalization,
we first introduce the concept of inferior and equivalent relations in the collection of intuitionistic
fuzzy sets. Then, we present a new formula for the similarity measure. Finally, the proposed
measure is illustrated through a pattern recognition problem to demonstrate its effectiveness and
practical value.

2020 Mathematics Subject Classifications: 03E72, 08A72, 28E10
Key Words and Phrases: Collection of Intuitionistic Fuzzy Sets, Equivalent Relation, General-
ization of Similarity Measure, Intuitionistic Fuzzy Sets, Inferior Relation

1. Introduction

Zadeh [1] first introduced the concept of fuzzy sets to solve the limitations of clas-
sical set theory. In fuzzy set theory, each element in a universal set is associated with
a membership degree that ranges in the interval [0, 1]. Research related to fuzzy sets
has been further developed by many researchers, such as [2], [3]. To extend the concept,
Atanassov [4] introduced the concept of intuitionistic fuzzy sets, in which each element of
the universal set is assigned a membership degree and a non-membership degree, both in
the interval [0, 1], such that the sum of these degrees does not exceed 1. Research related
to intuitionistic fuzzy sets has been further developed by many researchers, such as [5],
[6], [7].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6514

Email addresses: dwinuryunianti@student.ub.ac.id (D. N. Yunianti), noorh@ub.ac.id (N. Hidayat),
radensulaiman@unesa.ac.id (R. Sulaiman), abdul rouf@ub.ac.id (A.R. Alghofari)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 2 of 17

Similarity measures between objects based on attributes such as shape, color, size, and
texture are critical in various scientific and engineering applications. Many researchers
have developed techniques and tools to measure the similarity between objects that relate
to scientific developments, including intuitionistic fuzzy sets. Several studies focusing on
the development of similarity measures for intuitionistic fuzzy sets have been developed
by [8], [9], [10], [11], [12], [13], [14], [15].

The existing similarity measures are limited to determining similarity between two
intuitionistic fuzzy sets. This becomes a problem when we want to compare complex
objects, each represented not by a single set, but by a collection or union of multiple
intuitionistic fuzzy sets. In many real- world applications, such as pattern recognition or
decision-making, representing objects as collections provides a flexible model. To overcome
this limitation, Yunianti et al.[16] were the first to define a collection of intuitionistic fuzzy
sets as A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X} where Aj is the intuitionistic fuzzy set in
the universe X, and µA(Aj), vA(Aj) denote the membership and non-membership degree
of Aj in the collection A, respectively.

Furthermore, a similarity measure for collections of intuitionistic fuzzy sets has been
introduced by Yunianti et al. [17]. An existing similarity measure for collections of in-
tuitionistic fuzzy sets assumes that the two collections being compared are defined over
the same universe of discourse. This limits their use when the collections come from dif-
ferent universes, which often happens in real-world cases. To overcome the limitations,
we propose a generalized similarity measure that can be applied to collections defined
over different universes. This new measure aims to overcome the limitations of existing
methods for comparing the similarity of collections when the universes of discourse are not
identical and allows for more flexible and realistic similarity comparisons in cases involving
heterogeneous data.

Before presenting the generalization, we introduce the inferior and equivalent relations,
as both are used to show that the proposed similarity measure satisfies the axioms of sim-
ilarity measures. Then, we present formula for the generalization. In addition, this paper
provides an example of the application of the proposed measure to a pattern recognition
problem.

2. Preliminaries

In this section, we review some basic theories related to intuitionistic fuzzy sets, dis-
tance of intuitionistic fuzzy sets, collection of intuitionistic fuzzy sets, and similarity mea-
sure for collection of intuitionistic fuzzy sets.

Definition 1. [4] Let X be a non empty and universal set. An intuitionistic fuzzy set A
in X is written as

A = {(x, µA(x), vA(x)) : x ∈ X}

where µA(x) and vA(x) respectively are the degree of membership and the degree of non-
membership of x in A and both belong to [0, 1], with 0 ≤ µA(x) + vA(x) ≤ 1. Moreover,
the hesitant degree of x in A is πA(x) = 1− µA(x)− vA(x).



