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닫기 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

## Transitive Closure of Vague Soft Set Relations and its Operators

Yousef Al-Qudah1 , Khaleed Alhazaymeh2, Nasruddin Hassan3, Hamza Qoqazeh1, Mohammad Almousa1, and Mohammad Alaroud1

1Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
2Department of Basic Sciences and Mathematics, Faculty of Science, Philadelphia University, Amman, Jordan
3School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Correspondence to :
Yousef Al-Qudah (alquyousef82@gmail.com)

Received: July 17, 2021; Revised: September 28, 2021; Accepted: October 12, 2021

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 , intuitionistic fuzzy set theory , 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  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 . Soft sets have been studied by many researchers, such as fuzzy soft sets , intuitionistic fuzzy soft sets , vague soft sets , multi-fuzzy soft sets , and vague soft set relations and functions . Babitha and Sunil  introduced the concept of soft set relations and functions. Agarwal et al.  discussed the concept of relations in generalized intuitionistic fuzzy soft sets. Ibrahim et al.  introduced the concept of soft set composition relations and the construction of transitive closure. Park et al.  studied some properties of the equivalence of soft set relations, while Su et al.  introduced the concept of intuitionistic fuzzy decision-making with similarity measures and the ordered weighted averaging (OWA) operator. In addition, Saxena and Tayal  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.

### 2.1 Vague Sets

A vague set  over U is characterized by a truth-membership function tν and false membership function fν,

tν:U[0,1]and fν:U[0,1],

where for any uiU, tν(ui) is a lower bound on the membership grade of ui derived from the evidence for ui; fν(ui) is a lower bound on the negation of ui derived from the evidence against ui; and tν(ui) + fν(ui) ≤ 1. The membership grade of ui in the vague set is bounded to a subinterval [tν(ui), 1 − fν(ui)] of [0, 1]. The vague values [tν(ui), 1−fν(ui)] indicate that the exact membership grade μν(ui) of ui may be unknown, but it is bounded by tν(ui) ≤ μν(ui) ≤ 1 − fν(ui), where tν(ui) + fν(ui) ≤ 1.

A review of the basic operations of the complement, intersection, and union of a vague set, as defined by Gau and Buehrer , are presented next.

Definition 1 

The complement of a vague set A is denoted by Ac and is defined by

tAc=fA,1-fAc=1-tA.
Definition 2 

The intersection of two vague sets A and B is a vague set C, denoted as C = AB, with truth-membership and false membership functions related to those of A and B by

tC=min(tA,tB),1-fC=min(1-fA,1-fB)=1-max(fA,fB).
Definition 3 

The union of two vague sets A and B is a vague set C, denoted as C = AB, with truth-membership and false membership functions related to those of A and B by

tC=max(tA,t(B),1-fC=max(1-fA,1-fB)=1-min(fA,fB).

### 2.2 Soft Sets

The soft set theory was proposed by Molodtsov  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 U be the universe of discourse, and let E be the universe of all possible parameters related to the objects in U.

Definition 4 

Let U be a universal set and let E be a set of parameters. Let P(U) denote the power set of U and AE. A pair (F,A) is called a soft set over U, where F is the mapping

F:AP(U).

Thus, a soft set over U is a parameterized family of subsets of universe U. For ɛA, F(ɛ) may be considered a set of ɛ-approximate elements of the soft set (F,A).

Definition 5 

Two soft sets (F,A) and (G,B) over a common universe U are said to be soft equal if (F,A) is a soft subset of (G,B) and (G,B) is a soft subset of (F,A).

Definition 6 

For two soft sets (F,A) and (G,B) over U, (F,A) is called a soft subset of (G,B) if

• (i) AB,

• (ii) ∀ɛA, F (ɛ) ⊆ G(ɛ).

This relationship is denoted as (F, A) ⊆̃ (G,B). In this case, (G,B) is called the soft superset of (F,A).

Definition 7

The complement of a soft set (F,A) is denoted by (F,A)c and defined by (F,A)c = (Fc, ⌉A), where Fc : ⌉AP (U) is a mapping given by

Fc(α)=U-F(α),αA.
Definition 8 

The union of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C) where C = AB, and ∀ɛC,

H(ɛ)={F(ɛ),if ɛA-B,G(ɛ),if ɛB-A,F(ɛ)G(ɛ),if ɛAB.

Ali et al.  proposed a definition of the extended intersection of soft sets as follows:

Definition 9 

The extended intersection of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C) where C = AB, and ∀ɛC,

H(ɛ)={F(ɛ),if ɛA-B,G(ɛ),if ɛB-A,F(ɛ)G(ɛ),if ɛAB.

### 2.3 Vague 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 U be a universe, E be a set of parameters, V (U) be the power set of vague sets on U, and AeqE.

Definition 10 

A pair ( F̃,A) is called a vague soft set over U, where is a mapping given by

F˜:AV(U).

In other words, a vague soft set over U is a parameterized family of vague sets of universe U. For ɛA, μ(ɛ) : U → [0, 1]2 is regarded as the set of ɛ approximate elements of the vague soft set (F̃,A).

Definition 11 

The union of two vague soft sets (F̃,A) and (G̃,B) over a universe U is a vague soft set denoted by (H̃,C), where C = AB, and

tH˜(e)(x)={tF˜(e)(x),if eA-B,tG˜(e)(x),if eB-A,max{(tF˜(e)(x),tG˜(e)(x)},if eAB,

and

1-fH˜(e)(x)={1-fF˜(e)(x),if eA-B,1-fG˜(e)(x),if eB-A,1-min {fF˜(e)(x),fG˜(e)(x)},if eAB,.

for all eC and xU. This is denoted as (F̃,A)∪̃( G̃,B) = (H̃,C).

Definition 12 

The intersection of two vague soft sets ( F̃,A) and (G̃,B) over a universe U is a vague soft set denoted by (H̃,C) where C = AB, and

tH˜(e)(x)={tF˜(e)(x),if eA-B,tG˜(e)(x),if eB-A,min{(tF˜(e)(x),tG˜(e)(x)},if eAB,

and

1-fH˜(e)(x)={1-fF˜(e)(x),if eA-B,1-fG˜(e)(x),if eB-A,1-max{fF˜(e)(x),fG˜(e)(x)},if eAB,.

for all eC and xU values. This is denoted by ( ,A)∩̃( , B) = (, C).

