A Study Of The Fundamentals Of Soft Set Theory

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132

A Study Of The Fundamentals Of Soft Set Theory

Onyeozili, I. A., Gwary T. M.

Abstract: In this paper, a systematic and critical study of the fundamentals of soft set theory, which include operations on soft sets and their properties, soft set relation and function, matrix representation of soft set among others, is carried out.

Index Terms - Soft set, soft subset, soft set operations, soft set relation and function, soft matrix.

————————————————————

1 INTRODUCTION

The concept of soft sets was first formulated by Molodtsov [1999] as a completely new mathematical tool for solving problems dealing with uncertainties. Molodtsov [1999] defines a soft set as a parameterized family of subsets of universe set where each element is considered as a set of approximate elements of the soft set. In the past few years, the fundamentals of soft set theory have been studied by various researchers. Maji et al. [2003] presented a detailed theoretical study of soft sets which includes subset and super set of a soft set, equality of soft sets, operations on soft sets such as union, intersection, AND and OR-operations among others. They also studied and discussed the basic properties of these operations. Pei and Miao [2005] redefined subset and intersection of soft sets and discussed the relationship between soft sets and information systems. Ali et al. [2009] introduced some new operations such as the restricted union, the restricted intersection, the restricted difference and the extended intersection of two soft sets and discussed their basic properties. Cagman and Enginoglu [2010] developed soft matrix theory and successfully applied it to a decision making problem. Babitha and Sunil [2010] introduced the concept of soft set relation and function and discussed many related concepts such as equivalence soft set relation, partition of soft sets, ordering on soft sets. In continuation of their work, Babitha and Sunil [2011] further worked on soft set relation and ordering by introducing the concept of anti-symmetric relation and transitive closure of a soft set relation. Yang and Guo [2011] introduced the notions of anti-symmetric closure of a soft set relation and obtained with proofs some results involving them. Sezgin and Atagun [2011], Ge and Yang [2011], Fuli [2011] etc., gave some modifications in the work of Maji et al. [2003] and also established some new results. Sezgin and Atagun [2011], also introduced the restricted symmetric difference of soft sets and investigated its properties with examples. Singh and Onyeozili [2012] obtained some results on distributive and absorption properties with respect to various operations on soft sets. Singh and Onyeozili [2012] proved that the operations defined on soft sets are equivalent to the corresponding operations defined on their soft matrices.

The rest of this paper is organized as follows: Section 2 gives some basic definitions and results on soft sets. Section 3 discusses in detail, various operations of soft sets. Section 4 states without proofs many properties of soft set operations. Section 5 focuses on soft set relations and functions .Finally section 6 which comprises of two subsections, first discusses soft matrices and their basic operations while the second subsection concentrates on their properties.

2 PRELIMINARIES

In this section, we give some basic definitions and results on soft sets and suitably exemplify them.

Definition 2.1. [10 ] (Soft Set)

Let U be an initial universe set and E a set of parameters or attributes with respect to U. Let P(U) denote the power set of U and

A



E

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

F A

:



P U

( ).

In other words, a soft set (F, A) over U is a parameterized family of subsets of U. For

e



A F e

,

( )

may be considered as the set of e-elements or e-approximate elements of the soft sets (F, A). Thus (F, A) is defined as





( , )

F A



F e

( )



P U

( ) :

e



E F e

,

( )

 

if

e



A

.

Example 2.1

Assume that

U





h h h h h h

₁

,

₂

,

₃

,

₄

,

₅

,

₆



be a universal set consisting of a set of six houses under consideration,



1

,

2

, ,

3 4

,

5



E



e e e e e

be a set of parameters with respect

to U, where each parameter

e i

_i

,



1, 2,



,5

stands for ‗expensive‘, ‗beautiful‘, ‗cheap‘, ‗modern‘, ‗wooden‘, respectively and

A





e e e

₁

,

₂

,

₃





E

. Suppose a soft set (F,A) describes the attractions of the houses, such that

  

1 2

,

4

   

,

2 1

,

3

,

5



F e



h h

F e



h h h

and

  