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 3 of 17

Next, we describe relations between intuitionistic fuzzy sets.

Definition 2. [4] Let A and B are intuitionistic fuzzy sets on X where
A = {(x, µA(x), vA(x)) : x ∈ X} and B = {(x, µB(x), vB(x)) : x ∈ X}
so we have

1. A ⊆ B if µA(x) ≤ µB(x) and vA(x) ≥ vB(x), ∀x ∈ X.

2. A = B if µA(x) = µB(x) and vA(x) = vB(x), ∀x ∈ X.

Because a similarity measure can be constructed based on distance measure, so we
review the definition of a distance measure between two intuitionistic fuzzy sets.

Definition 3. A function d : E × E → [0, 1] is said distance measure between two intu-
itionistic fuzzy sets if it satisfies the following

1. 0 ≤ d(A,B) ≤ 1

2. d(A,B) = d(B,A)

3. d(A,B) = 0 iff A = B

4. If A ⊆ B ⊆ C, then d(A,B) ≤ d(A,C), and d(B,C) ≤ d(A,C).

The following example is a distance measure developed based on the measure proposed
by Atanassov in [4].

Example 1. [4] Let Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} and Bk = {(xi, µBk
(xi), vBk

(xi)) :
xi ∈ X} are intuitionistic fuzzy sets on the universal set X = {x1, x2, . . . xn} with j, k =
1, 2, ..,m.

dI(Aj , Bk) =
1

2n

n∑
i=1

|µAj (xi)− µBk
(xi)|+ |vAj (xi)− vBk

(xi)|

is a distance measure between two intuitionistic fuzzy sets.

Next, we define a collection of intuitionistic fuzzy sets that was constructed by Yunianti
et al [16].

Definition 4. [16] Let Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} is intuitionistic fuzzy set on
X = {xi : i = 1, 2, . . . , n} with j = 1, 2, ..,m.
A collection of intuitionistic fuzzy sets on X = {Aj : j = 1, 2, . . . ,m} can state as

A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X}

where µA : X → [0, 1] is membership function A on X and vA : X → [0, 1] is non
membership function A on X .
Moreover, µA(Aj) can be described as the membership degree of Aj on A and vA(Aj) can
be described as the non-membership degree Aj on A where

0 ≤ µA(Aj) + vA(Aj) ≤ 1

. The hesitancy degree of Aj on A is stated as

πA(Aj) = 1− µA(Aj)− vA(Aj)



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 4 of 17

For understanding the definition, here, we give an example of collection of intuitionistic
fuzzy sets.

Example 2. A consumer will choose to recommend one restaurant out of three based on
three criteria: price (p′), food variety (q′), and restaurant facilities (r′).
Let S = {p′, q′, r′} as a set of criteria .
The ratings of the first, second, and third restaurants are represented as intuitionistic fuzzy
sets as follows:
P1 = {(p′, 0.7, 0.2), (q′, 0.8, 0.1), (r′, 0.8, 0.1)},
P2 = {(p′, 0.1, 0.6), (q′, 0.2, 0.5), (r′, 0.3, 0.6)},
P3 = {(p′, 0.5, 0.2), (q′, 0.6, 0.1), (r′, 0.7, 0.1)}
The consumer provides an overall rating for the three restaurants based on P1, P2, P3 and
this overall rating is represented as a collection of intuitionistic fuzzy sets P .
Consider X = {P1, P2, P3} be universal set. A collection of intuitionistic fuzzy sets on X
is given by

P = {(P1, 0.8, 0.2), (P2, 0.2, 0.6), (P3, 0.6, 0.3)}
. Here, the membership degree of 0.8 for P1 indicates that the first restaurant is highly
recommended by the consumer compared to P2 and P3, while the non-membership degree of
0.2 for P1 indicates a low tendency for the first restaurant not to be recommended relative
to the others.

