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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 4, APRIL 2014 ISSN 2277-8616

132 IJSTR©2014 www.ijstr.org

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.

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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

( , ) ( ) ( ) : , ( ) ifF A F e P U e E F e e A .

Example 2.1

Assume that 1 2 3 4 5 6, , , , ,U h h h h h h 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 , 1,2, ,5ie i stands for

‗expensive‘, ‗beautiful‘, ‗cheap‘, ‗modern‘, ‗wooden‘,

respectively and 1 2 3, ,A e e e 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 : 1,2,3iF e i of subset of U

defined as 1 2 3( , ) , ,F A F e F e F e , 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:

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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 AF or ,AF E .

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) , ( ) and ( )e A F e 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 1 2 3 4 5 6, , , , ,U u u u u u u 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 1 2 3 5, , and , , ,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 1 2 3 4, , ,U u u u u be a universe and 1 2 3, ,E e e e

be a set of parameters such that if 1 1 3, ,A e B e e

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 1 1, ( , )e F e F A but

1 1, ( , )e F e G B .

Definition 2.4 [9 ] (Not Set)

Let 1 2 3, , , , nE e e e e be a set of parameters. The ‗Not

set of E‘, denoted by E is defined by

1 2 3, , , , nE e e e e , where ie means not

1,2,3, ,ie 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 = 2 3 4e ,e ,e 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 , ∅ ,

),(e3 , ∅ , )}, 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 .CU

If D = E such that F(e1) = U , Eie , i = 1,2,3,4, Then

the soft set (F,D) = EU is an absolute soft set.



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 ( , )CF A is

defined as ( , ) ,C CF A F A where

: ( )CF A P U is a mapping given by

( ) ( )CF 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

( , )rF A is defined by ( , ) ,r rF A F A where rF

: ( )A P U is a mapping given by

( ) ( ),rF U F A .

In view of the above discussion, we present the following example: Example 2.4

Let 1 2 3 4 5, , , ,U u u u u u 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. C

AU = A = r

AU

iv. C

A = 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

andC 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 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

. 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 is a soft set (H,C), where C A B

and , ( ) ( ) ( )e C H e F e G e . Ali et al. [2] later

introduced the following operations. Definition 3.4 [2] Let (F,A) and (G,B) be two soft sets over U. Then

(i) the extended 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



( ), 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

biintersection 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 .

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

denoted ( , ) ( , )RF A G B is a soft set (H,C),

where andC A B e C ,

( ) ( ) ( )H e F e G e .

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

denoted ( , ) ( , )RF A 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 is a soft set defined by

( , ) ( , ) ( , ) ( , ) (( , ) ( , )) R RF A G B F A G B F A G B or

( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B F A G B G B 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 is a soft set (H,C), where

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 1 2 3 4 5 6, , , , ,U h h h h h h be a universe,

1 2 3 4 5, , , ,E e e e e e be the parameter set with respect to

U, and 1 2 3, ,A e e e 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 3 4 5, ,B e e e 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) 1 2 4 2 1 3 5 3 1 2 3 4 5( , ) ( , ) , , , , , , , , , , , ,F A G B e h h e h h h e h h h h h

4 2 3 6 5 2 3 4, , , , , , , .e h h h e h h h

(ii) 3 1 2 3 4 5( , ) ( , ) , , , , ,RF A G B e h h h h h

(iii) 3 3( , ) ( , ) ,F A G B e h

(iv) 1 2 4 2 1 3 5 3 3( , ) ( , ) , , , , , , , ,F A G B e h h e h h h e h

4 2 3 6 5 2 3 4, , , , , , ,e h h h e h h h

(v) 3 3 5( , ) ( , ) , ,RF A G B e h h

(vi) 3 1 2 4 5( , ) ( , ) , , , ,F A G B e h h h h

(vii) 1 3 2 1 4 2 1 5 4( , ) ( , ) , , , , , , , ,F A G B e e h e e h e e h

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) 1 3 1 2 3 4( , ) ( , ) , , , , , ,F A G B e e h h h h

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 h

e e h h h h h

e e h h h h

4. PROPERTIES OF SOFT SET OPERATIONS Various results on properties of soft set operations have been established by many authors. We will state without proofs the basic properties on soft set operations, some of which we have established and published.