3 3

,

4

,

5



F e



h h h

. Then the soft set (F, A) is a

parameterized family



F e

 

i

:

i



1, 2,3



of subset of U defined as

( , )

F A





F e

     

₁

,

F e

₂

,

F e

₃



, i.e.,



 





2 4 1 3 5 3 4 5



( , )

F A



h h

,

h h h

,

h h h

,

. The soft set

(F,A) can also be represented as a set of ordered pairs as follows:

___________________________

 Department of Mathematics, University of Abuja – Nigeria

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 







 





 





1 1 2 2 3 3



( , )

F A



e F e

,

e F e

,

e F e

,

i.e.,



























1 2 4 2 1 3 5 3 3 4 5



( , )

F A



e h h

,

e

,

h h h

, ,

,

e

,

h h h

, ,

Other notations for (F, A) are

F

_A or



F E

A

,



.

Definition 2.2 [9] (Soft subset/soft equal)

Let (F,A) and (G,B) be two soft sets over a common universe U, we say that

(a) (F,A) is a soft subset of (G,B) denoted

( , )

F A





( , )

G B

if

(i)

A



B

, and

(ii)

 

e

A F e

, ( ) and

G e

( )

are identical approximations.

(b) (F, A) is soft equal set to (G,B) denoted by (F,A) = (G,B) if

( , )

F A





( , )

G B

and

( , )

G B





( , )

F A

.

Pei and Miao [11] pointed out that generally in (a) (ii) F(e) and G(e) may not be identical and so modified the definition of soft subset in the following way

Definition 2.3 [11] (Soft subset redefined)

For two soft sets (F, A) and (G, B) over a universe U, we say that (F, A) is a soft subset of (G,B) if

(i)

A



B

, and

(ii)

 

e

A F e

,

( )



G e

( )

.

Example 2.2

Let

U





u u u u u u

₁

,

₂

,

₃

,

₄

,

₅

,

₆



be a universe set and



1

,

2

, ,

3 4

,

5



E



e e e e e

be a set of parameters. Let



1

, ,

3 5



and



1

,

2

, ,

3 5



A



e e e



E

B



e e e e



E

. Suppose (F,A) and (G,B) are two soft sets over U where

  

1 2

,

4

   

,

2 3

,

4

,

5

    

,

5 1

F e



u u

F e



u u u

F e



u

and

  

1 2

,

4

   

,

2 1

, , ,

3 4 5

   

,

3 3

, ,

4 5

    

,

5 1

,

4

G e



u u

G e



u u u u

G e



u u u

G e



u u

. Then

( , )

F A





( , )

G B

since

A



B

and

( )

F e



G e

 

e

A

. But

( , )

G B





( , )

F A

. Hence

( , )

F A



( , )

G B

.

Remark 2.1

Let (F, A) and (G, B) be soft sets over a common universe U.

( , )

F A





( , )

G B

does not imply that every element of (F,A) is an element of (G,B). Therefore, the definition of classical subset does not hold for soft subset. For example,

let

U





u u u u

1

,

2

,

3

,

4



be a universe and

E





e e e

1

,

2

,

3



be a set of parameters such that if

A



 

e

1

,

B





e e

1

,

3



and











1 2 4







1



2 3 4







3



1 5







( , )

F A



e

,

u u

,

, ( , )

G B



e

,

u u u

,

e

,

u u

,

, then

 

e

A F e

,

( )



G e

( )

and

A



B

. Hence

( , )

F A





( , )

G B

.Clearly



e F e

1

,

 

1





( , )

F A

but

 



e F e

1

,

1





( , )

G B

.

Definition 2.4 [9 ] (Not Set)

Let

E





e e e

1

,

2

, ,

3



,

e

n



be a set of parameters. The ‗Not

set of E‘, denoted by



E

is defined by



1

,

2

,

3

,

n



E

e

    





, where



e

imeans not

1, 2,3,

,

i

e

 

i



n

Proposition 2.1[9]

Let E be a universal parameter set, A , B



E, then

i)



(



A) = A

ii)



(A ⋃ B) =



A ⋃



B

iii)



(A ∩ B) =



A ∩



B

Remark 2.2

It has been proved in [14] that



A ≠ Ac and that



A



E and so proposition 2.1 above hold. But Ge and Yang[8] made the assumption that



A



E and came up with the following proposition.

