A New Trend of Bipolar-Valued Fuzzy Cartesian Products, Relations,
and Functions
FADI M. A. AL-ZU’BI1,3, ABDUL GHAFUR AHMAD1,
ABD ULAZEEZ ALKOURI2, MASLINA DARUS1
1Department of Mathematical Sciences, Faculty of Science & Technology,
Universiti Kebangsaan Malaysia,
43600 UKM Bangi, Selangor,
MALAYSIA
2Department of Mathematics Department, Faculty of Science,
Ajloun National University, P.O. 43, Ajloun- 26810,
JORDAN
3College of Natural and Health Sciences,
Zayed University,
Abu Dhabi,
UNITED ARAB EMIRATES
Abstract: - A bipolar-valued fuzzy set (BVFS) is a generalization of the fuzzy set (FS). It has been applied to a
wider range of problems that cannot be represented by FS. New forms of the bipolar-valued fuzzy Cartesian
product (BVFCP), bipolar-valued fuzzy relations (BVFRs), bipolar-valued fuzzy equivalence relations
(BVFERs), and Bipolar-valued fuzzy functions (BVFFs) are constructed to be a cornerstone of creating new
approach of BVF group theory. Unlike other approaches, the definition of BVFCP “A×B” is exceptionally
helpful at reclaiming again the subset A and B by using a fitting lattice. Also, the present approach reduced the
calculations and numerical steps in contrast to fuzzy and classical BVF cases. Results relating to those on
relations, equivalence relations, and functions in the fuzzy cases are proved for BVFRs, BVFERs, and BVFFs.
Key-Words: - Bipolar Valued Fuzzy Cartesian Product, Bipolar Valued Fuzzy Relation, Bipolar Valued Fuzzy
Equivalence Relations, Bipolar valued Fuzzy Functions, Fuzzy Cartesian Product, Fuzzy
Relation, Fuzzy Equivalence Relations, Fuzzy Functions.
Received: October 15, 2023. Revised: May 13, 2024. Accepted: July 4, 2024. Published: July 29, 2024.
1 Introduction
In 1965, Fuzzy sets, [1], were introduced as a
generalization of an ordinary set. The concepts of
similarity relations and fuzzy orderings were
fundamental in many branches of pure and
practical research, [2]. The most modern electronic
machines help people save energy, time, water, and
effort by using the notion of fuzzy sets. Regarding
people's needs the notions of fuzzy sets, [1], and
fuzzy logic control systems were established and
applied in the deepness of industry. Several
applications are used and applied fuzzy logic
control in the industry such as washing machines,
subway trains, cars, coffee machines, etc, [3], [4],
[5], [6], [7], [8], [9], [10].
Some limitations and obstacles in conveying
human information and inherited experience to
mathematical tools led to the incorporation of the
perception of FS. For instance, [11], established the
notion of BVFS as an enlargement of FSs in which
the codomain of membership degree is expanded
from unit interval “[0, 1]” to the “[-1, 1]”. In
BVFS, the “0” value of the membership degree
represents elements that are irrelevant to the
identical property, the value of the membership
degree lies in (0, 1] represents elements that
partially satisfy the property, While the value of
membership degree lies in [-1, 0) represents
elements that partially satisfy to essentially
connected with opposite-property. In terms of
BVFS applications, numerous researchers proposed
applications for decision-making problems, In [12]
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applied bipolar-valued rough fuzzy sets to decision
information systems. In 2022, [13], introduced the
assessment for choosing the best alternative fuel by
using bipolar-valued fuzzy sets. Also, In 2019,
[14], incorporated Hesitant fuzzy sets and bipolar-
valued fuzzy sets to solve the problem of multi-
attribute group decision-making. In 2020, [15],
defined bipolar Fuzzy Graphs with Applications.
In 2023, [16], defined the bipolar interval-
valued fuzzy hypergraph. Their notion can
represent fuzzy data structure. Their research
discovered the inner relationship of fuzzy data and
gave some characterizations of it. In 2018, [17],
introduced the notion of bipolar fuzzy matroids and
applied it to graph theory and linear algebra. Also,
they applied several applications in decision
support systems and network analysis by using
bipolar fuzzy matroids. The first generalization of
bipolar fuzzy sets to the realm of complex numbers
is highlighted by introducing the notion of bipolar
complex fuzzy sets, [18].