Definition 5. [17] Given

A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X}

and
B = {(Aj , µB(Aj), vB(Aj)) : Aj ∈ X}

respectively collection of intuitionistic fuzzy sets on X = {Aj : j = 1, 2, . . . ,m}.
We define that

i) A ⊆ B if and if µA(Aj) ≤ µB(Aj) and νA(Aj) ≥ νB(Aj)

ii) A = B if and if µA(Aj) = µB(Aj) and νA(Aj) = νB(Aj)

The proposed similarity measure is a generalization of the similarity measure intro-
duced in [17], which determines the similarity between collections of intuitionistic fuzzy
sets with identical elements. For completeness, we restate the similarity measure from [17]
below.

Theorem 1. [17] Let Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} be intuitionistic fuzzy sets
on X = {x1, x2, . . . , xn} with j = 1, 2, ..,m. A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X}
and B = {(Aj , µB(Aj), vB(Aj)) : Aj ∈ X} are collections of intuitionistic fuzzy sets on
X = {Aj : j = 1, 2, . . . ,m}.

SI(A,B) = 1− 1

2m

m∑
j=1

|µA(Aj)− µB(Aj)|+ |vA(Aj)− vB(Aj)|

is a similarity measure between two collections of intuitionistic fuzzy sets.



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 5 of 17

The following example is example for using Theorem 1.

Example 3. Let A = {(A1, 0.7, 0.2), (A2, 0.8, 0.1), (A3, 0.6, 0.4), (A4, 0.4, 0.3)} and
B = {(A1, 0.7, 0.1), (A2, 0.8, 0.1), (A3, 0.6, 0.4), (A4, 0.5, 0.3)} are collections of intuitionis-
tic fuzzy set on X = {A1, A2, A3, A4}.
Similarity measure between A and B is

SI(A,B) = 1− 1

2m

4∑
j=1

|µA(Aj)− µB(A)|+ |vA(Aj)− vB(Aj)| = 0.975

3. Results

In this section, we present a similarity measure designed to evaluate the similarity
between collections of intuitionistic fuzzy sets defined over distinct universal sets. Be-
fore introducing the similarity measure, we first provide definitions of the inferiority and
equivalence relations on collections of intuitionistic fuzzy sets.

3.1. A New Approach of Relation on Collection of Intuitionistic Fuzzy
Sets

In this sub section, we define inferiority and equivalence relations on collections of
intuitionistic fuzzy sets. The inferiority and equivalence relations are a new approach of
subset and equality relations on collection of intuitionistic fuzzy sets.

Definition 6. Consider Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} and
Bk = {(xi, µBk

(xi), vBk
(xi)) : xi ∈ X} are intuitionistic fuzzy sets on X = {x1, x2, . . . , xn}

with j, k = 1, 2, ..,m.
A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X} is a collection of intuitionistic fuzzy set on X =
{Aj : j = 1, 2, . . . ,m} and B = {(Bk, µB(Bk), vB(Bk)) : Bk ∈ Y} is a collection of
intuitionistic fuzzy set on

Y = {Bk : k = 1, 2, . . . ,m}

.
A is inferior to B, or we write A⊆̃B if:

i. There exists an injective function f from X to Y such that Aj ⊆ f(Aj)

ii. For every j , there exists k such that µA(Aj) ≤ µB(Bk) and vA(Aj) ≥ vB(Bk)

Example 4. We have the following intuitionistic fuzzy sets defined on X = {x1, x2, x3}
A1 = {(x1, 0.4, 0.5), (x2, 0.6, 0.4), (x3, 0.3, 0.4)}
A2 = {(x1, 0.3, 0.7), (x2, 0.5, 0.4), (x3, 0.5, 0.5)}
B1 = {(x1, 0.3, 0.6), (x2, 0.5, 0.3), (x3, 0.5, 0.2)}
B2 = {(x1, 0.5, 0.4), (x2, 0.7, 0.3), (x3, 0.3, 0.1)}
Also let A = {(A1, 0.5, 0.4), (A2, 0.4, 0.6)} be a collection of intuitionistic fuzzy set on