Proposition 2.2[8]

i)



(A ∪ B) =



A ∩



B (De Morgan‘s Law) i)



_{(A ∩ B) =}



_A_∪



_B_{(De Morgan‘s law)}

Definition 2.5 [2]

Let U be a universe, E be a set of parameters and A ⊆ E.

a) (F, A) is called a relative null soft set with respect

to A, denoted





A, if F(e) = ∅ ,

 

e

A

.

b) (F , A) is called a relative whole soft set or

A-universal with respect to A, denoted

U



A , if F(e)

= U ,

 

e

A

.

c) The relative whole soft set with respect to E

denoted

U



E is called the absolute soft set over

U_.

Example 2.3

Let E={e1,e2 e3, e4 }. If A = 

e , e , e

2 3 4



such that F(e2)

= {u2 , u4}, F(e3) = ∅ , , F(e4) = U, then the soft set (F,A)

= {(e2 , {u2 , u4}) , (e4 , U)}.

If B = {e1 , e3} such that the soft set (G , B) = { (e1 , ∅ ,

),_(e

3 , ∅ , )}, then the soft set (G,B) is a relative null soft

set , ie (G,B) =





B.

If C = {e1 , e2} such that H(e1) = U , H(e2) = U, then the soft

set (H, C) is a relative whole soft set

U



C.

If D = E such that F(e1) = U ,

 

e

i

E

, i = 1,2,3,4, Then

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134

Proposition 2.3 [13]

Let U be a universe, E a set of parameters, A, B , C ⊂ E. If (F,A) , (G,B) and (H,C) are soft sets over U , Then

i) (F,A)





U



A.

ii)





A





(F,A) iii) (F,A)





(F,A)

iv) (F,A)





(G,B) and (G,B)





(H,C) implies

(F,A)





(H,C)

v) (F,A) = (G,B) and (G,B) = (H,C) implies

(F,A) = (H,C)

Definition 2.6 [9 ] (Complement)

The complement of a soft set (F, A) denoted by

( , )

F A

C is defined as

( , )

F A

C





F

C

,



A



where

:

( )

C

F

 

A

P U

is a mapping given by

( )

(

)

C

F



  

U

F



 



A

.

Later Ali et al.[ 2 ] introduced a new notion of complement called relative complement which is defined in the next definition.

Definition 2.7 [2 ] (Relative Complement)

The relative complement of a soft set (F,A) denoted by

( , )

F A

r is defined by

( , )

F A

r





F

r

,

A



where

F

r

:

A



P U

( )

is a mapping given by

( )

( ),

r

F



 

U

F



 



A

.

In view of the above discussion, we present the following example:

Example 2.4

Let

U





u u u u u

₁

,

₂

,

₃

,

₄

,

₅



be a universe set and



1

,

2

, ,

3 4



E



e e e e

be a set of parameters. Suppose A = {e2 , e3 , e4} ⊂ E such that the soft set (F,A)={e2 , {u2 , u4}) ,

(e4 , U)}, then

i. (F,A)c = {(



e2 , {u1 , u3 , u5}) , (



e3 , U)}

ii. (F,A)r = {(e2 , {u1 , u3 , u5}) , (e3 , U)}

Proposition 2.4

Let (F,A) be a soft set over a universe U. Then

i. (F,A)c)c = (F,A)

ii. ((F,A)r)r = (F,A)

iii.

U



CA =





A =

r A

U



iv.





CA =

U



A =

r A





3. SOFT SET OPERATIONS

Definition 3.1 [9 ]

Let (F, A) and (G, B) be two soft sets over a common universe U. Then:

(i) the union of (F,A) and (G,B), denoted

( , )

F A





( , )

G B

is a soft set (H,C), where

and

C

 

A

B

 

e

C

( ),

( )

( ),

( )

( ),

F e

e

A B

H e

G e

e

B

A

F e

G e e

A

B

 







_

 





 



(ii) the intersection of (F,A) and (G,B) denoted

( , )

F A





( , )

G B

is a soft set (H,C) where C=A∩B and



e



C, H(e) = F(e) or G(e) (as both are same set).