Definition 13 

Let ( F̃,A) and (G̃,B) be two vague soft sets over U. The restricted union of ( F̃,A) and (G̃,B) is defined as: the vague soft set 〈H̃,C〉, where C = AB and

tH˜(e)(x)=max {(tF˜(e)(x),tG˜(e)(x)},1-fH˜(e)(x)=1-min {fF˜(e)(x),fG˜(e)(x)},

for all eC, and xU if ∈ ABφ; otherwise, (H̃,C) ≠ φφ. This is denoted by (H̃,C) = (F̃,A) ⋓ ( G̃,B).

Definition 14 

Let ( F̃,A) and (G̃,B) be two vague soft sets over U. The restricted intersection of ( F̃,A) and (G̃,B) is defined as the vague soft set 〈H̃,C〉, where C = AB and

tH˜(e)(x)=min {(tF˜(e)(x),tG˜(e)(x)},1-fH˜(e)(x)=1-max {fF˜(e)(x),fG˜(e)(x)},

for all eC, and xU if ∈ ABφ; otherwise, (H̃,C) ≠ φφ. This is denoted by (H̃,C) = (F̃,A) ⋒ (G̃,B).

Definition 15 

If a pair ( F̃,A) and (G̃,B) are two vague soft sets over U, then the Cartesian product of ( F̃,A) and (G̃,B) is defined as, (F̃,A) × (G̃,B) = (H̃,A × B), where : A × BV (U × U) and (a, b) = F(a) × G(b), where (a, b) ∈ A × B i.e., H(a, b) = {(hi, hj) : hiF(a) and hjG(b)}.

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 ( 1,A) × ( 2,A) ×× ( n,A) of the non-empty vague soft sets ( 1,A), ( 2,A),…, ( n,A) is the vague soft set of all ordered n-tuples (h1, h2,…, hn), where hii(a).

Definition 16 

Let ( F̃,A) and (G̃,B) be two vague soft sets over U. Then, the relation between ( F̃,A) and (G̃,B) is a vague soft subset of ( F̃,A) × (G̃,B). The relation between ( F̃,A) and (G̃B) is of the form (1, S), where SA×B and 1(a, b)∀a, bS. Any subset of ( F̃,A) × ( F̃,A) is called a relation on ( F̃,A) in parameterized form as follows:

If (F˜,A)={F˜(a),F˜(b),},thenF˜(a)F˜(b)if and only if F˜(a)×F˜(b).

### 3. Transitive Closure of Vague Soft Set

Suppose that ℜ is a relation on a vague soft set (F,A), then ℜ may or may not have some property ρ, such as reflexivity, symmetry, or transitivity. If there is a relation S with property ρ containing ℜ, such that S is a sub-soft set of every relation with property ρ containing ℜ, then S is called the closure of ℜ with respect to ρ. The closure of a relation with respect to a property may or may not exist.

In this section, the concepts of reflexive closure, symmetry closure, and transitive closure of soft sets proposed by Ibrahim et al.  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.

### 3.1 Reflexive Closure

Definition 17 

The reflexive closure of R equals R ∪ ϒ, where ϒ = {(F(a), F(a)) : F(a) ∈ (F, a)} is the diagonal relation on (F,A).

Note that R is the relation on the soft set. The relation of the vague soft set ℜ with respect to its reflexive closure can be formed by adding to ℜ all pairs of the form (F(a), F(a)) with F(a) in(F,A), not already in ℜ. The addition of these pairs produces a new relation that is reflexive, contains ℜ, and is contained within any reflexive relation that contains ℜ. Using ℜ instead of R in Definition 13, the following new definition of the reflexive closure of a vague soft set is obtained.

Definition 18

Let ℜ be a vague soft set relation on (F,A). The minimal reflexive vague soft set relation containing ℜ is called the reflexive closure of ℜ, denoted by (ℜ).

Now, the notion of reflexive closure of a vague soft set is illustrated using the following example.

Example 1

Let U = {u1, u2, u3}, A = {e1, e2, e3}. The vague soft set (F,A) is given by

F(e1)={u10,1,0.2,u20.3,0.7,u30.1,0.1},F(e2)={u10.2,0.6,u20.1,0.3,u30.8,0.9},F(e3)={u10.4,0.6,u20.3,0.6,u30.3,0.3}.

Consider the vague soft set relation ℜ defined on (F,A) as

={F(e1)×F(e2)u10.1,0.6,u20.1,0.7,u30.3,0.9,F(e2)×F(e3)u10.2,0.6,u20.1,0.6,u30.3,0.9,and F(e3)×F(e3)u10.4,0.6,u20.3,0.6,u30.3,0.3}.

Then

r¯()={F(e1)×F(e1),F(e2)×F(e2),F(e3)×F(e3)}={F(e1)×F(e2),F(e2)×F(e3),F(e3)×F(e3)}{F(e1)×F(e1),F(e2)×F(e2),F(e3)×F(e3)}={F(e1)×F(e1)u10.1,0.2,u20.3,0.7,u30.1,0.1,F(e1)×F(e2)u10.1,0.6,u20.1,0.7,u30.3,0.9,F(e2)×F(e2)u10.2,0.6,u20.1,0.3,u30.8,0.9,F(e2)×F(e3)u10.2,0.6,u20.1,0.6,u30.3,0.9,and F(e3)×F(e3)u10.4,0.6,u20.3,0.6,u30.3,0.3},

### 3.2 Symmetry Closure

The symmetry closure of a relation ℜ is constructed by adding all ordered pairs of the form (F(b), F(a)), where (F(a), F(b)) is a relation that is not already present in ℜ. Adding these pairs produces a symmetric relation that contains ℜ.

Definition 19 

The symmetric closure of a relation is obtained by taking the union of a relation with its inverse, i.e., RR−1 where R−1 = {(F(b), F(a)) : (F(a), F(b)) ∈ R}.

The definition of symmetric closure on soft sets by Ibrahim et al.  is extended to the symmetric closure of a vague soft set below, followed by an example to illustrate its operation.

Definition 20

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 = {(F(b), F(a)): (F(F(a), F(b)) ∈ ℜ}.

In other words, let ℜ be a vague soft set relation on (F,A). The minimal symmetric vague soft set relation containing ℜ is called the symmetric closure of ℜ, denoted by (ℜ).

Example 2

Consider Example 1

s¯()=-1={F(e1)×F(e2),F(e2)×F(e3),F(e3)×F(e3)}{F(e2)×F(e1),F(e3)×F(e2),F(e3)×F(e3)}={F(e1)×F(e2),F(e2)×F(e3),F(e2)×F(e1),F(e3)×F(e2),F(e3)×F(e3)}.