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 6 of 17

X = {A1, A2}and
B = {(B1, 0.4, 0.5), (B2, 0.6, 0.2)} be a collection of intuitionistic fuzzy set on Y = {B1, B2}.
Because

i. A1 ⊆ B2 and A2 ⊆ B1.
So we have there exists an injective function f : X → Y so A1 ⊆ f(A1) and A2 ⊆
f(A2).

ii. µA(A1) ≤ µB(B2) and vA(A1) ≥ vB(B2)
µA(A2) ≤ µB(B1) and vA(A2) ≥ vB(B1)

Thus, we can conclude that A is inferior to B or A⊆̃B.

The following definition describes about equivalence relation between two collection of
intuitionistic fuzzy sets.

Definition 7. Consider Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} and Bk = {(xi, µBk
(xi), vBk

(xi)) :
xi ∈ X} are intuitionistic fuzzy sets on a non-empty universal set X = {x1, x2, . . . , xn}
with j, k = 1, 2, ..,m.
A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X} is a collection of intuitionistic fuzzy set in the uni-
verse of discourse X = {Aj : j = 1, 2, . . . ,m} and B = {(Bk, µB(Bk), vB(Bk)) : Bk ∈ Y}
is a collection of intuitionistic fuzzy set in the universe of discourse

Y = {Bk : k = 1, 2, . . . ,m}

. A is equivalent to B, or we write A=̃B if A⊆̃B and B⊆̃A.

Definition 8. Let Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} and
Bk = {(xi, µBk

(xi), vBk
(xi)) : xi ∈ X} be intuitionistic fuzzy sets that defined in

X = {xi : i = 1, 2, . . . , n} respectively with j, k = 1, 2, ..,m.
A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X} is a collection of intuitionistic fuzzy set on X =
{Aj : j = 1, 2, . . . ,m} and B = {(Bk, µB(Bk), vB(Bk)) : Bk ∈ Y} is a collection of
intuitionistic fuzzy set on Y = {Bk : k = 1, 2, . . . ,m}.
We say that A=̃B if only if

i. There exists an injective function f from X to Y such that f(Aj) = Bk

ii. For every j, there exists k such that µA(Aj) = µB(Bk) and vA(Aj) = vB(Bk)

Definitions 7 and 8 are equivalent. By using the concept of inferior relation, we can
derive Definition 8 from Definition 7. Next, we provide an example to illustrate the
equivalence relation between two collections of intuitionistic fuzzy sets.

Example 5. We have the following intuitionistic fuzzy sets defined on X = {x1, x2, x3}
A1 = {(x1, 0.1, 0.8), (x2, 0.2, 0.5), (x3, 0.3, 0.4)}
A2 = {(x1, 0.2, 0.7), (x2, 0.3, 0.6), (x3, 0.4, 0.5)}
B1 = {(x1, 0.2, 0.7), (x2, 0.3, 0.6), (x3, 0.4, 0.5)}



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 7 of 17

B2 = {(x1, 0.1, 0.8), (x2, 0.2, 0.5), (x3, 0.3, 0.4)}
Let X = {A1, A2}, Y = {B1, B2} be universal sets,
Define collections of intuitionistic fuzzy sets on these universes as
A = {(A1, 0.1, 0.8), (A2, 0.3, 0.6)} be a collection of intuitionistic fuzzy set on X ,
B = {(B1, 0.3, 0.6), (B2, 0.1, 0.8)} be a collection of intuitionistic fuzzy set on Y.
Because

i. A1 = B2 and A2 = B1.
So we have there exists an injective function f : X → Y so f(A1) = B2 and f(A2) =
B1

ii. µA(A1) = µB(B2) and vA(A1) = vB(B2)
µA(A2) = µB(B1) and vA(A2) = vB(B1)

Thus, we can conclude that A=̃B

3.2. Generalization of Distance and Similarity Measure Between Collec-
tions of Intuitionistic Fuzzy Sets

In this section, we describe generalization of distance and similarity measure between
collections of intuitionistic fuzzy sets that have been constructed. First, we declare the
formula of distance measure. Because similairt measure is dual of distance, so we get
similarity measure based on the distance’s formula .