(iii) the AND-operation of (F,A) and (G,B) denoted (F,A) AND (G,B) or

( , )

F A



( , )

G B

is a soft set defined by

( , )

F A



( , )

G B



( ,

H A B



)

where

( , )

( )

( ),

( , )

H a b



F a



G b



a b

 

A B

.

(iv) the OR-operation of (F,A) and (G,B) denoted (F,A) OR (G,B) or

( , )

F A



( , )

G B

( , )

F A



( , )

G B



( ,

H A B



)

where

( , )

( )

( ),

( , )

H a b



F a



G b



a b

 

A B

.

Pei and Miao [11] pointed out that in Definition 3.1 (ii), F(e) and G(e) may not be the same set and thus revised the definition as follows:

Definition 3.2 [11 ] (Intersection redefined)

Let (F,A) and (G,B) be two soft sets over U. The intersection (also called bi-intersection by Feng et al. [6]) of (F,A) and (G,B) denoted

( , )

F A





( , )

G B

is a soft set

(H,C) where

C

 

A

B

and

,

( )

e

C H e

F e

G e

 





. Moreover, Ahmad and

Kharal [1 ] pointed out that in the above Definition 3.2,

A



B

must be non-empty to avoid the degenerate case and hence improved the definition as follows:

Definition 3.3 [1 ] (Intersection redefined)

Let (F,A) and (G,B) be two soft sets over U with

A

  

B

. The intersection of (F,A) and (G,B) denoted

( , )

F A





( , )

G B

C

 

A

B

and

 

e

C H e

,

( )



F e

( )



G e

( )

. Ali et al. [2] later introduced the following operations.

Let (F,A) and (G,B) be two soft sets over U. Then

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135

( ), if

( )

( ), if

( )

( ), if

.

F e

e

A B

H e

G e

e

B

A

F e

G e

e

A

B

 







_

 





 



(ii) the restricted intersection (also called intersection by Pei and Miao [10 ] and

biintersection by Feng et al. [6 ]) of (F,A) and (G,B), denoted

( , )

F A





( , )

G B

C

 

A

B

and

,

( )

e

C H e

F e

G e

 





.

(iii) the restricted union of (F,A) and (G,B), denoted

( , )

F A



_R

( , )

G B

is a soft set (H,C),

where

C

 

A

B

and

 

e

C

,

( )

H e



F e



G e

.

(iv) the restricted difference of (F,A) and (G,B) denoted

( , )

F A



_R

( , )

G B

is a soft set (H,C)

where

C

 

A

B

and

,

( )

e

C H e

F e

G e

 





.

Sezgin and Atagun [13] in 2011, defined the following operation;

Definition 3.5 [13] (Restricted symmetric difference)

The restricted symmetric difference of (F,A) and (G,B) denoted

( , ) ( , )

F A





G B





( , ) ( , )

F A





G B



( , )

F A



_R

( , )

G B



_R

(( , )

F A





( , ))

G B

or



 



( , ) ( , )

F A





G B



( , )

F A



R

( , )

G B



R

( , )

G B



R

( , )

F A

.

The above definition (3.5) can also be defined as follows:

Definition 3.6

The restricted symmetric difference of (F,A) and (G,B), denoted

( , ) ( , )

F A





G B

C

 

A

B

and

 

e

C H e

,

( )



F e

( )



G e

( )

(the symmetric difference of F(e) and G(e)).

Example 3.1

Let

U





h h h h h h

₁

,

₂

,

₃

,

₄

,

₅

,

₆



be a universe,



1

,

2

, ,

3 4

,

5



E



e e e e e

be the parameter set with respect to

U, and

A





e e e

₁

,

₂

,

₃





E

.