1. Idempotent properties

(i) ( , ) ( , ) ( , ) ( , ) ( , )RF A F A F A F A F A

(ii) ( , ) ( , ) ( , ) ( , ) ( , )F A F A F A F A F A

2. Identity Properties

(i) ( , ) ( , ) ( , ) RF A F A F A

(ii) ( , ) ( , ) ( , )F A U F A F A U

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

(iv) ( , ) ( , ) ( , ) ( , )RF A F A F A F A

3. Domination Properties

(i) ( , ) ( , ) RF A U U F A U

(ii) ( , ) ( , )F A F A

4. Complementation Properties

(i) C rU

(ii) C rU U

5. Double Complementation (Involution) Property

( , ) ( , ) ( , )C r

C rF A F A F A

6. Exclusion Properties

( , ) ( , ) ( , ) ( , )r r

RF A F A U F A F A

7. Contradiction Properties

( , ) ( , ) ( , ) ( , )r rF 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) ( , ) ( , ) ( , ) ( , )C C CF A G B F A G B

(ii) ( , ) ( , ) ( , ) ( , )C C CF A G B F A G B

(iii) ( , ) ( , ) ( , ) ( , )r r r

RF A G B F A G B

(iv) ( , ) ( , ) ( , ) ( , )r r r

RF A G B F A G B

(v) ( , ) ( , ) ( , ) ( , )C C CF A G B F A G B

(vi) ( , ) ( , ) ( , ) ( , )C C CF A G B F A G B

(vii) ( , ) ( , ) ( , ) ( , )r r rF A G B F A G B

(viii) ( , ) ( , ) ( , ) ( , )

r r rF A G B F A G B

(ix) ( , ) ( , ) ( , ) ( , )r r rF A G B F A G B

(x) ( , ) ( , ) ( , ) ( , )r r rF A G B F A G B

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. ( , ) ( , ) ( , ) ( , )RF A F A G B F A

iv. ( , ) ( , ) ( , ) ( , )RF A F A G B F A

Remark 4.3

(i) and do not absorb over each other[15]

(ii) andR do not absorb over each other[15]

10. Commutative Properties

(i) ( , ) ( , ) ( , ) ( , )F A G B G B F A

(ii) ( , ) ( , ) ( , ) ( , )R RF A G B G B 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) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B H C F A G B 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

12. Distributive properties



(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) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A G B F A H C

(iv) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R RF A G B H C F A G B F A H C

(v) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A G B F A H C

(vi) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R RF A G B H C F A G B F A H C

(vii) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A G B F A H C

(viii) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A G B F A H C

(ix) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C

(x) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A G B F A H C

(xi) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R RF A G B H C F A G B 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) , andR do not distribute over R

(iv) , , ,R and R do not distribute over

and

(v) R distribute 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,

( , ) , : ( ) and ( )i j i jH 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

1, ( , )D A F A where

1 : ( , ) ,for someA 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

1, ( , )E B G B where

1 1and : ( , ) for someB B B b B H a b R a A

and 1 1 1 1E b G b b B

(c) the inverse of R denoted by 1Ris 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 1 2 3 4, , ,A a a a a 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

describes people with different professions and the soft set (G,B) given by



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) 1 1 2 3 3 2 1 4, , ,R F a G b F a G b F a G b F a G b

(ii) 1dom , ,R D A where

1 1 2 3 1, , and ( ) ( )A a a a A D a F a a A

(iii) 1ran , ,R E B where

1 1 2 3 4 1, , , and ( ) ( )B b b b b E b G b b B

(iv)

1 1 2 31

3 2 4 1

, ,

,

G b F a G b F aR

G b F a G b F a

.

Definition 5.4 [3] Let R, Q be two soft set relations on a soft set (F,A)

(a) if , , ( ) ( ) ( ) ( )R Q a b A F a F b R F a F b Q

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

( ) ( ) : ( ) ( ) , ,CR 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 F a Then

1 2 2 2, .CR F a 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 F a then

1 1 2 1 2 2, ,R Q F a F a 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 1 2,A a a . If a

relation R on (F, A) is defined by

1 2 2 1

1 1 2 2

, ,

,

F a F a F a F aR

F a F a F a 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

composition of R and S denoted by SoR is defined as

follows: If ( ) ( , )and (c) ( , )F a F A H H C , then

( ) (c)F a H SoR

( ) ( ) and ( ) (c) ,F a G b R G b H S for some

( ) ( , )G b G B .



