EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2198-2207
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Properties of Operations in the Collection of
Intuitionistic Fuzzy Sets : A Novel Approach

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. In this paper, we introduce collection of intuitionistic fuzzy sets as a developing and
expanding intuitionistic fuzzy theory. A collection of intuitionistic fuzzy sets is a set whose members
of the universe set are intuitionistic fuzzy sets. We present intersection and union operation in the
collection of of intuitionistic fuzzy sets and show that the operations hold commutative, assosiative,
idempotent, and De Morgan’s laws properties.

2020 Mathematics Subject Classifications: 03E72, 03G25

Key Words and Phrases: Intuitionistic fuzzy sets, Collection of intuitionistic fuzzy sets, Inter-
section, Union

1. Introduction

Zadeh [1] first introduced the concept of fuzzy sets which provides a solution to the
weaknesses of classical set theory. The fuzzy set assigns a membership value from 0 to
1 to all elements of the considered universal set. Many researchers have done researches
on fuzzy mathematics in [2], [3], [4], [5], [6], [7], [8]. Atanassov [9] proposed the idea of
generalizing fuzzy sets and it called intuitionistic fuzzy sets. Intuitionistic fuzzy sets have
membership and non-membership values respectively on interval [0,1] that assigned to all
elements of the universal set. The developing about algebra structures on intuitionistic
fuzzy sets have been explored by Ejegwa et al. [10], Macodi-Ringia and Petalcorin [11],
Roh et al. [12]. Whereas, the properties of arithmetic, algebraic, model operators, and
normalization of intuitionistic fuzzy sets have been researched by Ejegwa et al. [13]. Beside
that, the concept of intuitionistic fuzzy sets can be applied to many fields such as medical
diagnosis, pattern recognition, multi criteria decision making (see [14], [15], [16], [17], [18],
[19], [20]).

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4939

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 2198 © 2023 EJPAM All rights reserved.



D. N. Yunianti et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2198-2207 2199

In real life, many contexts that we compare are collections of objects which already
in a fixed state. Intuitionistic fuzzy sets can not be completely used to compared the
contexts. So, new concept of intuitionistic fuzzy set must be defined whose the members
of the universe set also have set members. In this paper, we introduce a new concept
of intuitionistic fuzzy sets and we call it a collection of intuitionistic fuzzy sets. The
philosophical background of the new concept is to develop and to expand the intuition-
istic fuzzy theory. A collection of intuitionistic fuzzy sets is a set whose members of the
universe set are intuitionistic fuzzy sets. Adopting definition of intuitionistic fuzzy set,
collection of intuitionistic fuzzy sets also give membership and non-membership degree for
every members of the universe set. This new concept is a theoretical basic for developing
applications relate to the comparison of the set collection as the universe set. Hence, in
this paper, we present definition about collection of intuitionistic fuzzy sets, and intro-
duce basic operations in the collection of intuitionistic fuzzy sets. Those operations are
intersection and union operation. Furthermore, we provide some properties that relate to
those operations.

2. Preliminaries

In this section, we review definition of intuitionistic fuzzy sets and their relations and
operations.

Definition 1. [9] Let Y be a non-empty and universal set. An intuitionistic fuzzy set A
of Y can be defined as:

A = {(x, µA(x), vA(x)) : x ∈ Y }

where µA(x) represents the membership degree of x in A and vA(x) represents the non-
membership degree of x in A, both satisfying:

µA : Y → [0, 1], vA : Y → [0, 1]

Moreover, µA(x) is defined as the membership degree of x in A, and vA(x) is defined as
the non-membership degree of x in A, where both of them are in the interval [0, 1], and
0 ≤ µA(x) + vA(x) ≤ 1.

Based on the membership and non-membership values, the hesitant degree of x in A
is defined as:

πA(x) = 1− µA(x)− vA(x)

Next, we describe relations and operations between intuitionistic fuzzy sets based on
Atanassov’s [9] explanation.

Definition 2. [9] Let A and B be intuitionistic fuzzy sets, respectively, of the universal
set Y . The following relations and operations on intuitionistic fuzzy sets.