Let a soft set (F,A) over U be given by



























1 2 4 2 1 3 5 3 3 4 5



( , )

F A



e

,

h h

,

e

,

h h h

,

e

,

h h h

,

. Suppose

B





e e e

₃

,

₄

,

₅



and (G,B) is a soft set over U

given by



























3 1 2 3 4 2 3 6 5 2 3 4



( , )

G B



e

,

h h h

, ,

,

e

,

h h h

, ,

,

e

,

h h h

, ,

. Then

(i)

( , ) ( , )

F A





G B







e h h

₁

,

 

₂

,

₄



,



e h h h

₂

,



₁

, ,

₃ ₅





,



e h h h h h

₃

,



₁

, , , ,

₂ ₃ ₄ ₅











e

4

,

h h h

2

,

3

,

6



,



e

5

,



h h h

2

,

3

,

4







.

(ii)

( , )

F A



R

( , )

G B







e

₃

,



h h h h h

₁

,

₂

,

₃

,

₄

,

₅







(iii)

( , )

F A





( , )

G B







e

₃

,

 

h

₃





(iv)

( , )

F A



_

( , )

G B







e h h

₁

,



₂

,

₄





,



e

₂

,



h h h

₁

, ,

₃ ₅





,



e

₃

,

 

h

₃









e

4

,

h h h

2

,

3

,

6



,



e

5

,



h h h

2

,

3

,

4







(v)

( , )

F A



_R

( , )

G B







e

₃

,



h h

₃

,

₅







(vi)

( , ) ( , )

F A





G B







e

₃

,



h h h h

₁

,

₂

,

₄

,

₅







(vii)

( , ) ( , )

F A



G B







   

e e

₁

,

₃

,

h

₂



,



   

e e

₁

,

₄

,

h

₂



,



   

e e

₁

,

₅

,

h

₄





 







_



  

_

_



  

_



  







  





 







2 3 1 3 2 4 3 2 5 5

3 3 3 3 4 3 3 5 3 4

,

e e

h h

e e

h

e e

h

e e

h

e e

h

e e

h h

(viii)

( , )

F A



( , )

G B







e e

₁

,

₃

 

,

h h h h

₁

,

₂

,

₃

,

₄





,



 









 









 









 









 









 









 









 









1 4 2 3 4 5

1 5 2 3 4

2 3 1 2 3 5

2 4 1 2 3 5 6

2 5 1 2 3 4 5

3 3 1 2 3 4 5

3 4 2 3 4 5 6

3 5 2 3 4 5

, , , , , ,

, , , , ,

, , , , , ,

, , , , , , ,

, , , , , .

e e h h h h

e e h h h

e e h h h h

e e h h h h h

e e h h h h h

e e h h h h

4. PROPERTIES OF SOFT SET OPERATIONS

(5)

136

1. Idempotent properties

(i)

( , )

F A





( , )

F A



( , )

F A



( , )

F A



_R

( , )

F A

(ii)

( , )

F A





( , )

F A



( , )

F A



( , )

F A



_

( , )

F A

2. Identity Properties

(i)

( , )

F A

 



( , )

F A



( , )

F A

 

_R



(ii)

( , )

F A

 



U



( , )

F A



( , )

F A



_

U



(iii)

( , )

F A

  

_R



( , )

F A



( , )

F A





(iv)

( , )

F A



_R

( , )

F A

  



( , ) ( , )

F A





F A

3. Domination Properties

(i)

( , )

F A

  



U



U



( , )

F A



_R

U



(ii)

( , )

F A

   



( , )

F A

 

_



4. Complementation Properties

(i)

   



C

U



r

(ii)

U



C

  



U



r

5. Double Complementation (Involution) Property



( , )

F A

C



C



( , )

F A





( , )

F A

r



r

6. Exclusion Properties

( , )

F A





( , )

F A

r

 

U



( , )

F A



_R

( , )

F A

r

7. Contradiction Properties

( , )

r

( , )

r

F A





F A

  



F A



_

F A

Remark 4.1

Exclusion and contradiction properties do not hold with respect to complement in Definition 2.6 [18]

8. De Morgan’s Properties

(i)



( , )

F A





( , )

G B



C



( , )

F A

C





( , )

G B

C (ii)



( , )

F A



_

( , )

G B



C



( , )