Definition 9. A function D : E ′×E ′ → [0, 1] is said distance measure between collections
of intuitionistic fuzzy sets, if it satisfies the following :

1. 0 ≤ D(A,B) ≤ 1

2. D(A,B) = D(B,A)

3. D(A,B) = 0 if only if A=̃B

4. If A⊆̃B⊆̃C, then D(A,B) ≤ D(A, C), and D(B, C) ≤ D(A, C).

The following theorems explain about the distance measure that we proposed.

Theorem 2. Let two intuitionitic fuzzy sets as Aj = {(xi, µAj (xi), vAj (xi)) : xi ∈ X} and
Bk = {(xi, µBk

(xi), vBk
(xi)) : xi ∈ X} on X = {x1, x2, . . . , xn} where j, k = 1, 2, ..,m.

A = {(Aj , µA(Aj), vA(Aj)) : Aj ∈ X} is a collection of intuitionistic fuzzy sets on
X = {Aj : j = 1, 2, . . . ,m} and B = {(Bk, µB(Bk), vB(Bk)) : Bk ∈ Y} is a collection
of intuitionistic fuzzy sets on Y = {Bk : k = 1, 2, . . . ,m}.

DI (A,B) = 1

2
(RAB +RAB)



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 8 of 17

is a distance measure between A and B, where

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+
m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

)

and

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}

+
m∑
k=1

min
j=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣})
Proof.

1. First, we want to prove that 0 ≤ DI (A,B) ≤ 1. Thus, we must prove that
0 ≤ RAB ≤ 1 and 0 ≤ RAB ≤ 1.
Because of 0 ≤ µA (Aj) , vA (Aj) ≤ 1 and 0 ≤ µB (Bk) , vB (Bk) ≤ 1, so we have
0 ≤ |µA (Aj)− µB (Bk)| ≤ 1 and 0 ≤ |vA (Aj)− vB (Bk)| ≤ 1.
Therefore, we have 0 ≤ |µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)| ≤ 2.
For some p = 1, 2, . . . ,m, we have

min
k=1,...,m

{|µA (A1)− µB (Bk)|+ |vA (A1)− vB (Bk)|}

≤ |µA (A1)− µB (Bp)|+ |vA (A1)− vB (Bp)| ≤ 2

It implies that

min
k=1,...,m

{|µA (A1)− µB (Bk)|+ |vA (A1)− vB (Bk)|}

+ min
k=1,...,m

{|µA (A2)− µB (Bk)|+ |vA (A1)− vB (Bk)|}

+ . . .+ min
k=1,...,m

{|µA (Am)− µB (Bk)|+ |vA (Am)− vB (Bk)|} ≤ 2m

Or we can say that

m∑
j=1

min
k=1,..,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|} ≤ 2m



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 9 of 17

Next, we have for some q = 1, 2, . . . ,m

min
j=1,...,m

{|µA (Aj)− µB (B1)|+ |vA (Aj)− vB (B1)|}

≤ |µA (Aq)− µB (B1)|+ |vA (Aq)− vB (B1)| ≤ 2

It implies that

min
j=1,...,m

{|µA (Aj)− µB (B1)|+ |vA (Aj)− vB (B1)|}

+ min
j=1,...,m

{|µA (Aj)− µB (B2)|+ |vA (Aj)− vB (B2)|}

+ . . .+ min
j=1,...,m

{|µA (Aj)− µB (Bm)|+ |vA (Aj)− vB (Bm)|} ≤ 2m

Or we can say that

m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|} ≤ 2m

Analogously, we get

m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|} ≤ 2m

Based on these results, we have

m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+
m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|} ≤ 4m

So

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+

m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

)
≤ 1

Based on the fact that |µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)| ≥ 0 and in a similar



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 10 of 17

manner, we can conclude that

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+
m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