### 3.3 Transitive Closure

Now, the definition of transitive closure of a soft set by Ibrahim et al.  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. R*is said to be the transitive closure of R, if it satisfies the following conditions:

• (i) R* is transitive.

• (ii) RR*.

• (iii) R* is the smallest transitive relation containing R.

Definition 21 

Let R be a relation on the soft set (F,A). Then, we define R*=i=1R.

Definition 22

Let ℜ be a relation on a vague soft set (F,A). Then, we define *=i=1.

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 (F,A) with m elements. Then

• (i) transitive (ℜ) = ℜ ∪ ℜ2 ∪ … ∪ ℜm,

• (ii) Mℜ* = MM2 ∪ … ∪ Mm,, where M is the matrix of the relation ℜ.

• (iii) M12 = M1M2, where ℜ1 and ℜ2 are relations on (F,A) with matrices M1 and M2.

The properties of the reflexive closure of a vague soft set can be illustrated by the following example.

Example 3

Suppose that ℜ is a relation on (F,A) with A = {a1, a2, a3}, where ℜ = {F(a1) × F(a2), F(a2) × F(a3), F(a3)×F(a3)}. The zero-one matrix for ℜ is given by

M=[0,11,10,10,10,11,10,10,11,1].

Thus M*=MM2M3 since n = 3.

Now

2=M2=M·M=[0,11,10,10,10,11,10,10,11,1]   [0,11,10,10,10,11,10,10,11,1]=[0,11,10,10,10,11,10,10,11,1],3=2·=M2·M=[0,10,11,10,10,11,10,10,11,1]   [0,11,10,10,10,11,10,10,11,1]=[0,10,11,10,10,11,10,10,11,1],M*=[0,11,10,10,10,11,10,10,11,1][0,10,11,10,10,11,10,10,11,1][0,10,11,10,10,11,10,10,11,1]=[0,11,11,10,10,11,10,10,11,1].

Reading from the zero-one matrix, we see that ℜ* = {F(a1)× F(a2), F(a1) × F(a3), F(a2) × F(a3), F(a3) × F(a3)} is the transitive (ℜ).

Theorem 1

The relation ℜ on a vague soft set (F,A) is transitive if and only if ℜn ⊆ ℜ for every nN.

Proof

Suppose that ℜn ⊆ ℜ for every nN. Specifically, ℜ2 ⊆ ℜ. To prove that ℜ is transitive, suppose that F(a) × F(b) ∈ ℜ and F(b) × F(c) ∈ ℜ; then, by the definition of composition, F(a) × F(c) ∈ ℜ2. Since ℜ2 ⊆ ℜ, it is inferred that F(a) × F(c) is in ℜ. Therefore, ℜ is transitive.

Conversely, suppose that ℜ is transitive. It can be proved that ℜn ⊆ ℜ by induction. This is true for n = 1. Assume that ℜn ⊆ ℜ for n. Then, we need to show that ℜn+1 ⊆ ℜ. To demonstrate this, assume that F(a) × F(b) ∈ ℜn+1. Since ℜn+1 = ℜn oℜ, there exists an element F(x) such that F(a) × F(x) ∈ ℜ and F(x) × F(b) ∈ ℜn. Now, ℜn ⊆ ℜ yields F(x) × F(b) ∈ ℜ. Furthermore, as ℜ is transitive and F(a) × F(x) ∈ ℜ, it follows that F(a) × F(b) is in ℜ. This shows that ℜn+1 ⊆ ℜ, thereby completing the proof.

### Theorem 2

If T and U are two vague soft set functions from (F,A) to (G,B), and ℜ and S are two vague soft set functions from (G,B) to (H,C), then ℜ ⊂ S and TU ⇒ ℜoTS oU.

Proof

Suppose that F(a)×H(c) ∈ RoT. This implies that there exists G(b) ∈ (G,B) such that F(a) × G(b) isinT and G(b) × H(c) is in ℜ.

Now ℜ ⊂ S ⇒; G(b) × H(c) ∈ S and TUF(a) × G(b) ∈ U. Then F(a) × H(c) ∈ S oU, showing that ℜoTS oU.

### Definition 23

Let ℜ be a binary relation on (F,A). The transitive closure of ℜ denoted by ℜ̃ is the smallest vague soft set relation containing ℜ that is transitive.

For the following definitions, lemmas, and proofs, dom is used to denote “domain of.”

### Definition 24

Let f and g be two vague soft set functions on (F,A) and (G,B), respectively. Then,

• (i) f and g are compatible if f(F(a)) = g(G(a)) for all F(a) ∈ domfdomg.

• (ii) A set of vague soft set functions Γ is a compatible system of functions if any two functions f and g from Γ are compatible.

### Lemma 1

• (a) vague soft set functions f and g are compatible if and only if fg is a function.

• (b) vague soft set functions f and g are compatible if and only if f/((domfdomg)) = g/((domfdomg)).

Proof

The result follows from Definition 20.

### Theorem 3

If Γ is a compatible system of functions, then ∪Γ is a function with dom ∪Γ = ∪{domf/f ∈ Γ}. Then, the function ∪Γ extends to all f ∈ Γ.

Proof

Clearly ∪Γ is a relation. We also prove that this is a function. If F(a)×F(b1) ∈ ∪ Γ and F(a)×F(b2) ∈ ∪ Γ, then there are functions f1, f2 ∈ Γ such that F(a)×F(b1) ∈ f1 and F(a) × F(b2) ∈ f2. However, f1 and f2 are compatible and f1domf1domf2. So F(b1) = f1(F(a)) = f2(F(a)) = F(b2).

It is trivial to show that F(x) ∈ dom ∪ Γ if and only if F(x) ∈ dom for some f ∈ Γ.

### 4. Properties of Closure

In this section, the properties of reflexive closure and symmetric closure of a vague soft set are introduced.

### Theorem 4

Let ℜ be a vague soft set relation on (F,A). Then

• (1) (ℜ) = ℜ ∪ I. Therefore, a mapping (called a reflexive closure operator) : RCℜ(F,A) → RCℜ(F,A) is obtained.

• (2) (ℜ) = ℜ ∪ ℜ−1. Therefore, a mapping (called the symmetric closure operator) : SCℜ(F,A) → SCℜ(F,A) is obtained.