F A

C





( , )

G B

C (iii)



( , )

F A



_R

( , )

G B



r



( , )

F A

r





( , )

G B

r (iv)



( , )



r

( , )

r

( , )

r

R

F A





G B



F A



G B

(v)



( , )

F A



( , )

G B



C



( , )

F A

C



( , )

G B

C (vi)



( , )

F A



( , )

G B



C



( , )

F A

C



( , )

G B

C (vii)



( , )

F A



( , )

G B



r



( , )

F A

r



( , )

G B

r (viii)



( , )



( , )

r _r _r

F A



G B



F A



G B

(ix)



( , )

F A





( , )

G B



r



( , )

F A

r





( , )

G B

r

(x)



( , )

F A





( , )

G B



r



( , )

F A

r





( , )

G B

r Remark 4.2

De Morgan‘s Property does not hold for restricted union and restricted intersection with respect to complement in Definition 2.6

i.e. ((F,A)



_R (G,B))c ≠ (F,A)c





(G, B)c [18] ((F,A)





(G,B))c ≠ (F,A)c



_R (G,B)c [18]

9. Absorption Properties

i.

( , )

F A







( , )

F A





( , )

G B





( , )

F A

ii.

( , )

F A







( , )

F A





( , )

G B





( , )

F A

iii.

( , )

F A



R



( , )

F A





( , )

G B





( , )

F A

iv.

( , )

F A







( , )

F A



R

( , )

G B





( , )

F A

Remark 4.3

(i)





and



_ do not absorb over each other[15]

(ii)



_R

and





do not absorb over each other[15]

10. Commutative Properties

(i)

( , )

F A





( , )

G B



( , )

G B





( , )

F A

(ii)

( , )

F A



_R

( , )

G B



( , )

G B



_R

( , )

F A

(iii)

( , )

F A





( , )

G B



( , )

G B





( , )

F A

(iv)

( , )

F A



_

( , )

G B



( , )

G B



_

( , )

F A

(v)

( , ) ( , )

F A





G B



( , ) ( , )

G B





F A

Remark 4.4

and





do not commute.

11. Associative Properties

(i)

( , )

F A







( , ) ( , )

G B





H C

 



( , ) ( , )

F A





G B







( , )

H C

(ii)

( , )

F A



_R



( , )

G B



_R

( , )

H C

 



( , )

F A



_R

( , )

G B





_R

( , )

H C

(iii)

( , )

F A







( , ) ( , )

G B





H C

 



( , ) ( , )

F A





G B







( , )

H C

(iv)

( , )

F A



_



( , )

G B



_

( , )

H C

 



( , )

F A



_

( , )

G B





_

( , )

H C

(v)

( , )

F A





( , ) ( , )

G B



H C

 



( , ) ( , )

F A



G B





( , )

H C

(vi)

( , )

F A





( , ) ( , )

G B



H C

 



( , ) ( , )

F A



G B





( , )

H C

(6)

137

(i)

( , ) ( , ) ( , )

F A







G B





H C

 



( , ) ( , )

F A





G B

 





( , ) ( , )

F A





H C



(ii)

( , )

F A







( , ) ( , )

G B





H C

 



( , ) ( , )

F A





G B

 





( , ) ( , )

F A





H C



(iii)

( , )

F A



_R



( , )

G B



_

( , )

H C

 



( , )

F A



_R

( , )

G B

 



_

( , )

F A



_R

( , )

H C



(iv)

( , )

F A



_



( , )

G B



_R

( , )

H C

 



( , )

F A



_

( , )

G B

 



_R

( , )

F A



_

( , )

H C



(v)

( , )

F A



R



( , ) ( , )

G B





H C

 



( , )

F A



R

( , )

G B

 





( , )

F A



R

( , )

H C



(vi)

( , )

F A







( , )

G B



_R

( , )

H C

 



( , ) ( , )

F A





G B

 



_R

( , ) ( , )

F A





H C



(vii)

( , )

F A



_R



( , ) ( , )

G B





H C

 



( , ) ( , )

F A



_R

G B

 





( , ) ( , )