Functions are unique types of relations in
standard set theory, while relations are subsets of
Cartesian products. Therefore, the standard theory
of relations and functions heavily relies on the
concept of the Cartesian product. Several
researchers, [19], [20], have dealt with bipolar-
valued fuzzy relations without referring to what
could be termed bipolar-valued fuzzy Cartesian
products; this concept has not yet been properly
explained.
In this stage, the priority of finding the basic
notions of mathematics such as Cartesian products,
relations, and equivalence relations based on BVFS
becomes essential. Several researchers, [21],
introduced fuzzy Cartesian products, fuzzy
relations, fuzzy equivalence relations, and fuzzy
functions. Also [22], [23], started their works by
introducing Cartesian products, relations,
equivalence relations, and functions under
intuitionistic fuzzy sets following the, [21],
approach. Later, [24], incorporated the complex
fuzzy sets and group theory by defining the
complex fuzzy Cartesian products, relations, and
functions according to [21]. All the mentioned
researchers extended their studies to the field of
algebra and used the [25], approach in fuzzy sets,
intuitionistic fuzzy sets, and complex fuzzy sets,
respectively, to build the fuzzy space, intuitionistic
fuzzy space, and complex fuzzy space. Therefore,
our contribution to defining the bipolar fuzzy
Cartesian products, relations, and functions can be
straightforward by following the [21], approach
under a new set of bipolar-valued fuzzy sets. The
problem appears in building a reasonable and
rational structure of BVFCP, BVFRs, and
BVFERs. Therefore, we start from ordinary set
principles. Ordinary functions are considered a
kind of ordinary relations as well as ordinary
relations are a collection of elements contained in
ordinary Cartesian products in ordinary set theory,
[26]. Thus, the Cartesian products highlighted a
magnificent need to build the basic theories of
functions and relations. The notion of fuzzy
relations was applied in several types of research,
[27], [28], [29], without referring to the notion of
fuzzy Cartesian products (FCPs). Later, the
generalized notions of FCPs, fuzzy relations (FR),
and fuzzy equivalent relations (FERs) were
reasonably achieved by [21]. In the same manner,
the notion of BVFCPs, BVFRs, and BVFER are
not yet correctly accomplished.
In 1991, [21], have avoided the inconvenience
of retrieving the fuzzy subsets A and B from the
A×B defined by [30], and reduced fuzzy Cartesian
products to the ordinary Cartesian product. In,
Reference, [21], They proposed FRs and FERs
based on his new finding of fuzzy Cartesian
products. [31], got several findings by employing
the concept of fuzzy relations. Consequently, [25],
introduced a new method to fuzzy group theory
based on fuzzy space and fuzzy binary operations.
His method was judged to reformulate and
generalize the fuzzy subgroups, [32]. The other
authors used and applied, [33], approach to
introducing Fuzzy ideals and bi-ideal in fuzzy
semigroups, fuzzy normal subgroups, [34],
intuitionistic fuzzy spaces, and intuitionistic fuzzy
groups, [22], [23], complex fuzzy groups, [24] and
others.
In this research, a reasonable development of
bipolar-valued fuzzy Cartesian products is
proposed. This notion avoids the inconvenience
that appeared in [30] and can reduce BVFCP to
fuzzy Cartesian products and consequently to an
ordinary Cartesian product. After that, some
reasonable notions such as BVFR and BVFER are
introduced. Some results corresponding to those on
crisp relations, fuzzy relations, and fuzzy
equivalence relations are studied and proved for
BVFRs, BVFERs and BVFFs.
2 Preliminaries
In this section, we recall some main theorems and
notions related to the present results.
Definition 2.1 [1], A fuzzy set can be written as
a membership function maps a universe of
discourse to a unit interval .
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Definition 2.2 [12]Let be a nonempty set.
