International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 59-68
Published online March 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.1.59
© The Korean Institute of Intelligent Systems
Yousef Al-Qudah^{1 }, Khaleed Alhazaymeh^{2}, Nasruddin Hassan^{3}, Hamza Qoqazeh^{1}, Mohammad Almousa^{1}, and Mohammad Alaroud^{1}
^{1}Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
^{2}Department of Basic Sciences and Mathematics, Faculty of Science, Philadelphia University, Amman, Jordan
^{3}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia
Correspondence to :
Yousef Al-Qudah (alquyousef82@gmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
A vague soft set is a mapping from a parameter set to the collection of vague subsets of the universal set. In this study, a vague soft relation is presented based on the Cartesian product of vague soft sets. The basic properties of these relations are studied to explain the concept of transitive closure of a vague soft relation. The symmetric, reflexive, and transitive closures of a vague soft set are introduced followed by examples to illustrate these relations. The concepts are further extended by proposing some of their properties. The existence and uniqueness of the transitive closure of a vague soft relation are established, and an algorithm to compute the transitive closure of a vague soft relation is also provided.
Keywords: Vague soft set, Transitive closure, Symmetric closure, Fuzzy set
Many researchers in economics, engineering, environmental sciences, social sciences, medical sciences, business, management, and numerous other fields encounter the modeling complexities presented by uncertain data on a daily basis. However, classical mathematical methods are not always effective because the uncertainties in these domains may be of various types. Probability theory, fuzzy set theory [1–4], intuitionistic fuzzy set theory [5], multi-fuzzy set theory [6,7], vague set theory [8,9], and interval mathematics [10,11] are often useful mathematical tools for describing uncertainty. Molodtsov [12] introduced the concept of a soft set for uncertain data. Maji and his colleagues [13,14] used the soft set theory in decision-making problems and introduced the concept of a fuzzy soft set [15]. Soft sets have been studied by many researchers, such as fuzzy soft sets [16–18], intuitionistic fuzzy soft sets [19–22], vague soft sets [23–28], multi-fuzzy soft sets [29–35], and vague soft set relations and functions [36]. Babitha and Sunil [37] introduced the concept of soft set relations and functions. Agarwal et al. [38] discussed the concept of relations in generalized intuitionistic fuzzy soft sets. Ibrahim et al. [39] introduced the concept of soft set composition relations and the construction of transitive closure. Park et al. [40] studied some properties of the equivalence of soft set relations, while Su et al. [41] introduced the concept of intuitionistic fuzzy decision-making with similarity measures and the ordered weighted averaging (OWA) operator. In addition, Saxena and Tayal [42] proposed the concept of normalization for the type-2 fuzzy relational data model based on fuzzy functional dependency, using fuzzy functions.
In the real world, vaguely specified data values exist in many applications, such as in data with fuzzy, imprecise, and uncertain properties. Fuzzy set (FS) theory was proposed to handle such vagueness by generalizing the notion of membership in a set. In a FS, each element is assigned a single value in the interval [0,1] reflecting its membership grade. This single value does not allow the separation of for membership evidence and against membership evidence. A vague set is a further generalization of FS. Instead of using point-based membership as in FSs, interval-based membership is used in a vague set. The interval-based membership is more expressive in capturing the vagueness of the data.
The remainder of this paper is organized as follows. In Section 2, basic notions about transitive closure of soft sets are reviewed. In Section 3, the transitive closure of vague soft sets is introduced; some theorems are proved; and examples are provided. In Section 4, certain properties of closure are studied on a vague soft set. The last section summarizes the contributions and highlights future research work.
In this section, some basic concepts of vague, soft, and vague soft sets are briefly reviewed.
A vague set [8] over
where for any
A review of the basic operations of the complement, intersection, and union of a vague set, as defined by Gau and Buehrer [8], are presented next.
The complement of a vague set
The intersection of two vague sets
The union of two vague sets
The soft set theory was proposed by Molodtsov [12] to provide an appropriate framework for uncertainty modeling. Molodtsov’s definitions of soft sets, soft subsets, complement, and the union of soft sets are presented below. Let
Let
Thus, a soft set over
Two soft sets (
For two soft sets (
(i)
(ii) ∀
This relationship is denoted as (
The complement of a soft set (
The union of two soft sets (
Ali et al. [43] proposed a definition of the extended intersection of soft sets as follows:
The extended intersection of two soft sets (
By combining a vague set and a soft set, Xu et al. [23,24] proposed a new concept called a vague soft set along with its operations of union and intersection, as specified in the following definitions. Let
A pair (
In other words, a vague soft set over
The union of two vague soft sets (
and
for all
The intersection of two vague soft sets (
and
for all
Let (
for all
Let (
for all
If a pair (
The Cartesian product of three or more non-empty vague soft sets can be defined by generalizing the definition of the Cartesian product of two vague soft sets. The Cartesian product (
Let (
Suppose that ℜ is a relation on a vague soft set (
In this section, the concepts of reflexive closure, symmetry closure, and transitive closure of soft sets proposed by Ibrahim et al. [39] are extended to those of vague soft sets, followed by examples to illustrate the operations of the newly defined relations. Then, a novel definition of transitive closure for a vague soft set is proposed along with its properties and an example to illustrate these properties.