1. A ⊆ B if and only if µA(x) ≤ µB(x) and vA(x) ≥ vB(x) for all x ∈ Y .

2. A = B if and only if µA(x) = µB(x) and vA(x) = vB(x) for all x ∈ Y .



D. N. Yunianti et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2198-2207 2200

3. Ac = {(x, vA(x), µA(x)) : x ∈ Y }. So, µAc(x) = vA(x) and vAc(x) = µA(x).

4. A ∩ B = {(x, µA∩B(x), vA∩B(x)) : x ∈ Y } with µA∩B(x) = min{µA(x), µB(x)} and
vA∩B(x) = max{vA(x), vB(x)}.

5. A ∪ B = {(x, µA∪B(x), vA∪B(x)) : x ∈ Y } with µA∪B(x) = max{µA(x), µB(x)} and
vA∪B(x) = min{vA(x), vB(x)}.

Example 1. Let Y be the universe set where Y = {a′, b′, c′}. We have A, B, and C be
three intuitionistic fuzzy sets where

A = {(a′, 0.4, 0.6), (b′, 0.7, 0.2), (c′, 0.6, 0.1)},

B = {(a′, 0.3, 0.7), (b′, 0.2, 0.2), (c′, 0.6, 0.0)},

C = {(a′, 0.4, 0.6), (b′, 0.7, 0.2), (c′, 0.1, 0.1)}.

We conclude that A ∩B is {(a′, 0.3, 0.7), (b′, 0.2, 0.2), (c′, 0.6, 0.1)} and
B ∪ C is {(a′, 0.4, 0.6), (b′, 0.7, 0.2), (c′, 0.6, 0.0)}.

Example 1 is an example of intuitionistic fuzzy sets whose elements of the universe set
are not sets. In the next section, we present the definition of collection of intuitionistic
fuzzy sets as a new concept of intuitionistic fuzzy sets.

3. Results

3.1. Collection Of Intuitionistic Fuzzy sets

By adopting definition of intuitionistic fuzzy sets and generalize the universe set, we
give the definition of collection of intuitionistic fuzzy sets.

Definition 3. Let X = {Ai : i = 1, 2, . . . , n} be the universe set and non empty set. Ai

is intuitionistic fuzzy set on Y . A collection of intuitionistic fuzzy sets on X is written as

A∗ = {(Ai, µA∗(Ai), vA∗(Ai)) : Ai ∈ X},

where µA∗ : X → [0, 1] is the membership function of A∗ and vA∗ : X → [0, 1] is the
non-membership function of A∗. Furthermore, µA∗(Ai) is the membership degree of Ai on
A∗ and vA∗(Ai) is the non-membership degree of Ai on A∗, where

0 ≤ µA∗(Ai) + vA∗(Ai) ≤ 1.

Example 2. A customer of three fast food restaurants A1, A2, A3 want to give best re-
comendation restaurant among them based on price, taste, and product variations. If we
give Y = {c1, c2, c3} where c1 is price, c2 is taste, and c3 is product variations, and suppose
that A1, A2, A3 are intuitionistic fuzzy sets in Y . So, A1, A2, A3 represented by

A1 = {(c1, 0.7, 0.2), (c2, 0.8, 0), (c3, 0.8, 0.1)}



D. N. Yunianti et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2198-2207 2201

A2 = {(c1, 0.2, 0.6), (c2, 0.4, 0.5), (c3, 0.3, 0.6)}

A3 = {(c1, 0.5, 0.5), (c2, 0.6, 0.2), (c3, 0.5, 0.4)}

Based on the intuitionistic fuzzy sets, a customer gives best recommendation of these
restaurants which represented by a collection of intuitionistic fuzzy sets. Let X = {A1, A2, A3}
be the universe set, the collection of intuitionistic fuzzy sets in X is

A∗ = {(A1, 0.8, 0.1), (A2, 0.2, 0.6), (A3, 0.4, 0.5)}

Obviously, a customer shows that A1 is better recommendation fast food restaurant than
A2 and A3.