Proof
• (1) ℜ ⊂ ℜ ∪ I. ∀aA, F(a) × F(a) ∈ I ⊂ ℜ ∪ I; therefore, ℜ ∪ I is reflexive. On the other hand, T has a reflexive vague soft set relation on (F,A) and ℜ ⊂ T. By the reflexivity of T, IT, if IT and ℜ ⊂ T, then ℜ ∪ IT. Therefore, (ℜ) = ℜ ∪ I.

• (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 (F,A), and ℜ ⊂ ℜ ∪ ℜ−1.

If T has a symmetric vague soft set relation on (F,A) and ℜ ⊂ T, then ℜ−1T−1. Since ℜ is symmetric iff ℜ = ℜ −1 and ℜ ∪ ℜ −1T then T = T−1. Therefore, (ℜ) = ℜ ∪ ℜ −1.

Next, some basic properties of the reflexive and symmetric closure operators are proposed.

### Theorem 5

The reflexive closure operator has the following properties:

• (1) (M) = M, r̄ (I) = I.

• (2) ∀ℜ ∈ RCℜ (F,A), ℜ ⊂ (ℜ).

• (3) ∀ℜ,QRCℜ (F,A), (ℜ ∪ Q) = ℜ ∪ r̄Q, r̄ (ℜ∩Q) = ℜ∩ r̄Q.

• (4) ∀ℜ,QRCℜ (F,A), if ℜ ⊂ Q, then (ℜ) ⊂ (Q).

• (5) ∀ℜ ∈ RCℜ (F,A), ( (ℜ)) = (ℜ).

Proof
• (1) By the reflexivity of M and I, (M) = M, r̄ (I) = I.

• (2) ∀ℜ ∈ RCℜ (F,A), by Theorem 4 (1) and (ℜ) = ℜ ∪ I, (ℜ) = ℜ ∪ I ⊃ ℜ.

• (3) ∀ℜ,QRCℜ (F,A), by Theorem 4, (ℜ ∪ Q) = (ℜ ∪ Q) ∪ I = (ℜ ∪ I) ∪ (QI) = (ℜ) ∪ (Q). (ℜ∩Q) = (ℜ ℜ∩Q) ∪ I = (ℜ∩I) ∩ (QI) = (ℜ) ∩ (Q).

• (4) ∀ℜ,QRCℜ (F,A), ℜ ⊂ Q by (3) and if ℜ ⊂ Q, then ℜ ∪ Q = Q and ℜ ∩ Q = ℜ, (Q) = (ℜ ∪ Q) = (ℜ) ∪ (Q) ⊃ (ℜ).

• (5) ∀ℜ ∈ RCℜ (F,A), by Theorem 4 (1), (ℜ) = ℜ ∪ I. Hence ( (ℜ)) = (ℜ ∪ I) = (ℜ ∪ I) ∪ I = ℜ ∪ I = (ℜ).

### Theorem 6

The symmetric closure operator has the following properties:

• (1) (M) = M, s̄ (I) = I.

• (2) ∀ℜ ∈ SCℜ (F,A), ℜ ⊂ (ℜ).

• (3) ∀ℜ,QSCℜ (F,A), (ℜ ∪ Q) = ℜ ∪ s̄Q, s̄ (ℜ∩Q) = ℜ∩ s̄Q.

• (4) ∀ℜ,QSCℜ (F,A), if ℜ ⊂ Q, then (ℜ) ⊂ (Q).

• (5) ∀ℜ ∈ SCℜ (F,A), ( (ℜ)) = (ℜ).

Proof
• (1) By symmetry of m,M, and I, (m) = m, s̄ (M) = M, s̄ (I) = I.

• (2) ∀ℜ ∈ SCℜ (F,A), by Theorem 5(2), ℜ ⊂ (ℜ).

• (3) ∀ℜ,QSCℜ (F,A), by Theorem 5 and (ℜ ∪ Q)−1 = ℜ−1Q−1 and (ℜ∩Q) −1 = ℜ−1Q−1, we have

s¯(Q)=(Q)(Q)-1=(Q)(-1Q-1)=(-1)(QQ-1)-1=s¯()s¯(Q),

and for (ℜ∩Q) = ℜ∩ s̄Q, we have

s¯(Q)=(Q)(Q)-1=(Q)(-1Q-1)=(-1)(QQ-1)-1=s¯()s¯(Q).

• (4) ∀ℜ, QSCℜ (F,A), ℜ ⊂ Q, by (3) and Theorem 5(4), (Q) = (ℜ ℜ ∪ Q) = (ℜ) ∪ (Q) ⊃ (ℜ).

• (5) ∀ℜ ∈ SCℜ (F,A), by Theorem 5(2), (ℜ) = ℜ ∪ ℜ−1. Hence

s¯(s¯())=s¯(-1)=(-1)(-1)-1=(-1)(-1(-1)-1)=(-1)(-1)=(-1)=s¯().

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.

### Conflict of Interest

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43. Ali, MI, Feng, F, Liu, X, Min, WK, and Shabir, M (2009). On some new operations in soft set theory. Computers & Mathematics with Applications. 57, 1547-1553. https://doi.org/10.1016/j.camwa.2008.11.009z Yousef Al-Qudah received the M.Sc. and Ph.D. degrees in mathematics from Universiti Kebangsaan Malaysia, Malaysia. He is currently an Assistant Professor in the Department of Mathematics, Amman Arab University, Jordan. His research interests include decision-making, fuzzy sets, fuzzy topology, fuzzy algebra, and complex fuzzy sets.

E-mail: alquyousef82@gmail.com Khaleed Alhazaymeh is an Associate Professor in the Department of Mathematics at Philadelphia University in Jordan. He received his M.Sc. and Ph.D. from the National University of Malaysia (UKM). He specializes in fuzzy sets, vague soft sets, and topics related to uncertainty, and has conducted extensive research in this field. Nasruddin Hassan received the B.Sc. degree in mathematics from Western Illinois University, USA, the M.Sc. degree in applied mathematics from Western Michigan University, USA, and the Ph.D. degree in applied mathematics from Universiti Putra Malaysia, Malaysia. His research interests include decision making, operations research, fuzzy sets, and numerical convergence. Hamza Qoqazeh received his M.Sc. from Al Al-Bayt University and Ph.D. from the University of Jordan. Since 2019, he has been at Amman Arab University. His research interests include topology and fuzzy topologies. Mohammad Almousa received his M.Sc. from Al Al-bayt University, and a Ph.D. degree in applied mathematics from Universiti Sains Malaysia, Malaysia. His research interests include numerical optimization and partial differential equations. Mohammad Alaroud received the M.Sc. and Ph.D. degrees in mathematics from Universiti Kebangsaan Malaysia, Malaysia. Since 2020, he is working as an Assistant Professor in the Faculty of Science and Arts, Amman Arab University. His research interests include numerical topology and partial differential equations.