Then, a set is called a bipolar-
valued fuzzy set in , where and
.
Definition 2.3 [12] Let be a nonempty set, and
let
(i)
is contained in , denoted by , if
and
(ii)
The form represents
the complement of , and it is a BFS in
defined as:
, where
(iii)
The form represents the intersection
of and , and it is a BFS in defined as:
(iv)
The form represents the union of
and , and it is a BFS in defined as:
Definition 2.4 [21] The FCP of two ordinary sets
and ,
, is the collection of all M-fuzzy
subsets of , where,
An element of
is then a function
, or
The FCP of a fuzzy subset of and a
fuzzy subset of is the M-fuzzy
subset of defined by:
It is clear that
is an element of
for
every and .
Definition 2.5 [21] An FR maps to is a
subset of the FCP . Then is a collection of
M-fuzzy subsets . An FR maps
to is called an FR in .
Definition 2.6 [21] Let be an FR in , that is
. Then is called:
-Reflexive in if and only if and ,
such that .
-Symmetric if and only if whenever
, such that
.
-Transitive if and only if whenever
and
, such that .
An FR in is called a FER in if and only if it is
satisfies all axioms above.
Definition 2.7 A fuzzy function from M to N is a
fuzzy relation G from M to N that meets the
conditions given below:
(i) For every element and membership
grade , there exist unique elements
and membership grade such that
belongs to some .
(ii) If (m, n, e1, t1) A G and (m, n', e2, t2) B
G, then n = n'.
(iii) If (m, n, e1, t1) A G and (m, n, e2, t2) B
G, then e1 > e2 indicates t1t2.
(iv) If , then e = 0 indicates t
= 0 and e = 1 indicates t = 1.
Conditions (i) and (ii) lead to the conclusion that
there exists a unique ordinary function G:M→ N
and for each element m M, there exists a unique
ordinary function gm: L→L. Conditions (iii) and
(iv) are equivalent to the following conditions:
( gm shows nondecreasing behaviour on the set L.
() gm(0) = 0 and gm (1) = 1.
3 New Cartesian Product between
BVFSs
In this section, the form of BVFCP is discussed,
and the structure of a suitable lattice is presented
below. The main definition is formulated in
Definition 3.1. the difference between our approach
and the previous approach is illustrated by Example
3.1. A justification after the example is discussed in
detail. Lastly, some relations of BVFCP union and
intersection are described in Proposition 3.1.
The totally ordered set is a
lattice concerning infimum and supremum
operations. Then is distributive but not
complemented lattice. Here a partial order "" on
, is defined on as follows:
(i)
iff
, and , whenever
and .
(ii)
whenever or .
for every and
.
The Cartesian product is then a
distributive but not complemented vector lattice.
The infimum and supremum operations in K are
characterized as follows:
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1. The operation of infimum in K is characterized
by
2. The operation of supremum in K is
characterized by
Note that the equality holds in the part “2”
whenand and .
A K-bipolar valued fuzzy subset associate
values of membership function from to the
lattice e, is thus a function from to
. In this research, the form
or, simply,
, where , ,
are used to represent a BVF subset of . Also, a
K-BVF subset of , a BVF subset of and a
K-BVF subset of are represented by
and ,
respectively. To each BVF subset
of and BVF subset of there
maps an K-BVF subset
of . Also,
the representation ; where
, where, .
Definition 3.1 The BVFCP of two ordinary sets
and , denoted by , is the collection of all K-
BVF subsets of that is,
An element of is then a function
, or
The BVFCP of a BVF subset
of and a BVF subset
of is the K-BVF subset of
defined by:
Therefore, is an element of ,
and .
Example 3.1 In this example we are going to
compare the ordinary approach with the present
approach, suppose a BVF subset
of
and a BVF subset
of . Then the K-
BVF subset of defined by:
But the ordinary (classical) cartesian product of
two subsets BVFS defined by:
where is any BVF t-norm, here we
apply
In Example 3.1, the difference clearly appeared
by providing the ability to recall the values of
positive and negative membership functions for
both objects , after computing
the K-BVF subset of In contrast to
the ordinary case of BVFS, the details were omitted
and new values appeared. The presented approach
identifies and recalls the original information
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before computations and simplification of the
information in any application.