The reflexive closure of
Note that
Let ℜ be a vague soft set relation on (
Now, the notion of reflexive closure of a vague soft set is illustrated using the following example.
Let
Consider the vague soft set relation ℜ defined on (
Then
The symmetry closure of a relation ℜ is constructed by adding all ordered pairs of the form (
The symmetric closure of a relation is obtained by taking the union of a relation with its inverse, i.e.,
The definition of symmetric closure on soft sets by Ibrahim et al. [39] is extended to the symmetric closure of a vague soft set below, followed by an example to illustrate its operation.
The symmetric closure of a relation is obtained by taking the union of the relation with its inverse, i.e., ℜ∪ℜ^{−1} where ℜ^{−1} = {(
In other words, let ℜ be a vague soft set relation on (
Consider Example 1
Now, the definition of transitive closure of a soft set by Ibrahim et al. [39] is extended to the transitive closure of a vague soft set.
The construction of the transitive closure of a relation is more complicated than that of reflexive or symmetric closure. The transitive closure of a relation can be determined by adding new ordered pairs that must be present and then repeating this process until no new ordered pairs are required.
(i)
(ii)
(iii)
Let
Let ℜ be a relation on a vague soft set (
Based on this duuality, the related properties of transitive closure of a vague soft set can also be investigated.
Let ℜ be a relation on a vague soft set (
(i) transitive (ℜ) = ℜ ∪ ℜ^{2} ∪ … ∪ ℜ
(ii)
(iii)
The properties of the reflexive closure of a vague soft set can be illustrated by the following example.
Suppose that ℜ is a relation on (
Thus
Now
Reading from the zero-one matrix, we see that ℜ* = {
The relation ℜ on a vague soft set (
Suppose that ℜ
Conversely, suppose that ℜ is transitive. It can be proved that ℜ
If
Suppose that
Now ℜ ⊂
Let ℜ be a binary relation on (
For the following definitions, lemmas, and proofs,
Let
(i)
(ii) A set of vague soft set functions Γ is a compatible system of functions if any two functions
(a) vague soft set functions
(b) vague soft set functions
The result follows from Definition 20.
If Γ is a compatible system of functions, then ∪Γ is a function with
Clearly ∪Γ is a relation. We also prove that this is a function. If
It is trivial to show that
In this section, the properties of reflexive closure and symmetric closure of a vague soft set are introduced.
Let ℜ be a vague soft set relation on (
(1)
(2)
(1) ℜ ⊂ ℜ ∪
(2) (ℜ ∪ ℜ^{−1}) ^{−1} = ℜ ^{−1} ∪ (ℜ^{−1})^{−1} = ℜ^{−1} ∪ ℜ = ℜ ∪ ℜ^{−1}, i.e. ℜ ∪ ℜ^{−1} is a symmetric vague soft set relation on (
If
Next, some basic properties of the reflexive and symmetric closure operators are proposed.
The reflexive closure operator
(1)
(2) ∀ℜ ∈
(3) ∀ℜ
(4) ∀ℜ
(5) ∀ℜ ∈
(1) By the reflexivity of
(2) ∀ℜ ∈
(3) ∀ℜ
(4) ∀ℜ
(5) ∀ℜ ∈
The symmetric closure operator
(1)
(2) ∀ℜ ∈
(3) ∀ℜ
(4) ∀ℜ
(5) ∀ℜ ∈
(1) By symmetry of
(2) ∀ℜ ∈
(3) ∀ℜ
and for
(4) ∀ℜ,
(5) ∀ℜ ∈
Reflexivity, symmetry, and transitivity are three of the most important properties of vague soft set relations. This work has shown how reflexive, symmetric, and transitive closure of vague soft set relations can be determined. After establishing some properties of the transitive closure of a vague soft set, some properties of the symmetric closure and reflexive closure operators were provided on a vague soft set. To extend this work, applications of transitive closure of vague soft sets in decision-making can be considered.
No potential conflict of interest relevant to this article was reported.