Clearly, every collection of intuitionistic fuzzy sets on X can be seen as intuitionistic
fuzzy set on X where the cardinality of X is one.

Definition 4. Let X = {Ai : i = 1, 2, . . . , n} be a non-empty universal set. Ai is an
intuitionistic fuzzy set on Y . The hesitant degree of Ai on A∗ is defined as

πA∗(Ai) = 1− µA∗(Ai)− vA∗(Ai).

Example 3. Based on Example 2, we have

πA∗(A1) = 0.1, πA∗(A2) = 0.2, πA∗(A3) = 0.1.

In the next theorem, we discuss about value of hesitant degree on the interval [0, 1].

Theorem 1. The hesitant degree πA∗(Ai) is in the interval [0, 1] for all Ai ∈ X.

Proof. Because

πA∗(Ai) = 1− µA∗(Ai)− vA∗(Ai) = 1− (µA∗(Ai) + vA∗(Ai))

and 0 ≤ µA∗(Ai) + vA∗(Ai) ≤ 1. So, we have −1 ≤ −(µA∗(Ai) + vA∗(Ai)) ≤ 0. Then,
0 ≤ 1− (µA∗(Ai) + vA∗(Ai)) ≤ 1. Hence, πA∗(Ai) ∈ [0, 1]

Definition 5. Let A∗ is collections of intuitionistic fuzzy sets on X, where

A∗ = {(Ai, µA∗(Ai), vA∗(Ai)) : Ai ∈ X}

Complement of A∗ is

A∗c = {(Ai, vA∗(Ai), µA∗(Ai)) : Ai ∈ X}

More over, we can see that µA∗c(Ai) = vA∗(Ai) and vA∗c(Ai) = µA∗(Ai)
Next, we present definition of intersection and union in the collections of intuitionistic

fuzzy sets as basic operations for further development.



D. N. Yunianti et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2198-2207 2202

Definition 6. Let A∗ and B∗ be collections of intuitionistic fuzzy sets on X respectively,
where

A∗ = {(Ai, µA∗(Ai), vA∗(Ai)) : Ai ∈ X}

and
B∗ = {(Ai, µB∗(Ai), vB∗(Ai)) : Ai ∈ X}.

The definition of intersection A∗ and B∗ is

A∗ ∩B∗ = {(Ai, µA∗∩B∗(Ai), vA∗∩B∗(Ai)) : Ai ∈ X}

with
µA∗∩B∗(Ai) = min{µA∗(Ai), µB∗(Ai)}

and
vA∗∩B∗(Ai) = max{vA∗(Ai), vB∗(Ai)}.

Example 4. Let A∗ and B∗ be collections of intuitionistic fuzzy sets on X = {A1, A2, A3, A4}
respectively, where

A∗ = {(A1, 0.9, 0.1), (A2, 0.1, 0.8), (A3, 0.1, 0.5), (A4, 0.1, 0.8)},

and
B∗ = {(A1, 0.5, 0.1), (A2, 0.3, 0.6), (A3, 0, 0.8), (A4, 0.7, 0.1)}.

The intersection of A∗ and B∗ is

A∗ ∩B∗ = {(A1, 0.5, 0.1), (A2, 0.1, 0.8), (A3, 0, 0.8), (A4, 0.1, 0.8)}.

Definition 7. Let A∗ and B∗ be collections of intuitionistic fuzzy sets on X respectively,
where

A∗ = {(Ai, µA∗(Ai), vA∗(Ai)) : Ai ∈ X},

and
B∗ = {(Ai, µB∗(Ai), vB∗(Ai)) : Ai ∈ X}.

The union of A∗ and B∗ is

A∗ ∪B∗ = {(Ai, µA∗∪B∗(Ai), vA∗∪B∗(Ai)) : Ai ∈ A∗}

with
µA∗∪B∗(Ai) = max{µA∗(Ai), µB∗(Ai)}

and
vA∗∪B∗(Ai) = min{vA∗(Ai), vB∗(Ai)}.