### Article

#### Original Article 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

## Transitive Closure of Vague Soft Set Relations and its Operators

Yousef Al-Qudah1 , Khaleed Alhazaymeh2, Nasruddin Hassan3, Hamza Qoqazeh1, Mohammad Almousa1, and Mohammad Alaroud1

1Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
2Department of Basic Sciences and Mathematics, Faculty of Science, Philadelphia University, Amman, Jordan
3School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Correspondence to:Yousef Al-Qudah (alquyousef82@gmail.com)

Received: July 17, 2021; Revised: September 28, 2021; Accepted: October 12, 2021

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.

### Abstract

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

### 1. Introduction

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 , intuitionistic fuzzy set theory , 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  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 . Soft sets have been studied by many researchers, such as fuzzy soft sets , intuitionistic fuzzy soft sets , vague soft sets , multi-fuzzy soft sets , and vague soft set relations and functions . Babitha and Sunil  introduced the concept of soft set relations and functions. Agarwal et al.  discussed the concept of relations in generalized intuitionistic fuzzy soft sets. Ibrahim et al.  introduced the concept of soft set composition relations and the construction of transitive closure. Park et al.  studied some properties of the equivalence of soft set relations, while Su et al.  introduced the concept of intuitionistic fuzzy decision-making with similarity measures and the ordered weighted averaging (OWA) operator. In addition, Saxena and Tayal  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.

### 2. Preliminaries

In this section, some basic concepts of vague, soft, and vague soft sets are briefly reviewed.

### 2.1 Vague Sets

A vague set  over U is characterized by a truth-membership function tν and false membership function fν,

$tν:U→[0,1] and fν:U→[0,1],$

where for any uiU, tν(ui) is a lower bound on the membership grade of ui derived from the evidence for ui; fν(ui) is a lower bound on the negation of ui derived from the evidence against ui; and tν(ui) + fν(ui) ≤ 1. The membership grade of ui in the vague set is bounded to a subinterval [tν(ui), 1 − fν(ui)] of [0, 1]. The vague values [tν(ui), 1−fν(ui)] indicate that the exact membership grade μν(ui) of ui may be unknown, but it is bounded by tν(ui) ≤ μν(ui) ≤ 1 − fν(ui), where tν(ui) + fν(ui) ≤ 1.

A review of the basic operations of the complement, intersection, and union of a vague set, as defined by Gau and Buehrer , are presented next.

Definition 1 

The complement of a vague set A is denoted by Ac and is defined by

$tAc=fA, 1-fAc=1-tA.$
Definition 2 

The intersection of two vague sets A and B is a vague set C, denoted as C = AB, with truth-membership and false membership functions related to those of A and B by

$tC=min(tA,tB),1-fC=min(1-fA,1-fB)=1-max(fA,fB).$
Definition 3 

The union of two vague sets A and B is a vague set C, denoted as C = AB, with truth-membership and false membership functions related to those of A and B by

$tC=max(tA,t(B),1-fC=max(1-fA,1-fB)=1-min(fA,fB).$

### 2.2 Soft Sets

The soft set theory was proposed by Molodtsov  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 U be the universe of discourse, and let E be the universe of all possible parameters related to the objects in U.

Definition 4 

Let U be a universal set and let E be a set of parameters. Let P(U) denote the power set of U and AE. A pair (F,A) is called a soft set over U, where F is the mapping

$F:A→P(U).$

Thus, a soft set over U is a parameterized family of subsets of universe U. For ɛA, F(ɛ) may be considered a set of ɛ-approximate elements of the soft set (F,A).

Definition 5 

Two soft sets (F,A) and (G,B) over a common universe U are said to be soft equal if (F,A) is a soft subset of (G,B) and (G,B) is a soft subset of (F,A).

Definition 6 

For two soft sets (F,A) and (G,B) over U, (F,A) is called a soft subset of (G,B) if

• (i) AB,

• (ii) ∀ɛA, F (ɛ) ⊆ G(ɛ).

This relationship is denoted as (F, A) ⊆̃ (G,B). In this case, (G,B) is called the soft superset of (F,A).

Definition 7

The complement of a soft set (F,A) is denoted by (F,A)c and defined by (F,A)c = (Fc, ⌉A), where Fc : ⌉AP (U) is a mapping given by

$Fc(α)=U-F(⌉α),∀α∈⌉A.$
Definition 8 

The union of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C) where C = AB, and ∀ɛC,

$H (ɛ)={F(ɛ),if ɛ∈A-B,G(ɛ),if ɛ∈B-A,F(ɛ)∪G(ɛ),if ɛ∈A∩B.$

Ali et al.  proposed a definition of the extended intersection of soft sets as follows:

Definition 9 

The extended intersection of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C) where C = AB, and ∀ɛC,

$H (ɛ)={F(ɛ),if ɛ∈A-B,G(ɛ),if ɛ∈B-A,F(ɛ)∩G(ɛ),if ɛ∈A∩B.$

### 2.3 Vague 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 U be a universe, E be a set of parameters, V (U) be the power set of vague sets on U, and AeqE.

Definition 10 

A pair ( F̃,A) is called a vague soft set over U, where is a mapping given by

$F˜:A→V(U).$

In other words, a vague soft set over U is a parameterized family of vague sets of universe U. For ɛA, μ(ɛ) : U → [0, 1]2 is regarded as the set of ɛ approximate elements of the vague soft set (F̃,A).

Definition 11 

The union of two vague soft sets (F̃,A) and (G̃,B) over a universe U is a vague soft set denoted by (H̃,C), where C = AB, and

$tH˜ (e)(x)={tF˜(e)(x),if e∈A-B,tG˜(e)(x),if e∈B-A,max{(tF˜(e)(x),tG˜(e)(x)},if e∈A∩B,$

and

$1-fH˜(e)(x)={1-fF˜(e)(x),if e∈A-B,1-fG˜(e)(x),if e∈B-A,1-min {fF˜(e)(x),fG˜(e)(x)},if e∈A∩B,.$

for all eC and xU. This is denoted as (F̃,A)∪̃( G̃,B) = (H̃,C).