E-mail: alquyousef82@gmail.com
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 59-68
Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.59
Copyright © The Korean Institute of Intelligent Systems.
Yousef Al-Qudah^{1 }, Khaleed Alhazaymeh^{2}, Nasruddin Hassan^{3}, Hamza Qoqazeh^{1}, Mohammad Almousa^{1}, and Mohammad Alaroud^{1}
^{1}Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
^{2}Department of Basic Sciences and Mathematics, Faculty of Science, Philadelphia University, Amman, Jordan
^{3}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia
Correspondence to:Yousef Al-Qudah (alquyousef82@gmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
A vague soft set is a mapping from a parameter set to the collection of vague subsets of the universal set. In this study, a vague soft relation is presented based on the Cartesian product of vague soft sets. The basic properties of these relations are studied to explain the concept of transitive closure of a vague soft relation. The symmetric, reflexive, and transitive closures of a vague soft set are introduced followed by examples to illustrate these relations. The concepts are further extended by proposing some of their properties. The existence and uniqueness of the transitive closure of a vague soft relation are established, and an algorithm to compute the transitive closure of a vague soft relation is also provided.
Keywords: Vague soft set, Transitive closure, Symmetric closure, Fuzzy set
Many researchers in economics, engineering, environmental sciences, social sciences, medical sciences, business, management, and numerous other fields encounter the modeling complexities presented by uncertain data on a daily basis. However, classical mathematical methods are not always effective because the uncertainties in these domains may be of various types. Probability theory, fuzzy set theory [1–4], intuitionistic fuzzy set theory [5], multi-fuzzy set theory [6,7], vague set theory [8,9], and interval mathematics [10,11] are often useful mathematical tools for describing uncertainty. Molodtsov [12] introduced the concept of a soft set for uncertain data. Maji and his colleagues [13,14] used the soft set theory in decision-making problems and introduced the concept of a fuzzy soft set [15]. Soft sets have been studied by many researchers, such as fuzzy soft sets [16–18], intuitionistic fuzzy soft sets [19–22], vague soft sets [23–28], multi-fuzzy soft sets [29–35], and vague soft set relations and functions [36]. Babitha and Sunil [37] introduced the concept of soft set relations and functions. Agarwal et al. [38] discussed the concept of relations in generalized intuitionistic fuzzy soft sets. Ibrahim et al. [39] introduced the concept of soft set composition relations and the construction of transitive closure. Park et al. [40] studied some properties of the equivalence of soft set relations, while Su et al. [41] introduced the concept of intuitionistic fuzzy decision-making with similarity measures and the ordered weighted averaging (OWA) operator. In addition, Saxena and Tayal [42] proposed the concept of normalization for the type-2 fuzzy relational data model based on fuzzy functional dependency, using fuzzy functions.
In the real world, vaguely specified data values exist in many applications, such as in data with fuzzy, imprecise, and uncertain properties. Fuzzy set (FS) theory was proposed to handle such vagueness by generalizing the notion of membership in a set. In a FS, each element is assigned a single value in the interval [0,1] reflecting its membership grade. This single value does not allow the separation of for membership evidence and against membership evidence. A vague set is a further generalization of FS. Instead of using point-based membership as in FSs, interval-based membership is used in a vague set. The interval-based membership is more expressive in capturing the vagueness of the data.
The remainder of this paper is organized as follows. In Section 2, basic notions about transitive closure of soft sets are reviewed. In Section 3, the transitive closure of vague soft sets is introduced; some theorems are proved; and examples are provided. In Section 4, certain properties of closure are studied on a vague soft set. The last section summarizes the contributions and highlights future research work.
In this section, some basic concepts of vague, soft, and vague soft sets are briefly reviewed.
A vague set [8] over
where for any
A review of the basic operations of the complement, intersection, and union of a vague set, as defined by Gau and Buehrer [8], are presented next.
The complement of a vague set
The intersection of two vague sets
The union of two vague sets
The soft set theory was proposed by Molodtsov [12] to provide an appropriate framework for uncertainty modeling. Molodtsov’s definitions of soft sets, soft subsets, complement, and the union of soft sets are presented below. Let
Let
Thus, a soft set over
Two soft sets (
For two soft sets (
(i)
(ii) ∀
This relationship is denoted as (
The complement of a soft set (
The union of two soft sets (
Ali et al. [43] proposed a definition of the extended intersection of soft sets as follows:
The extended intersection of two soft sets (
By combining a vague set and a soft set, Xu et al. [23,24] proposed a new concept called a vague soft set along with its operations of union and intersection, as specified in the following definitions. Let
A pair (
In other words, a vague soft set over
The union of two vague soft sets (
and
for all
The intersection of two vague soft sets (
and
for all
Let (
for all
Let (
for all
If a pair (
The Cartesian product of three or more non-empty vague soft sets can be defined by generalizing the definition of the Cartesian product of two vague soft sets. The Cartesian product (
Let (
Suppose that ℜ is a relation on a vague soft set (
In this section, the concepts of reflexive closure, symmetry closure, and transitive closure of soft sets proposed by Ibrahim et al. [39] are extended to those of vague soft sets, followed by examples to illustrate the operations of the newly defined relations. Then, a novel definition of transitive closure for a vague soft set is proposed along with its properties and an example to illustrate these properties.