F A



_R

H C



(viii)

( , )

F A



_R



( , ) ( , )

G B



_

H C

 



( , ) ( , )

F A



_R

G B

 



_

( , ) ( , )

F A



_R

H C



(ix)

( , )

F A



R



( , ) ( , )

G B



R

H C

 



( , ) ( , )

F A



R

G B

 



R

( , ) ( , )

F A



R

H C



(x)

( , )

F A



_R



( , ) ( , )

G B





H C

 



( , ) ( , )

F A



_R

G B

 





( , ) ( , )

F A



_R

H C



(xi)

( , )

F A







( , ) ( , )

G B



R

H C

 



( , ) ( , )

F A





G B

 



R

( , ) ( , )

F A





H C



(xii)

( , )

F A







( , ) ( , )

G B





H C







( , ) ( , )

F A





G B

 





( , ) ( , )

F A





H C



Remark 4.5

(i)





and



_do not distribute over each other

(ii)



and



do not distribute over each other

(iii)

 



,

_R

and



_do not distribute over



_R

(iv)

   



,

_R

,

_

,



and



_Rdo not distribute over

and





(v)



_Rdistribute over





but the converse is false

(vi)





distribute over



_but the converse is false

5. SOFT SET RELATION AND FUNCTION

Definition 5.1 [3] (Cartesian Product of Soft Set)

Let (F,A) and (G,B) be two soft sets over a common universe U. Then the Cartesian product of (F,A) and (G,B) denoted by

( , ) ( , )

F A



G B

is a soft set

( ,

H A B



)

where

:

(

)

H A B

 

P U U



and

( , )

( )

( , )

H a b



F a



G b



a b

 

A B

,i.e,









( , )

i

,

j

:

i

( ) and

j

( )

H a b



h h

h



F a

h



G b

.

Definition 5.2 [3] (Soft Set Relation)

Let (F,A) and (G,B) be two soft sets over a common universe U. Then a relation from (F,A) to (G,B) called a soft

set relation (R,C) or simply R is a soft subset of

( , ) ( , )

F A



G B

where

C

 

A B

and



( , )

a b



C

. R(a,b) = H(a,b), where

( ,

H A B



)



( , ) ( , )

F A



G B

.

A soft set relation on (F, A) is a soft subset of

( , ) ( , )

F A



F A

. In an equivalent way, we can define a relation R on the soft set (F, A) in the parameterized form as follows:

If

( , )

F A





F a F b

( ), ( ),





,then

( )

( ) iff

( )

F a RF b

F a



F b



R

.

Definition 5.3

Let R be a soft set relation from (F, A) to (G,B) such that

( , ) ( , )

F A



G B



( ,

H A B



)

. Then

(a) the domain of R (domR) is the soft set



D A

,

1







( , )

F A

where





1

:

( , )

,for some

A

 

a

A H a b



R

b



B

and

 

1

 

1

,

1 1

D a



F a



a



A

.

(b) the range of R (ran R) is a soft set



E B

,

1







( , )

G B

where





1

and

1

:

( , )

for some

B



B

 

b B H a b



R

a



A

and

E b

 

₁



G b

 

₁

 

b

₁

B

₁

(c) the inverse of R denoted by

R

1is a soft set relation from

( , ) to( , )

G B

F A

defined by





1

( )

( ) :

( )

R





G b



F a

F a RG b

.

Example 5.1

Let U denote a set of ten people given by



1

,

2

,

3

,

4

,

5

,

6

,

7

,

8

,

9

,

10



U



p p p p p p p p p p

.

Let A denote different professionals given by A = {Accountants, Doctors, Engineers, Teachers} represented by

A





a a a a

₁

,

₂

,

₃

,

₄



respectively.

Let B denote the qualification of people given by

B = {B.Sc., B.Tech.,MBBS, M.Sc.} represented by



1

,

2

, ,

3 4



B



b b b b

respectively.