Definition 12 

The intersection of two vague soft sets ( F̃,A) and (G̃,B) over a universe U is a vague soft set denoted by (H̃,C) where C = AB, and

$tH˜(e)(x)={tF˜(e)(x),if e∈A-B,tG˜(e)(x),if e∈B-A,min{(tF˜(e)(x),tG˜(e)(x)},if e∈A∩B,$

and

$1-fH˜(e)(x)={1-fF˜(e)(x),if e∈A-B,1-fG˜(e)(x),if e∈B-A,1-max{fF˜(e)(x),fG˜(e)(x)},if e∈A∩B,.$

for all eC and xU values. This is denoted by ( ,A)∩̃( , B) = (, C).

Definition 13 

Let ( F̃,A) and (G̃,B) be two vague soft sets over U. The restricted union of ( F̃,A) and (G̃,B) is defined as: the vague soft set 〈H̃,C〉, where C = AB and

$tH˜(e)(x)=max {(tF˜(e)(x),tG˜(e)(x)},1-fH˜(e)(x)=1-min {fF˜(e)(x),fG˜(e)(x)},$

for all eC, and xU if ∈ ABφ; otherwise, (H̃,C) ≠ φφ. This is denoted by (H̃,C) = (F̃,A) ⋓ ( G̃,B).

Definition 14 

Let ( F̃,A) and (G̃,B) be two vague soft sets over U. The restricted intersection of ( F̃,A) and (G̃,B) is defined as the vague soft set 〈H̃,C〉, where C = AB and

$tH˜(e)(x)=min {(tF˜(e)(x),tG˜(e)(x)},1-fH˜(e)(x)=1-max {fF˜(e)(x),fG˜(e)(x)},$

for all eC, and xU if ∈ ABφ; otherwise, (H̃,C) ≠ φφ. This is denoted by (H̃,C) = (F̃,A) ⋒ (G̃,B).

Definition 15 

If a pair ( F̃,A) and (G̃,B) are two vague soft sets over U, then the Cartesian product of ( F̃,A) and (G̃,B) is defined as, (F̃,A) × (G̃,B) = (H̃,A × B), where : A × BV (U × U) and (a, b) = F(a) × G(b), where (a, b) ∈ A × B i.e., H(a, b) = {(hi, hj) : hiF(a) and hjG(b)}.

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 ( 1,A) × ( 2,A) ×× ( n,A) of the non-empty vague soft sets ( 1,A), ( 2,A),…, ( n,A) is the vague soft set of all ordered n-tuples (h1, h2,…, hn), where hii(a).

Definition 16 

Let ( F̃,A) and (G̃,B) be two vague soft sets over U. Then, the relation between ( F̃,A) and (G̃,B) is a vague soft subset of ( F̃,A) × (G̃,B). The relation between ( F̃,A) and (G̃B) is of the form (1, S), where SA×B and 1(a, b)∀a, bS. Any subset of ( F̃,A) × ( F̃,A) is called a relation on ( F̃,A) in parameterized form as follows:

$If (F˜,A)={F˜(a),F˜(b),…},thenF˜(a)ℜF˜(b) if and only if F˜(a)×F˜(b)∈ℜ.$

### 3. Transitive Closure of Vague Soft Set

Suppose that ℜ is a relation on a vague soft set (F,A), then ℜ may or may not have some property ρ, such as reflexivity, symmetry, or transitivity. If there is a relation S with property ρ containing ℜ, such that S is a sub-soft set of every relation with property ρ containing ℜ, then S is called the closure of ℜ with respect to ρ. The closure of a relation with respect to a property may or may not exist.

In this section, the concepts of reflexive closure, symmetry closure, and transitive closure of soft sets proposed by Ibrahim et al.  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.

### 3.1 Reflexive Closure

Definition 17 

The reflexive closure of R equals R ∪ ϒ, where ϒ = {(F(a), F(a)) : F(a) ∈ (F, a)} is the diagonal relation on (F,A).

Note that R is the relation on the soft set. The relation of the vague soft set ℜ with respect to its reflexive closure can be formed by adding to ℜ all pairs of the form (F(a), F(a)) with F(a) in(F,A), not already in ℜ. The addition of these pairs produces a new relation that is reflexive, contains ℜ, and is contained within any reflexive relation that contains ℜ. Using ℜ instead of R in Definition 13, the following new definition of the reflexive closure of a vague soft set is obtained.

Definition 18

Let ℜ be a vague soft set relation on (F,A). The minimal reflexive vague soft set relation containing ℜ is called the reflexive closure of ℜ, denoted by (ℜ).

Now, the notion of reflexive closure of a vague soft set is illustrated using the following example.

Example 1

Let U = {u1, u2, u3}, A = {e1, e2, e3}. The vague soft set (F,A) is given by

$F (e1)={u1〈0,1,0.2〉,u2〈0.3,0.7〉,u3〈0.1,0.1〉},F (e2)={u1〈0.2,0.6〉,u2〈0.1,0.3〉,u3〈0.8,0.9〉},F (e3)={u1〈0.4,0.6〉,u2〈0.3,0.6〉,u3〈0.3,0.3〉}.$

Consider the vague soft set relation ℜ defined on (F,A) as

$ℜ={F(e1)×F(e2)u1〈0.1,0.6〉,u2〈0.1,0.7〉,u3〈0.3,0.9〉,F(e2)×F(e3)u1〈0.2,0.6〉,u2〈0.1,0.6〉,u3〈0.3,0.9〉,and F(e3)×F(e3)u1〈0.4,0.6〉,u2〈0.3,0.6〉,u3〈0.3,0.3〉}.$

Then

$r¯(ℜ)=ℜ∪{F(e1)×F(e1),F(e2)×F(e2),F(e3)×F(e3)}={F(e1)×F(e2),F(e2)×F(e3),F(e3)×F(e3)}∪ {F(e1)×F(e1),F(e2)×F(e2),F(e3)×F(e3)}={F(e1)×F(e1)u1〈0.1,0.2〉,u2〈0.3,0.7〉,u3〈0.1,0.1〉,F(e1)×F(e2)u1〈0.1,0.6〉,u2〈0.1,0.7〉,u3〈0.3,0.9〉,F(e2)×F(e2)u1〈0.2,0.6〉,u2〈0.1,0.3〉,u3〈0.8,0.9〉,F(e2)×F(e3)u1〈0.2,0.6〉,u2〈0.1,0.6〉,u3〈0.3,0.9〉,and F(e3)×F(e3)u1〈0.4,0.6〉,u2〈0.3,0.6〉,u3〈0.3,0.3〉},$

### 3.2 Symmetry Closure

The symmetry closure of a relation ℜ is constructed by adding all ordered pairs of the form (F(b), F(a)), where (F(a), F(b)) is a relation that is not already present in ℜ. Adding these pairs produces a symmetric relation that contains ℜ.