)
≥ 0

So, we have 0 ≤ RAB ≤ 1.
Now, we want to prove that 0 ≤ RAB ≤ 1.
By taking
0 ≤ µAj

(xi) , vAj (xi) ≤ 1 and 0 ≤ µBk
(xi) , vBk

(xi) ≤ 1, so we can get

0 ≤
∣∣µAj (xi)− µBk

(xi)
∣∣ ≤ 1 and 0 ≤

∣∣vAj (xi)− vBk
(xi)

∣∣ ≤ 1

Therefore, 0 ≤
∣∣µAj (xi)− µBk

(xi)
∣∣+ ∣∣vAj (xi)− vBk

(xi)
∣∣ ≤ 2

Hence,

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}

≤ 1

n

n∑
i=1

∣∣µAj (xi)− µBp (xi)
∣∣+ ∣∣vAj (xi)− vBp (xi)

∣∣ ≤ 1

n
.2n = 2

It implies that

m∑
j=1

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣} ≤ 2m

Analogously, we can obtain that

m∑
k=1

min
j=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}≤ 2m

Thus,

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}

+
m∑
k=1

min
j=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}) ≤ 1



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 11 of 17

By using the same approach, we also obtain that 0 ≤ RAB ≤ 1.
Moreover,0 ≤ 1

2 (R+R) ≤ 1 or 0 ≤ DI (A,B) ≤ 1.

2. Because of

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+
m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

)

=
1

4m

(
m∑
k=1

min
j=1,...,m

{|µB (Bk)− µA (Aj)|+ |vB (Bk)− vA (Aj)|}

+

m∑
j=1

min
k=1,...,m

{|µB (Bk)− µA (Aj)|+ |vB (Bk)− vA (Aj)|}

)

and

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}

+

m∑
k=1

min
j=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣})

=
1

4m

(
m∑
k=1

min
j=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}

+
m∑
j=1

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣})

So, we get DI (A,B) = DI (B,A).

3. Let A=̃B, so

i) There exists an injective function f from X to Y such that f (Aj) = Bk.

ii) For all j, there exits k such that µA (Aj) = µB (Bk) and vA (Aj) = vB (Bk).

From i), we know that for all j, there exists a unique k such that Aj = Bk.
It implies that µAj (xi) = µBk

(xi) and vAj (xi) = vBk
(xi), thus

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 12 of 17

+

m∑
k=1

min
j=1,...,m

{
1

n

n∑
i=1

∣∣µAj (xi)− µBk
(xi)

∣∣+ ∣∣vAj (xi)− vBk
(xi)

∣∣}) = 0

From ii), we know that

RAB =
1

4m

(
m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+
m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

)
= 0

Thus, DI (A,B) = 0.
The other side, if we have DI (A,B) = 1

2(RAB + RAB) = 0, then RAB = 0 and
RAB = 0.
For RAB = 0, we get for every j there exists k such that µA (Aj) = µB (Bk) and
vA (Aj) = vB (Bk).
It satisfies the statement of ii).
For RAB = 0, we get for every j there exists k such that µAj (xi) = µBk

(xi) and
vBk

(xi) = vAj (xi).
And for every k there exists j such that µAj (xi) = µBk

(xi) and vBk
(xi) = vAj (xi).

Hence, we can conclude that that there exists a one to one correspondence between
X and Y such that µAj (xi) = µBk

(xi) and vBk
(xi) = vAj (xi).

Therefore, we have Aj = Bk and we can say that there exists an injective function f
from X to Y such that f (Aj) = Bk.
Based on these results, we can conclude that A=̃B

4. For proving that DI (A,B) ≤ DI (A, C) so we must prove that RAB ≤ RAC and
RAB ≤ RAC .
Let A⊆̃B⊆̃C, by definition we have

µA (Aj) ≤ µB (Bk) ≤ µC (Cr) and vAj (xi) ≥ vBk
(xi) ≥ vCr (xi) ,∀xi ∈ X

Since
µB (Bk) ≤ µC (Cr), then

−µB (Bk) ≥ −µC (Cr)