Example 5.4

Let 1 2 3 1 2 1 2, , , , and ,A a a a B b b C c c . Let R

and S be soft set relations over a common universe, defined respectively from (F,A) to (G,B) and (G,B) to (H,C) such that

1 1 2 2 3 2, ,R F a G b F a G b F a G b , and

1 1 2 2,S G b H c G b H c Then

1 1 2 2 3 2, ,SoR F a H c F a H c F a H c .

Suppose 1 2 3, ,A B C e e e such that

1 2 2 3 3 1, , ,R F e F e F e F e F e F e

1 3 2 2 3 3, , .S F e F e F e F e F e F e Then

1 2 2 3 3 3, ,SoR F e F e F e F e F e F e and

1 1 2 3 3 1, , .RoS F e F e F e F e F e F e

Thus in general, SoR RoS

Definition 5.7 [3] (Soft Set Function)

Let (F,A) and (G,B) be two non-empty soft sets over U. Then a soft set relation f from (F,A) to (G,B) written

: ( , ) ( , )f F A G B is called a soft set function if every

element in the domain of f has a unique element in the

range of f. If F(a) f G(b), i.e., ( ) ( )F a G b f , then we

write ( ) ( )f F a G b .

Example 5.5

Let 1 2 3 4 1 2, , , and ,A a a a a B b b . Consider the soft

sets (F,A) and (G,B) over a universe U. Then a soft set function f from (F, A) to (G, B) can be given by

(i)

1 1 2 1

3 3 4 2

, ,

,

F a G b F a G bf

F a G b F a G b

(ii)

1 1 2 1

3 1 4 1

, ,

,

F a G b F a G bf

F a G b F a G b

.

But

1 1 1 2 2 2 3 1, , ,F a G b F a G b F a G b F a G b

is not a soft set function.

Definition 5.8 [3]

A function f from (F,A) to (G,B) is called

(i) Injective (one-to-one) if

( ) ( ) ( ) ( )F a F b f F a f F b

(ii) Surjective (onto) if range f = (G,B) (iii) Bijective (one- to-one and onto) if f is both

injective and surjective.

Example 5.6

Consider the function f in Example (5.5), (i) is onto but (ii) is not.

Definition 5.9 [3] (Identity Soft Set Function)

The identity soft set function I on a soft set (F, A) is defined

by : ( , ) ( , )I F A F A such that

( ) ( ) ( ) ( , )I F a F a F a F A

6. MATRIX REPRESENTATION OF SOFT SET We present the matrix representation of soft sets, their basic operations and properties with illustrative examples. Definition 6.1 [5] (Soft Matrix) Let U be a universe, E a set of parameters with respect to U

and A ⊆ E. Let ,Af E be a soft set over U. Then a

subset AR of U E ,uniquely defined as

( , ) : , ( ) ,A AR u e e A u f e is called a relation form

of the soft set ,Af E .

The characteristic functionAR of AR is defined as

where

1, ( , ) ;( , )

0, ( , ) .A

A

R

A

u e Ru e

u e R

Now if 1 2 1 2, , , , , ,m nU u u u and E e e e then

the soft set (fA , E) can be represented by a matrix ija

called an m× n ―soft matrix‖ of the soft set (fA , E) over U as follows

11 12 1

21 22 2

1 2

n

n

ij m n

m m mn

a a a

a a Aa

a a a

where ,

Aij R i ja u e

In other words, a soft set is uniquely represented by its corresponding soft matrix. Example 6.1

Let 1 2 3 4 5, , , ,U u u u u u be a universe set, and

: {0,1},AR U E



1 2 3 4, , ,E e e e e be a set of all parameters with

respect to U.

Let 1 3 4 1 3 4 3, , , , ,A AA e e e f e u u f e , and

4 1 3 5, , .Af e u u u Then the soft set ,Af E is given

by

1 3 4 4 1 3 5, , , , , , , .Af E e u u e u u u

The relation form AR of ,Af E is given by

3 1 4 1 1 4 3 4 5 4, , , , , , , , ,AR u e u e u e u e u e .