D. N. Yunianti et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2198-2207 2203

Example 5. Let A∗ and B∗ be collections of intuitionistic fuzzy sets on X = {A1, A2, A3, A4}
respectively, where

A∗ = {(A1, 0.9, 0.1), (A2, 0.1, 0.8), (A3, 0.1, 0.5), (A4, 0.1, 0.8)},

and
B∗ = {(A1, 0.5, 0.1), (A2, 0.3, 0.6), (A3, 0, 0.8), (A4, 0.7, 0.1)}.

The union of A∗ and B∗ is

A∗ ∪B∗ = {(A1, 0.9, 0.1), (A2, 0.3, 0.6), (A3, 0.1, 0.5), (A4, 0.7, 0.1)}.

3.2. 3.2 Some Properties Of Intersection And Union In The Collection
Of Intuitionistic Fuzzy Sets

In this section, we show that intersection and union operation hold commutative,
assosiative, idempotent, and De Morgan’s laws properties.

Theorem 2. If A∗,∅ are collections of intuitionistic fuzzy sets on X then A∗ ∩ ∅ = ∅
and A∗ ∪∅ = A∗.

Proof. Let ∅ = {(Ai, 0, 1) : Ai ∈ X} so clearly that

A∗ ∩∅ = ∅ and A∗ ∪∅ = A∗.

.

Theorem 3. Let A∗ and B∗ be collections of intuitionistic fuzzy sets on X respectively.
Then,

(i) A∗ ∩B∗ = B∗ ∩A∗

(ii) A∗ ∪B∗ = B∗ ∪A∗

Proof.

(i) A∗ ∩B∗ = {(Ai, µA∗∩B∗(Ai), vA∗∩B∗(Ai)) : Ai ∈ X}
= {(Ai,min{µA∗(Ai), µB∗(Ai)},max{vA∗(Ai), vB∗(Ai)}) : Ai ∈ X}
= {(Ai,min{µB∗(Ai), µA∗(Ai)},max{vB∗(Ai), vA∗(Ai)}) : Ai ∈ X}
= B∗ ∩A∗

(ii) A∗ ∪B∗ = {(Ai, µA∗∪B∗(Ai), vA∗∪B∗(Ai)) : Ai ∈ X}
= {(Ai,max{µA∗(Ai), µB∗(Ai)},min{vA∗(Ai), vB∗(Ai)}) : Ai ∈ X}
= {(Ai,max{µB∗(Ai), µA∗(Ai)},min{vB∗(Ai), vA∗(Ai)}) : Ai ∈ X}
= B∗ ∪A∗

Based on that theorem, the intersection and union operations in the collections of
intuitionistic fuzzy sets satisfy the commutative property.



D. N. Yunianti et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2198-2207 2204

Theorem 4. If A∗, B∗, and C∗ are collections of intuitionistic fuzzy sets on X respectively
then:

(i) (A∗ ∩B∗) ∩ C∗ = A∗ ∩ (B∗ ∩ C∗)

(ii) (A∗ ∪B∗) ∪ C∗ = A∗ ∪ (B∗ ∪ C∗)

Proof.

(i) (A∗ ∩B∗) ∩ C∗ = {(Ai, µ(A∗∩B∗)∩C∗(Ai), v(A∗∩B∗)∩C∗(Ai)) : Ai ∈ X}
= {(Ai,min(µA∗∩B∗(Ai), µC∗(Ai)),max(vA∗∩B∗(Ai), vC∗(Ai))) : Ai ∈ X}
= {(Ai,min(min(µA∗(Ai), µB∗(Ai)), µC∗(Ai)),max(max(vA∗(Ai), vB∗(Ai)), vC∗(Ai))) :
Ai ∈ X}
= {(Ai,min(µA∗(Ai),min(µB∗(Ai), µC∗(Ai))),max(vA∗(Ai),max(vB∗(Ai), vC∗(Ai)))) :
Ai ∈ X}
= A∗ ∩ (B∗ ∩ C∗)