µA (Aj)− µB (Bk) ≥ µA (Aj)− µC (Cr)

− (µA (Aj)− µB (Bk)) ≤ − (µA (Aj)− µC (Cr))

|µA (Aj)− µB (Bk)| ≤ |µA (Aj)− µC (Cr)|

Since vB (Bk) ≥ vC (Cr), then

−vB (Bk) ≤ −vC (Cr)



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 13 of 17

vA (Aj)− vB (Bk) ≤ vA (Aj)− vC (Cr)

|vA (Aj)− vB (Bk)| ≤ |vA (Aj)− vC (Cr)|

Hence, for all j, k, r we can get

|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)| ≤ |µA (Aj)− µC (Cr)|+ |vA (Aj)− vC (Cr)|

It follows that, for all j, we have

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

≤ min
r=1,...,m

{|µA (Aj)− µC (Cr)|+ |vA (Aj)− vC (Cr)|}

and for all r, we have

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

≤ min
j=1,...,m

{|µA (Aj)− µC (Cr)|+ |vA (Aj)− vC (Cr)|}

These results imply that

1

4m

(
m∑
j=1

min
k=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

+

m∑
k=1

min
j=1,...,m

{|µA (Aj)− µB (Bk)|+ |vA (Aj)− vB (Bk)|}

)

≤ 1

4m

(
m∑
j=1

min
r=1,...,m

{|µA (Aj)− µC (Cr)|+ |vA (Aj)− vC (Cr)|}

+

m∑
r=1

min
j=1,...,m

{|µA (Aj)− µC (Cr)|+ |vA (Aj)− vC (Cr)|}

)

Thus RAB ≤ RAC .
By the definition of A⊆̃B⊆̃C, and in a similar manner, we can prove that

RAB ≤ RAC

It is obvious that

DI (A,B) = 1

2
(RAB +RAB) ≤

1

2
(RAC +RAC) = DI (A, C)



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 14 of 17

.
Analogue for proving that D1 (B, C) ≤ D1 (A, C).
So, the distance measure DI (A,B) satisfies all properties of distance measure.

Next, based on the proposed distance measure, we derive a generalized similarity mea-
sure between two collections of intuitionistic fuzzy sets.

Definition 10. A function S : E ′ × E ′ → [0, 1] is said similarity measure between two
collections of intuitionistic fuzzy sets, if it satisfies the following :

1. 0 ≤ S(A,B) ≤ 1

2. S(A,B) = 1 if only if A=̃B.

3. S(A,B) = S(B,A)

4. If A⊆̃B⊆̃C, then S(A,B) ≥ S(A, C), and S(B, C) ≥ S(A, C).

Theorem 3.
SI(A,B) = 1−DI(A,B)

is a similarity measure between A and B where DI(A,B) is the distance measure defined
in Theorem 2.

Proof. DI(A,B) is the distance measure defined in Theorem 2. Therefore, we have

1. 0 ≤ DI(A,B) ≤ 1

2. DI(A,B) = DI(B,A)

3. DI(A,B) = 0 if only if A=̃B

4. If A⊆̃B⊆̃C, then DI(A,B) ≤ DI(A, C), and DI(B, C) ≤ DI(A, C).

Therefore, based on 1 until 4 we have

a.
−1 ≤ −DI(A,B) ≤ 0

so
0 ≤ SI(A,B) ≤ 1

b. A=̃B gives that
SI(A,B) = 1−DI(A,B) = 1− 0 = 1

and
SI(A,B) = 1−DI(A,B) = 1

gives that DI(A,B) = 0, so A=̃B.

c. SI(A,B) = 1−DI(A,B) = 1−DI(B,A) = SI(B,A)



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 15 of 17

d. If A⊆̃B⊆̃C then

SI(A,B) = 1−DI(A,B) ≥ 1−D1(A, C) = SI(A, C)

SI(B, C) = 1−D1(B, C) ≥ 1−D1(A, C) = SI(A, C)

Therefore
SI(A,B) ≥ SI(A, C) and SI(B, C) ≥ SI(A, C)

In the following section, we present an illustrative example for solving a pattern recog-
nition problem using the proposed similarity measure.