Remarks: (1) When the ordinary sets U and V are
considered as bipolar-valued fuzzy subsets of
themselves, i.e.
, the notions of bipolar-valued
fuzzy Cartesian product and both fuzzy Cartesian
products and ordinary Cartesian products of U and
V equal, i.e. .
(2) It is easy to generalize the previous definition
and statements by substituting a random completely
distributive lattice for the W. The following
proposition can be easily verified if one considers
the properties of the lattice K.
Proposition 3.1 For all nonempty BVF subsets
of and nonempty fuzzy subsets C, D of ,
we have:
(1)
(2)
(3)
(4)
(5)
Proof. Trivial.
4 Bipolar Valued Fuzzy- Relations
and Equivalence Relations
In this section, the main definitions of BVFR and
BVER are proposed and discussed. The most
fundamental theorems and results on ordinary
relations, [26] and fuzzy relations, [21], are
investigated and developed to be reasonable under
the notion of BVFR and BVFER.
Definition 4.1 A BVFR maps to is a subset
of the BVFCP . In other words, is a
member of K-BVF subsets . A
BVFR from to is said to be a BVFR in .
Note 4.1 From Definition 3.1, we may see that the
BVFCP is itself a BVFR from to .
Note 4.2 The BVFCP is called the complete
BVFR in .
Note 4.3 The BVFR is called the null
BVFR.
Note 4.4 The identity BVFR lies between complete
and null BVFR, denoted by . that is
contains in
K-BVF subset.
Definition 4.2 Let and : to be two
BVFRs. We call that is containing , denoted
by if and only if when
, there
exists such that
. If
and , then and are equal, that is
.
Note. 4.5 We may associate each K-BVF subset
of to a
K-BVF subset of defined by
Definition 4.3 Let be a BVFR. The
inverse of = is the BVFR defined by
.
Definition 4.4 Let and be two
BVFRs. The composition of and , denoted
, is a BVFR defined by
}. Where a K-BVF subsets
defined by:
if and only
if such that
and
for some and
Example 4.1 Suppose I, N, A, Z, M and Q
represent names of cities and there are three sets
labelled as and
. Then the ordinary Cartesian product of
and are defined as
, and
. For example, let
be a relation called “the first city is
warmer than the second city” and let be a
relation called “the first city is more modern than
the second city”. The relations and
can be presented by the opinion of
tourists who have visited and/or had enough
knowledge about these cities. The following
relational matrices may evaluate the relations
by using a bipolar fuzzy mathematical
method as:
=
For illustration, the relation between cities M
and I (the element ) may be
represented with membership value of
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where the values “”,
and “” have pointed the opinion of tourists that
the cities M, and I, respectively, are warm (satisfied
the property) and the values “”, and “”
have pointed the opinion of tourists that the cities
M, and I, respectively, are not warm (not satisfied
the property).
=
For illustration, the relation between cities I
and K (the element ) may be
represented with membership value of
where the values “”,
and “” have pointed the opinion of tourists that
the cities I, and N, respectively, are considered as
modern city and the values “”, and “”
have pointed the opinion of tourists that the cities I,
and N, respectively, are not considered as a modern
city. Then, the composition presented the
current approach may be running as follows:
=
Clearly, Definition 4.4 has the same algebraic
structure with ordinary composition relation and
composition fuzzy relation. Therefore, no need to
do an additional process to evaluate the
membership values of the composition of two
fuzzy relations as in, [1], [26].
Theorem 4.1 For any fuzzy relations
defined on the appropriate sets,
we have:
(6)
and (7)
(8)
(9)
(10)
and (11)
(12)
(13)
Proof. Straightforward from Definition 2.3, 4.2,
4.3, and 4.4.
Definition 4.5 Let be a BVFR in , i.e.
. Then
1. is called reflexive in if and only if
and , such that
, that is if
and only if .
2. is called symmetric if and only if whenever
,
such that ,
that is if and only if .