The reflexive closure of
Note that
Let ℜ be a vague soft set relation on (
Now, the notion of reflexive closure of a vague soft set is illustrated using the following example.
Let
Consider the vague soft set relation ℜ defined on (
Then
The symmetry closure of a relation ℜ is constructed by adding all ordered pairs of the form (
The symmetric closure of a relation is obtained by taking the union of a relation with its inverse, i.e.,
The definition of symmetric closure on soft sets by Ibrahim et al. [39] is extended to the symmetric closure of a vague soft set below, followed by an example to illustrate its operation.
The symmetric closure of a relation is obtained by taking the union of the relation with its inverse, i.e., ℜ∪ℜ^{−1} where ℜ^{−1} = {(
In other words, let ℜ be a vague soft set relation on (
Consider Example 1
Now, the definition of transitive closure of a soft set by Ibrahim et al. [39] is extended to the transitive closure of a vague soft set.
The construction of the transitive closure of a relation is more complicated than that of reflexive or symmetric closure. The transitive closure of a relation can be determined by adding new ordered pairs that must be present and then repeating this process until no new ordered pairs are required.
(i)
(ii)
(iii)
Let
Let ℜ be a relation on a vague soft set (
Based on this duuality, the related properties of transitive closure of a vague soft set can also be investigated.
Let ℜ be a relation on a vague soft set (
(i) transitive (ℜ) = ℜ ∪ ℜ^{2} ∪ … ∪ ℜ
(ii)
(iii)
The properties of the reflexive closure of a vague soft set can be illustrated by the following example.
Suppose that ℜ is a relation on (
Thus
Now
Reading from the zero-one matrix, we see that ℜ* = {
The relation ℜ on a vague soft set (
Suppose that ℜ
Conversely, suppose that ℜ is transitive. It can be proved that ℜ
If
Suppose that
Now ℜ ⊂
Let ℜ be a binary relation on (
For the following definitions, lemmas, and proofs,
Let
(i)
(ii) A set of vague soft set functions Γ is a compatible system of functions if any two functions
(a) vague soft set functions
(b) vague soft set functions
The result follows from Definition 20.
If Γ is a compatible system of functions, then ∪Γ is a function with
Clearly ∪Γ is a relation. We also prove that this is a function. If
It is trivial to show that
In this section, the properties of reflexive closure and symmetric closure of a vague soft set are introduced.
Let ℜ be a vague soft set relation on (
(1)
(2)
(1) ℜ ⊂ ℜ ∪
(2) (ℜ ∪ ℜ^{−1}) ^{−1} = ℜ ^{−1} ∪ (ℜ^{−1})^{−1} = ℜ^{−1} ∪ ℜ = ℜ ∪ ℜ^{−1}, i.e. ℜ ∪ ℜ^{−1} is a symmetric vague soft set relation on (
If
Next, some basic properties of the reflexive and symmetric closure operators are proposed.
The reflexive closure operator
(1)
(2) ∀ℜ ∈
(3) ∀ℜ
(4) ∀ℜ
(5) ∀ℜ ∈
(1) By the reflexivity of
(2) ∀ℜ ∈
(3) ∀ℜ
(4) ∀ℜ
(5) ∀ℜ ∈
The symmetric closure operator
(1)
(2) ∀ℜ ∈
(3) ∀ℜ
(4) ∀ℜ
(5) ∀ℜ ∈
(1) By symmetry of
(2) ∀ℜ ∈
(3) ∀ℜ
and for
(4) ∀ℜ,
(5) ∀ℜ ∈
Reflexivity, symmetry, and transitivity are three of the most important properties of vague soft set relations. This work has shown how reflexive, symmetric, and transitive closure of vague soft set relations can be determined. After establishing some properties of the transitive closure of a vague soft set, some properties of the symmetric closure and reflexive closure operators were provided on a vague soft set. To extend this work, applications of transitive closure of vague soft sets in decision-making can be considered.
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