Example 6. An expert in the health sector wants to determine which region, between
region B and region C exhibits malnutrition characteristics similar to those of region A.
The region A is made up of three subregions, indicated by A1, A2, A3. Each subregion
represents a specific part of the region A and is characterized by its own level of nutritional
indicators based on fuzzy intuitionistic values. To make a fair comparison, the regions
B and C are also divided into three subregions, denoted B1, B2, B3 and C1, C2, C3,
respectively.
The goal is to measure the similarity between these regions and region A using a proposed
similarity measure between collections of intuitionistic fuzzy sets.

Let the universal set be X = {x1, x2, x3} where x1 is poverty rate, x2 is low education
rate, and x3 is the ease of access to health.
Given the following collections of intuitionistic fuzzy sets. These are
A = {(A1, 0.6, 0.3), (A2, 0.5, 0.5), (A3, 0.6, 0.4)} where

A1 = {(x1, 0.5, 0.2), (x2, 0.6, 0.3), (x3, 0.1, 0.6)}
A2 = {(x1, 0.55, 0.1), (x2, 0.6, 0.4), (x3, 0.4, 0.5)}
A3 = {(x1, 0.7, 0.1), (x2, 0.5, 0.2), (x3, 0.2, 0.6)}

B = {(B1, 0.6, 0.3), (B2, 0.6, 0.2), (B3, 0.7, 0.2)} where

B1 = {(x1, 0.55, 0.1), (x2, 0.7, 0.3), (x3, 0.4, 0.6)}
B2 = {(x1, 0.65, 0.2), (x2, 0.5, 0.3), (x3, 0.4, 0.5)}
B3 = {(x1, 0.7, 0.3), (x2, 0.6, 0.4), (x3, 0.3, 0.5)}

C = {C1, 0.5, 0.5), (C2, 0.7, 0.3), (C3, 0.4, 0.4) where

C1 = {(x1, 0.6, 0.2), (x2, 0.8, 0.1), (x3, 0.1, 0.8)}
C2 = {(x1, 0.6, 0.2), (x2, 0.8, 0.2), (x3, 0.3, 0.6)}
C3 = {(x1, 0.5, 0.3), (x2, 0.6, 0.2), (x3, 0.3, 0.7)}

Using the proposed distance measure and similarity measure between collections of intu-
itionistic fuzzy sets, we get

DI(A,B) = 0.11681



Dwi Nur Yunianti et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6514 16 of 17

DI(A, C) = 0.10883

Therefore
SI(A,B) = 1−DI(A,B) = 1− 0.11681 = 0.88319

SI(A, C) = 1−DI(A, C) = 1− 0.10883 = 0.89117

Based on the calculation, region C has a higher degree of similarity to region A than B.
Thus, we conclude that region C has more similar characteristics of malnutrition with
region A.

4. Conclusions

In this paper, we propose a generalized similarity measure for collections of intuitionis-
tic fuzzy sets considering the differences in their universal sets. Unlike an existing similar-
ity measure, which typically assumes that the collections of intuitionistic fuzzy sets share
an identical universal set, the proposed similarity measure overcomes this limitation by
considering the different of the universal set of collections of intuitionistic fuzzy sets. The
proposed similarity measure, developed using a distance-based approach by assuming com-
plete knowledge of the membership degree, and nonmembership degree. It offers a more
flexible and comprehensive method for comparing such collections. Furthermore, we intro-
duce formal definitions of inferiority and equivalence relations to enhance the structural
understanding of such collections. These relations also support the theoretical foundation
of the proposed measure, ensuring that it satisfies the standard axioms of similarity mea-
sures. Through illustrative example as pattern recognition, we show that the proposed
measure can effectively support the decision-making process. This approach opens further
opportunities for application in real-world problems such as medical diagnosis, pattern
recognition, and multi-criteria decision analysis.

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