3. is called transitive if and only if whenever
and
such that
, that is
if and only if .
A BVFR in is called a BVFER in if and
only if it is satisfied all three axioms above.
Example 4.1 For any set and are
BVFER in .
Theorem 4.2 Let and be BVFRs of a
nonempty set . As a result,
)i) If is reflexive, then it follows that and
are also reflexive. (This applies to both
symmetric and transitive cases).
(iii) If is reflexive, then is a subset of
(iv) If is symmetric, then both the union and
intersection between and are symmetric and
their composition is commutative.
(v) If and are reflexive, then their intersection
is reflexive. (Holds for both symmetric and
transitive)
(vi) If and are symmetric, then their union is
symmetric.
Regarding properties 5.1 part (i) and (v), we
may deduce that , and are BVFER
in , if and are BVFR in
Theorem 4.3 Let be a BVFER in . Then,
(i) induces a FER, in
defined by:
and
for some
(ii) induces an equivalence
relation, in the ordinary case, in
defined by
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for some
Proof. (i) We want to prove that is,
• Reflexive: , since is reflexive,
we get for
some , therefore the elements on the form
.
• Symmetric: if ,
then for
some . But is symmetric, then
for some
. Therefore the elements in the form
.
• Transitive: if .
and , then
and
for some
. However, since is transitive, then
for some
. Therefore, the elements in the form
.
(ii) the proof of (ii) is like (i).
5 Bipolar Valued Fuzzy Function
To extend an ordinary theory of fuzzy relation, we
mean to introduce the notion of BVF function,
since functions are considered as a kind of relation
in the ordinary set theory. In this section, we
identify BVF functions similarly to a kind of
BVFRs.
Definition 5.1 Let and be nonempty sets. A
BVF function from to can be described as a
function from to characterized by the
ordered pair uU, where
is a function from to and
uU is a family of functions
that satisfy the
following conditions:
1. are nondecreasing on ,
and
2.
and
In such a way that the image of any bipolar
valued fuzzy subset M of U under F results in the
bipolar valued fuzzy subset F(M) of V, defined as:
=
if
if
(14)
for every , We write
uU or, simply,
to represent a BVF
function from U to V, and we refer to the
individual functions as follows:
, the comembership
functions associated to .
Two BVF functions
and from to are
considered equal, denoted as , iff
for every . we have:
Theorem 5.1 Two BVF functions
and
from to are equal iff
and
, where
for every .
Proof: It is evident that if and , for
every , then .
Alternatively, assuming . If , then
there exists an element such that
. Now, let us consider the bipolar-valued
fuzzy subset M of U defined by:
if
if (15)
Then we have
if
if
(16)
and
if
if
(17)
Now, if , then , This
Refutes the assertion that F = G.
Alternatively, if then there exist
and such that
. In such a case, let us consider
the bipolar- valued fuzzy subset of U.
if
if (18)
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then and
This implies that .
Hence, the theorem is proven.
Let be a BVF
function. The inverse image under of a bipolar
valued fuzzy subset of , denoted by , is
a bipolar valued fuzzy subset of defined by:
(19)
If the comembership functions ,
, are surjective, then, taking the properties of
into account, we get
and
where Δ is any W subset. In this instance, the
preceding definition is equal to:
Proposition 5.1 Assume that
be any BVF
function whose comembership
functions are onto. For every
bipolar valued fuzzy subset of we have
=
(20)
where the supremum is taken over the set of values
Proof. Let , and be
the bipolar valued fuzzy subset of denoted by
. We show that .
For simplicity, let H= N(F(u)). For each , we
have:
Hence, , Which implies that .
Alternatively, assume there exists an element
such that
for every
We have thus showed that each such that
is a subset of . This proves that .
Theorem 5.2
Let be a bipolar
valued fuzzy (BVF) function. For every bipolar
valued fuzzy subsets , of and for every l
bipolar valued fuzzy subsets of , the
following holds:
a. (21)
b. if is onto, (22)
c. if then (23)
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d. (24)
e.