Hence the soft matrix ija of the soft set ,Af E is

given by

0 0 0 1

0 0 0 0

, 1,2, ,5; 1,2, , 4.1 0 0 1

1 0 0 0

0 0 0 1

ija i j

As noted earlier, 3Af e , since there is no element in

U related to the parameter 3e A , so it does not appear in

the aforesaid description of the soft set ,Af E .

Definition 6.2 [5](Special Soft Matrices) Let the set of all mxn soft matrices over U be denoted SM (U)mxn or just SM(U)

Let ( )ija SM U . Then ija is called

(a) A zero soft matrix, denoted 0 , if

0 and ;ija i j

(b) An A-universal soft matrix, denoted ija , if

1 :ij A ja j I j e A and i.

(Note that it is so called, since 1ija only for the

parameters appearing in the set );A E and

(c) A universal soft matrix denoted I , if

1 andija i j

Example 6.2

Let 1 2 3 4 1 2 3 4, , , , , , ,U u u u u E e e e e and

4 4, , ( )ij ij ija b c SM U . If

1 2 3 1 2 3, , , A A AA e e e f e f e f e then

0ija is a zero soft matrix given by

0 0 0 0

0 0 0 00 .

0 0 0 0

0 0 0 0

If 2 4 2 4, , ,B BB e e f e U f e then ijb is a B-

universal soft matrix given by

0 1 0 1

0 1 0 1.

0 1 0 1

0 1 0 1

ijb

If , e iC E f e U for each i, then ijc I

is a

universal soft matrix given by

1 1 1 1

1 1 1 1.

1 1 1 1

1 1 1 1

I

Definition 6.3 [5](Soft Sub matrices)

Let , ( ).ij ijM a N b SM U Then we define the

following:

(i) M is a soft sub matrix of N, denoted M N if

ij ija b for each i and j.

In this case, we also say that M is dominated by N or N

dominates M. Note that similar to ( 1),kR k the k-

dimensional real space, holds without the holding of either < or =. We define M and N comparable, denoted

, iff ;M N M N or N M

(ii) M is a proper soft sub matrix of N, denoted

, if ij ijM N a b and for at least one term

ij ija b for all i and j. In this case, we say that M is

properly dominated by N. (iii) M is a strictly proper soft sub matrix of N,

denoted M N , if M N and ij ija b , for each i and

j. In this case we say that M is strictly dominated by N. (iv) M and N are soft equal matrices denoted

if ij ijM N a b for each i and j. Equivalently, if

M N and ,N M then

.M N It is immediate to see that is a partial ordering

(reflexive, anti-symmetric and transitive) on the class of soft matrices.



6.2 Operations On Soft Matrices We discuss the operations of union, intersection complement, difference and products of soft matrices and their basic properties. Definition 6.4 [5] (Soft Matrix Operations)

Let , ( )ij ijM a N b SM U . Then a soft matrix

( )ijP c SM U is called the

(i) union of M and N, denoted M N , if

max , for all and ;ij ij ijc a b i j

(ii) intersection of M and N, denoted M N

min , for all and ;ij ij ijc a b i j

(iii) complement of M, denoted 0M , if

1 for all and ;ij ijc a i j

(iv) difference of N from M, also called the relative complement of N in M, denoted M – N or M\N if

0P M N .

In view of the (ii) above, M and N are said to be disjoint if

0M N .

Example 6.3 Let

0 0 1 1 1 0 0 0

1 0 1 0 0 1 0 1

and .0 1 0 1 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 1 0 0 0 0

M N

Then,

(i)

1 0 1 1

1 1 1 1

;1 1 0 1

0 1 0 0

0 0 0 1

M N

(ii)

0 0 0 0

0 0 0 0

00 0 0 0

0 0 0 0

0 0 0 0

M N

which implies

that M and N are disjoint;

(iii)

0

0

0

1 1 0 0

0 1 0 1and ,

1 0 1 00 ;

1 1 1 1

1 1 1 0

M M IM

M M

(iv) 0

0 1 1 1

1 0 1 0

;0 1 1 1

1 0 1 1

1 1 1 1

N

(v) 0

0 0 1 1

1 0 1 0

;and0 1 0 1

0 0 0 0

0 0 0 1

M N M N

(vi) 0

1 0 0 0

0 1 0 1

.1 0 0 0

0 1 0 0

0 0 0 0

N M N M

Proposition 6.1: Properties of Soft Matrix Operations

Let , , ( )ij ij ijM a N b P c SM U .