Then the soft set (F,A) given by

  

   





  



_

1 1 2 2 4 5 3 7 9

4 3 4 7

( , )

,

F A

F a

p p

F a

p p

F a

p p

F a

p p p



(7)

138

  

   





  



_

1 1 2 6 8 10 2 3 6 7 9 3 3 4 5 8

4 1 2 3 8

( , )

, , , ,

,

, , ,

,

, , ,

,

, , ,

,

G B

G b

p p p p p

G b

p p p p G b

p p p p

G b

p p p p



describes peoples‘ qualifications. If we define a relation R from (F,A) to (G,B) as follows:

( )

( ) iff

( )

F a RG b

F a



G b

, then

(i)

R





F a

               

₁



G b F a

₁

,

₂



G b F a

₃

,

₃



G b F a

₂

,

₁



G b

₄



(ii)

dom

R





D A

,

₁



,

where





1 1

, ,

2 3

and ( )

( )

1

A



a a a



A

D a



F a

 

a A

(iii)

ran

R





E B

,

₁



,

where





1 1

, , ,

2 3 4

and ( )

( )

1

B



b b b b

E b



G b

 

b B

(iv)

       

1 1 2 3

1

3 2 4 1

,

G b

F a

G b

F a

R

G b

F a

G b

F a



_{ }





















.

Let R, Q be two soft set relations on a soft set (F,A)

(a)

R Q



if



a b A F a F b

,



, ( )



( )

 

R

F a F b Q

( )



( )



(b) The complement of R denoted as RC is defined by



( )

( ) :

( )

, ,



C

R



F a



F b

F a



F b



R a b



A

(c) The union of R and Q, denoted as

R



Q

is defined by



( )

( ) : ( )

( )

or ( )

( )



R Q

 

F a F b F a F b





R

F a F b





Q

(d) The intersection of R and Q denoted as

R



Q

is defined by



( )

( ) : ( )

( )

and ( )

( )



.

R Q

 

F a F b F a F b





R

F a F b





Q

Example 5.2

Consider a soft set (F, A) over U, where



1

,

2

,

3

,

4



,



1

,

2



U



u u u u

A



a a

and

  

1 1

,

2

   

,

2 2

,

3

,

4



.

F a



u u

F a



u u u

If a soft set relation R on (F, A) is defined as

 

   

 



1 1

,

2 1



,

R



F a



F a



F a

Then

 

   

 



1 2

,

2 2



.

C

R



F a



F a



F a

If another soft set relation Q on (F, A) is defined as

 

   

 



1 1

,

2 2



,

Q



F a



F a



F a

then

           



1 1

,

2 1

,

2 2



R Q

 

F a



F a



F a



F a

 



1 1



.

R

 

Q

F a



F a

It is easy to verify that the union and the intersection of soft set relations satisfy commutative, associative and distributive properties.

Definition 5.5[ 3 ]

Let R be a soft set relation on (F, A), then

(i) R is reflexive if

F a

( )



F a

( )

  

R

a

A

(ii) R is symmetric if

( )

,

( , )

F a

F b

R

F b

F a

R

a b

A A



 







 

(iii) R is transitive if

F a

( )



F b

( )



R

and

( )

, ,

F b



F c

 

R

F a



F c

 

R

a b c



A

(iv) R is equivalence if it is reflexive, symmetric and transitive

(v) R is an identity if

a



b F a

, ( )



F b

( )



R

but

( )

,

F a



F b



R



a b



A

, i.e.,

( )

,

F a



F b

   

R

a

b

a b



A

, e.g.,



( )

( ), ( )

( ) ,



, ,

.

R

F a

F a F b

F b F c

F c

a b c

A









Example 5.3

Consider a soft set (F, A) over U, where

A





a a

₁

,

₂



. If a relation R on (F, A) is defined by

 

   

 

   

 

1 2 2 1

1 1 2 2

,

F a

R

F a









 











, then

R is a soft set equivalence relation.

Note that here

R



( , ) ( , )

F A



F A

.

Definition 5.6 [3] (Composition of Soft Set Relations)

Let (F, A), (G, B) and (H, C) be three soft sets over a common universe. Let R be a soft set relation from (F, A) to (G, B) and S be a soft set relation from (G, B) to (H, C). Then, a new soft set relation from (F, A) to (H,C) called the