(equality holds if is (25)
f. (26)
g. if then (27)
h. (28)
(equality holds if is bijective
i. If is onto for all , then:
j. ( is an index
set), (29)
k. (equality holds if
is ), (30)
l. , (31)
m. , (32)
n. (equality holds if is onto).
(33)
o. If
for every ,
then:
if is onto. (34)
p. {Equality is achieved when F is a bijective
function and if
.}
If is bijective and if
, then:
(35)
Proof. To exemplify the employed technique, we
shall solely demonstrate the proofs for parts (e), (l),
and (p).
(e)
=
If F is a one-to-one function, the operation is
not used in the above steps, and the equality holds.
(l) It is evident, based on property (g), that
Now,
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(p)
,
since
and is bijective
The composition of two bipolar valued fuzzy
functions and
is the bipolar
valued fuzzy function defined by
, for all .
Let , be onto, for all . Then,
F
since the are onto
This means that
where , evidently satisfy the
conditions and of comembership functions.
Let be a
bipolar valued fuzzy (BVF) function. F is
considered injective or one-to-one if, for any
bipolar valued fuzzy (BVF) subsets and of
, implies . The
definitions of surjective and bijective bipolar
valued fuzzy functions can be established in a
comparable manner.
Establishing that F is one-to-one (respectively,
onto) is not a challenging task, as it can be
demonstrated that F and ,
are one-to-one (resp. onto).
A bipolar valued fuzzy function
is said to be
invertible if there exists a bipolar valued fuzzy
function such
that and , where
The bipolar valued fuzzy function is
called the inverse of and is denoted by .
Theorem 5.3
Let and
be bipolar valued
fuzzy functions. Let be onto, for
all . Then
(i) The composition of and is
given by:
(ii) is one-to-one (resp.
onto) iff and , are one-
to-one (respectively, onto).
(iii) is invertible iff
and are invertible. The inverse
of is given by
.
6 Conclusion
A novel structure of bipolar-valued fuzzy Cartesian
products was introduced, indicating all parts of its
structure. So, analogously to the basic use of crisp
Cartesian products, BVF-relation, BVF-
equivalence relations and BVFFs were proposed.
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Some Results and numerical examples of BVFR,
BVFER and BVFF related to ordinary and fuzzy
relations were studied, distinguished and proved.
What distinguishes our research is the logical
structure that coincides with the basic structure of
algebra. (i.e. the BVFCP “A×B” can reclaim the
subset A and B without losing or omitting some
values as in ordinary algebraic structure), The
limitation of this study appears in the disability of
representing two-dimensional phenomena using the
form of BVFCP, BVFR, and BVF function.
Therefore, complex bipolar-valued fuzzy Cartesian
products, relations, and functions can handle this
limitation by extending the range of BVFCP from
[-1, 1] to the complex form [-1,1] + i [-1, 1]. As
future research, our results will be a cornerstone to
build the BVF-equivalence class, BVF partial
order. Also, the concept of bipolar valued fuzzy
function can be utilized to start an attractive
journey to introduce the BVF group and ring.
References:
[1] L. A. Zadeh, “Fuzzy sets”, Information and
Control, vol. 8, no. 3, pp. 338–353, Jun.
1965, doi: 10.1016/S0019-9958(65)90241-
X.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Fadi Al-Zubi prepared, created and presented this
work, specifically writing the initial draft.
- Abdul Ghafoor Ahmed and Maslina Darus
Supervision and leadership responsibility for the
implementing research activities.
- Abd Ulazeez Alkouri is responsible for project
management and coordination for planning and
implementing the research activity.
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DOI: 10.37394/23206.2024.23.53
Fadi M. A. Al-Zu’bi, Abdul Ghafur Ahmad,
Abd Ulazeez Alkouri, Maslina Darus
E-ISSN: 2224-2880
513
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Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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DOI: 10.37394/23206.2024.23.53
Fadi M. A. Al-Zu’bi, Abdul Ghafur Ahmad,
Abd Ulazeez Alkouri, Maslina Darus
E-ISSN: 2224-2880
514
Volume 23, 2024