(i) ; (Idempotent laws)M M M M M M

(ii) 0 ; (Identity laws)M M M I M

(iii) ; 0 0 (Domination laws)M I I M

(iv) 0 0

0 ; 0 (De Morgan's laws)I I

(v) 0 0; 0 (De Morgan's laws)M M I M M

(vi)

0 00 0 0 0;

(De Morgan's laws)

M N M N M N M N

(vii) 0

0 for all (Involution law)M M M

(viii)

; (Commutative laws)M N N M M N N M

(ix)

;

(Associative laws)

M N P M N P

M N P M N P

(x) ;M N P M N M P



.M N P M N M P (Distributive

Laws) Proof: Most of the proofs follow from definitions. Let us, for example, prove the first parts of (vi), (ix) and (x). (vi) For each i and j,

00

0

0 0

0 0

max ,

1 max ,

min 1 , 1

.

ij ij

ij ij

ij ij

ij ij

ij ij

M N a b

a b

a b

a b

a b

M N

(ix)

max ,max ,

max max , ,

.

ij ij ij

ij ij ij

ij ij ij

ij ij ij

M N P a b c

a b c

a b c

a b c

M N P

(x)

,

max ,min ,

min max , ,max

.

ij

ij ij ij

ij ij ij

ij ij ij c

ij ij ij ij

M N P a b c

a b c

a b a

a b a c

M N M P

Definition 6.5 [5] ( Product of Soft Matrices)

Let , ( )ij ik m nM a N b SM U . Then

(i) AND-product of M and N, denoted M N is defined

: ( ) ( )m n m nSM U SM U 2( )m n

SM U

such

that ,ij ik ipa b c where min ,ip ij ikc a b

and ( 1)P n j k .

(ii) OR-product of M and N, denoted M N is defined

: ( ) ( )m n m nSM U SM U 2( )m n

SM U

such

that ,ij ik ipa b c where max ,ip ij ikc a b

and ( 1)P n j k .

(iii) AND-NOT-product of M and N, denoted M N is

defined : ( ) ( )m n m nSM U SM U 2( )m n

SM U

such that ,ij ik ipa b c where

min ,1ip ij ikc a b and ( 1)P n j k .

(iv) OR-NOT-product of M and N, denoted M N is

defined : ( ) ( )m n m nSM U SM U 2( )m n

SM U

such that ,ij ik ipa b c where

max ,1ip ij ikc a b and ( 1)P n j k

Note: Products of soft matrices hold if the two matrices are of the same order or have the same number of rows Proposition 6.2 [5] (Properties of Product of Soft Matrices)

Let , ( )ij ikM a N b SM U . Then the following

hold:

(i) 0 0 0 0 0 0( ) ; ( )M N M N M N M N

(De Morgan‘s laws)

(ii) 0 0 0 0 0 0( ) ; ( )M N M N M N M N

(De Morgan‘s laws) Proof: The proofs follow from definitions. Example 6.4

Let 4 4, ( )ij ikM a N b SM U given by

0 1 0 1 0 0 1 1

0 1 1 1 1 0 0 1and

0 1 1 0 0 0 1 1

0 1 0 0 0 0 1 1

M N

Then

4 16

0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1

0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1

0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0

0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

ij ik ipM N a b c

Similarly, the other products ,M N M N and

M N can be found.

Also,



1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0

1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0( ) ,

1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1

1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1

M N

1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0

1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0

1 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1

1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1

M N

Thus,0 0 0( )M N M N . Note that the commutative

laws are not valid for products of soft matrices.

CONCLUSION AND FUTURE WORK In this paper, we have discussed in detail the fundamentals of soft set theory such as soft subsets, soft set operations and their properties, soft set relation and function, soft matrices among others, and exemplified them. It was observed that some properties on classical sets do not hold for soft set operations. Similar study could be extended to related concepts such as fuzzy soft set, intuitionistic fuzzy soft set, soft multi set among others.

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