(ii) (A∗ ∪B∗) ∪ C∗ = {(Ai, µ(A∗∪B∗)∪C∗(Ai), v(A∗∪B∗)∪C∗(Ai)) : Ai ∈ X}
= {(Ai,max(µA∗∪B∗(Ai), µC∗(Ai)),min(vA∗∪B∗(Ai), vC∗(Ai))) : Ai ∈ X}
= {(Ai,max(max(µA∗(Ai), µB∗(Ai)), µC∗(Ai)),min(min(vA∗(Ai), vB∗(Ai)), vC∗(Ai))) :
Ai ∈ X}
= {(Ai,max(µA∗(Ai),max(µB∗(Ai), µC∗(Ai))),min(vA∗(Ai),min(vB∗(Ai), vC∗(Ai)))) :
Ai ∈ X}
= A∗ ∪ (B∗ ∪ C∗)

Based on that theorem, the intersection and union operations in the collections of
intuitionistic fuzzy sets satisfy the associative property.

Theorem 5. If A∗ is a collection of intuitionistic fuzzy sets on X, then:

(i) A∗ ∩A∗ = A∗

(ii) A∗ ∪A∗ = A∗

Proof.

(i) A∗ ∩A∗ = {(Ai, µA∗∩A∗(Ai), vA∗∩A∗(Ai)) : Ai ∈ X}
= {(Ai,min(µA∗(Ai), µA∗(Ai)),max(vA∗(Ai), vA∗(Ai))) : Ai ∈ X}
= {(Ai, µA∗(Ai), vA∗(Ai)) : Ai ∈ X}
= A∗

(ii) A∗ ∪A∗ = {(Ai, µA∗∪A∗(Ai), vA∗∪A∗(Ai)) : Ai ∈ X}
= {(Ai,max(µA∗(Ai), µA∗(Ai)),min(vA∗(Ai), vA∗(Ai))) : Ai ∈ X}
= {(Ai, µA∗(Ai), vA∗(Ai)) : Ai ∈ X}
= A∗



REFERENCES 2205

Based on that theorem, the intersection and union operations in the collections of
intuitionistic fuzzy sets satisfy the idempotent property.

Theorem 6. If A∗ and B∗ are collections of intuitionistic fuzzy sets on X respectively,
then:

(i) (A∗ ∩B∗)c = A∗c ∪B∗c

(ii) (A∗ ∪B∗)c = A∗c ∩B∗c

Proof.

(i) (A∗ ∩B∗)c = {(Ai, vA∗∩B∗(Ai), µA∗∩B∗(Ai)) : Ai ∈ X}
= {(Ai,max(vA∗(Ai), vB∗(Ai)),min(µA∗(Ai), µB∗(Ai))) : Ai ∈ X}
= {(Ai,max(µA∗c(Ai), µB∗c(Ai)),min(vA∗c(Ai), vB∗c(Ai))) : Ai ∈ X}
= A∗c ∪B∗c

(ii) (A∗ ∪B∗)c = {(Ai, vA∗∪B∗(Ai), µA∗∪B∗(Ai)) : Ai ∈ X}
= {(Ai,min(vA∗(Ai), vB∗(Ai)),max(µA∗(Ai), µB∗(Ai))) : Ai ∈ X}
= {(Ai,min(µA∗c(Ai), µB∗c(Ai)),max(vA∗c(Ai), vB∗c(Ai))) : Ai ∈ X}
= A∗c ∩B∗c

Based on that theorem, the intersection and union operations in the collections of
intuitionistic fuzzy sets satisfy De Morgan’s law.

4. Conclusions

In this article, we give definition collection of intuitionistic fuzzy sets. Definition
and some properties of intersection and union in the collection of intuitionistic fuzzy
sets are given too. We can say the properties of set operations in the set theory such
as commutative, assosiative, idempotent, and De Morgan’s law are also hold too in the
collection of intuitionistic fuzzy sets. Moreover, by using this concept, we can explore the
others properties of relations and develop tools for applications related the comparison of
object collection.

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