Definition 19 

The symmetric closure of a relation is obtained by taking the union of a relation with its inverse, i.e., RR−1 where R−1 = {(F(b), F(a)) : (F(a), F(b)) ∈ R}.

The definition of symmetric closure on soft sets by Ibrahim et al.  is extended to the symmetric closure of a vague soft set below, followed by an example to illustrate its operation.

Definition 20

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 = {(F(b), F(a)): (F(F(a), F(b)) ∈ ℜ}.

In other words, let ℜ be a vague soft set relation on (F,A). The minimal symmetric vague soft set relation containing ℜ is called the symmetric closure of ℜ, denoted by (ℜ).

Example 2

Consider Example 1

$s¯(ℜ)=ℜ∪ℜ-1={F(e1)×F(e2),F(e2)×F(e3),F(e3)×F(e3)}∪{F(e2)×F(e1),F(e3)×F(e2),F(e3)×F(e3)}={F(e1)×F(e2),F(e2)×F(e3),F(e2)×F(e1),F(e3)×F(e2),F(e3)×F(e3)}.$

### 3.3 Transitive Closure

Now, the definition of transitive closure of a soft set by Ibrahim et al.  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. R*is said to be the transitive closure of R, if it satisfies the following conditions:

• (i) R* is transitive.

• (ii) RR*.

• (iii) R* is the smallest transitive relation containing R.

Definition 21 

Let R be a relation on the soft set (F,A). Then, we define $R*=∪i=1∞R$.

Definition 22

Let ℜ be a relation on a vague soft set (F,A). Then, we define $ℜ*=∪i=1∞ℜ$.

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 (F,A) with m elements. Then

• (i) transitive (ℜ) = ℜ ∪ ℜ2 ∪ … ∪ ℜm,

• (ii) Mℜ* = MM2 ∪ … ∪ Mm,, where M is the matrix of the relation ℜ.

• (iii) M12 = M1M2, where ℜ1 and ℜ2 are relations on (F,A) with matrices M1 and M2.

The properties of the reflexive closure of a vague soft set can be illustrated by the following example.

Example 3

Suppose that ℜ is a relation on (F,A) with A = {a1, a2, a3}, where ℜ = {F(a1) × F(a2), F(a2) × F(a3), F(a3)×F(a3)}. The zero-one matrix for ℜ is given by

$Mℜ=[〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉].$

Thus $Mℜ*=Mℜ∨Mℜ2∨Mℜ3$ since n = 3.

Now

$ℜ2=Mℜ2=Mℜ·Mℜ=[〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉] [〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉]=[〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉],ℜ3=ℜ2·ℜ=Mℜ2·Mℜ=[〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉] [〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉]=[〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉],Mℜ*=[〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉]∨ [〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉]∨[〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉]=[〈0,1〉〈1,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉〈0,1〉〈0,1〉〈1,1〉].$

Reading from the zero-one matrix, we see that ℜ* = {F(a1)× F(a2), F(a1) × F(a3), F(a2) × F(a3), F(a3) × F(a3)} is the transitive (ℜ).

Theorem 1

The relation ℜ on a vague soft set (F,A) is transitive if and only if ℜn ⊆ ℜ for every nN.

Proof

Suppose that ℜn ⊆ ℜ for every nN. Specifically, ℜ2 ⊆ ℜ. To prove that ℜ is transitive, suppose that F(a) × F(b) ∈ ℜ and F(b) × F(c) ∈ ℜ; then, by the definition of composition, F(a) × F(c) ∈ ℜ2. Since ℜ2 ⊆ ℜ, it is inferred that F(a) × F(c) is in ℜ. Therefore, ℜ is transitive.

Conversely, suppose that ℜ is transitive. It can be proved that ℜn ⊆ ℜ by induction. This is true for n = 1. Assume that ℜn ⊆ ℜ for n. Then, we need to show that ℜn+1 ⊆ ℜ. To demonstrate this, assume that F(a) × F(b) ∈ ℜn+1. Since ℜn+1 = ℜn oℜ, there exists an element F(x) such that F(a) × F(x) ∈ ℜ and F(x) × F(b) ∈ ℜn. Now, ℜn ⊆ ℜ yields F(x) × F(b) ∈ ℜ. Furthermore, as ℜ is transitive and F(a) × F(x) ∈ ℜ, it follows that F(a) × F(b) is in ℜ. This shows that ℜn+1 ⊆ ℜ, thereby completing the proof.

### Theorem 2

If T and U are two vague soft set functions from (F,A) to (G,B), and ℜ and S are two vague soft set functions from (G,B) to (H,C), then ℜ ⊂ S and TU ⇒ ℜoTS oU.

Proof

Suppose that F(a)×H(c) ∈ RoT. This implies that there exists G(b) ∈ (G,B) such that F(a) × G(b) isinT and G(b) × H(c) is in ℜ.

Now ℜ ⊂ S ⇒; G(b) × H(c) ∈ S and TUF(a) × G(b) ∈ U. Then F(a) × H(c) ∈ S oU, showing that ℜoTS oU.

### Definition 23

Let ℜ be a binary relation on (F,A). The transitive closure of ℜ denoted by ℜ̃ is the smallest vague soft set relation containing ℜ that is transitive.

For the following definitions, lemmas, and proofs, dom is used to denote “domain of.”

### Definition 24

Let f and g be two vague soft set functions on (F,A) and (G,B), respectively. Then,

• (i) f and g are compatible if f(F(a)) = g(G(a)) for all F(a) ∈ domfdomg.

• (ii) A set of vague soft set functions Γ is a compatible system of functions if any two functions f and g from Γ are compatible.

### Lemma 1

• (a) vague soft set functions f and g are compatible if and only if fg is a function.

• (b) vague soft set functions f and g are compatible if and only if f/((domfdomg)) = g/((domfdomg)).

Proof

The result follows from Definition 20.

### Theorem 3

If Γ is a compatible system of functions, then ∪Γ is a function with dom ∪Γ = ∪{domf/f ∈ Γ}. Then, the function ∪Γ extends to all f ∈ Γ.

Proof

Clearly ∪Γ is a relation. We also prove that this is a function. If F(a)×F(b1) ∈ ∪ Γ and F(a)×F(b2) ∈ ∪ Γ, then there are functions f1, f2 ∈ Γ such that F(a)×F(b1) ∈ f1 and F(a) × F(b2) ∈ f2. However, f1 and f2 are compatible and f1domf1domf2. So F(b1) = f1(F(a)) = f2(F(a)) = F(b2).

It is trivial to show that F(x) ∈ dom ∪ Γ if and only if F(x) ∈ dom for some f ∈ Γ.

### 4. Properties of Closure

In this section, the properties of reflexive closure and symmetric closure of a vague soft set are introduced.

### Theorem 4

Let ℜ be a vague soft set relation on (F,A). Then

• (1) (ℜ) = ℜ ∪ I. Therefore, a mapping (called a reflexive closure operator) : RCℜ(F,A) → RCℜ(F,A) is obtained.

• (2) (ℜ) = ℜ ∪ ℜ−1. Therefore, a mapping (called the symmetric closure operator) : SCℜ(F,A) → SCℜ(F,A) is obtained.

Proof
• (1) ℜ ⊂ ℜ ∪ I. ∀aA, F(a) × F(a) ∈ I ⊂ ℜ ∪ I; therefore, ℜ ∪ I is reflexive. On the other hand, T has a reflexive vague soft set relation on (F,A) and ℜ ⊂ T. By the reflexivity of T, IT, if IT and ℜ ⊂ T, then ℜ ∪ IT. Therefore, (ℜ) = ℜ ∪ I.

• (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 (F,A), and ℜ ⊂ ℜ ∪ ℜ−1.

If T has a symmetric vague soft set relation on (F,A) and ℜ ⊂ T, then ℜ−1T−1. Since ℜ is symmetric iff ℜ = ℜ −1 and ℜ ∪ ℜ −1T then T = T−1. Therefore, (ℜ) = ℜ ∪ ℜ −1.

Next, some basic properties of the reflexive and symmetric closure operators are proposed.

### Theorem 5

The reflexive closure operator has the following properties:

• (1) (M) = M, r̄ (I) = I.

• (2) ∀ℜ ∈ RCℜ (F,A), ℜ ⊂ (ℜ).

• (3) ∀ℜ,QRCℜ (F,A), (ℜ ∪ Q) = ℜ ∪ r̄Q, r̄ (ℜ∩Q) = ℜ∩ r̄Q.

• (4) ∀ℜ,QRCℜ (F,A), if ℜ ⊂ Q, then (ℜ) ⊂ (Q).

• (5) ∀ℜ ∈ RCℜ (F,A), ( (ℜ)) = (ℜ).

Proof
• (1) By the reflexivity of M and I, (M) = M, r̄ (I) = I.

• (2) ∀ℜ ∈ RCℜ (F,A), by Theorem 4 (1) and (ℜ) = ℜ ∪ I, (ℜ) = ℜ ∪ I ⊃ ℜ.

• (3) ∀ℜ,QRCℜ (F,A), by Theorem 4, (ℜ ∪ Q) = (ℜ ∪ Q) ∪ I = (ℜ ∪ I) ∪ (QI) = (ℜ) ∪ (Q). (ℜ∩Q) = (ℜ ℜ∩Q) ∪ I = (ℜ∩I) ∩ (QI) = (ℜ) ∩ (Q).

• (4) ∀ℜ,QRCℜ (F,A), ℜ ⊂ Q by (3) and if ℜ ⊂ Q, then ℜ ∪ Q = Q and ℜ ∩ Q = ℜ, (Q) = (ℜ ∪ Q) = (ℜ) ∪ (Q) ⊃ (ℜ).

• (5) ∀ℜ ∈ RCℜ (F,A), by Theorem 4 (1), (ℜ) = ℜ ∪ I. Hence ( (ℜ)) = (ℜ ∪ I) = (ℜ ∪ I) ∪ I = ℜ ∪ I = (ℜ).

### Theorem 6

The symmetric closure operator has the following properties:

• (1) (M) = M, s̄ (I) = I.

• (2) ∀ℜ ∈ SCℜ (F,A), ℜ ⊂ (ℜ).

• (3) ∀ℜ,QSCℜ (F,A), (ℜ ∪ Q) = ℜ ∪ s̄Q, s̄ (ℜ∩Q) = ℜ∩ s̄Q.

• (4) ∀ℜ,QSCℜ (F,A), if ℜ ⊂ Q, then (ℜ) ⊂ (Q).

• (5) ∀ℜ ∈ SCℜ (F,A), ( (ℜ)) = (ℜ).

Proof
• (1) By symmetry of m,M, and I, (m) = m, s̄ (M) = M, s̄ (I) = I.

• (2) ∀ℜ ∈ SCℜ (F,A), by Theorem 5(2), ℜ ⊂ (ℜ).

• (3) ∀ℜ,QSCℜ (F,A), by Theorem 5 and (ℜ ∪ Q)−1 = ℜ−1Q−1 and (ℜ∩Q) −1 = ℜ−1Q−1, we have

$s¯(ℜ∪Q)=(ℜ∪Q)∪(ℜ∪Q)-1=(ℜ∪Q)∪(ℜ-1∪Q-1)=(ℜ∪ℜ-1)∪(Q∪Q-1)-1=s¯(ℜ)∪s¯(Q),$

and for (ℜ∩Q) = ℜ∩ s̄Q, we have

$s¯(ℜ∩Q)=(ℜ∩Q)∩(ℜ∩Q)-1=(ℜ∩Q)∩(ℜ-1∩Q-1)=(ℜ∩ℜ-1)∩(Q∩Q-1)-1=s¯(ℜ)∩s¯(Q).$

• (4) ∀ℜ, QSCℜ (F,A), ℜ ⊂ Q, by (3) and Theorem 5(4), (Q) = (ℜ ℜ ∪ Q) = (ℜ) ∪ (Q) ⊃ (ℜ).

• (5) ∀ℜ ∈ SCℜ (F,A), by Theorem 5(2), (ℜ) = ℜ ∪ ℜ−1. Hence

$s¯(s¯(ℜ))=s¯(ℜ∪ℜ-1)=(ℜ∪ℜ-1)∪(ℜ∪ℜ-1)-1=(ℜ∪ℜ-1)∪(ℜ-1∪(ℜ-1)-1)=(ℜ∪ℜ-1)∪(ℜ-1∪ℜ)=(ℜ∪ℜ-1)=s¯(ℜ).$

### 5. Conclusion

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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