A Study of Properties of Soft Set and its Applications

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A STUDY OF PROPERTIES OF SOFT SET AND ITS APPLICATIONS

Shamshad Husain

1

_{, Km. Shivani}

2

1_{M.Phil Student, Pure mathematics, Shobhit University, Meerut, Uttar Pardesh, India.} 2_{Psot Graduate, Mathematics, CCS University, Meerut Uttar Pardesh, India.}

---***---Abstract -In this paper the authors study the theory of soft sets initiated by Molodtsov. Equality of two soft sets, subset, superset of a soft set. Complement of soft set, null soft set, and absolute of soft set and examples. Soft set operation like OR, AND, union, intersection relative intersection, relative union, symmetric difference, relative symmetric difference and examples. Properties of soft sets, De Morgan’s law (with proof by particular example) associative, commutative, distributive, etc.

Key Words: – Soft set, subset and superset of soft set, approximation, complement, relative Complement, NOT set , Null set , Soft set operations,

1. INTRODUCTION

Most of the traditional tools for formal modelling, and computing are crisp, deterministic, and precise in character. However, there are many complicated problems in economics, engineering environment, social science, medical science, etc. that involve data which are not always crisp. We cannot successfully use classical methods because of various types of uncertainties present in these problems. There are theories theory of probability theory, fuzzy set theory, intuitionistic fuzzy sets theory, vague set theory, interval mathematics theory, rough set theory, which can considered mathematical tools for dealing with uncertainties. But all these theories have their inherent difficulties as pointed out .the reason for these difficulties is, possibly, the inadequacy of the parameterization tools of theories. Consequently, Molodtsov initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties which is free from the above difficulties.

(We are aware of the soft sets defined by Pawlak, which is different concept and useful to solve some other type of problems). Soft set theory has a rich potential for application in several directions, few of which had been shown by Molodtsov in his pioneer work in the present paper, we make a theoretical study of the Soft Set Theory in more detail.

2. SOFT SET

Let

U

be an initial universe set

E

is the set of parameters or attributes with respect to

U

. Let P(U)Denote the power set of

U

and

A



E

, then pair (F,A)is called a soft set over

U

, where

F

is a mapping given byF:AP(U).

In other words:- A soft set (F,A)over Uis parameterized

family of subsets

U

for eA,F(e), may be consider as the soft set of

e



elements or

e



approximate Elements of the soft set.(F,A), thus (F,A)is defined as

} )

( , : ) ( ) ( { ) ,

(F A  F e PU eE F e  if eA

Example 2.1- Assume that U{c₁,c₂,c₃,c₄,c₅,c₆}be universal

set constricting of a set of Six Cars under sale. Now

} , , , ,

{e₁ e₂ e₃ e₄ e₅

E be the set of parameters with respect to

U

. Where each parameters e_i,i1,2,3,4,5stands for {Expensive,

Good design, Mileage, Modern, Space capacity} respectively andA{e₁,e₂,e₃,e₄,e₅}E suppose a soft set(F,A).

Describe the attraction by costumer for cars.

} , { )

(e1 c1 c4

F  ,F(e₂){c₁,c₃,c₅},F(e₃){c₃,c₄,c₅}, }

, { ) (e₄ c₂ c₃

F  _,F(e₅){c₂,c₄,c₆}

Then the soft set (F,A)is a parameters family

5 , 4 , 3 , 2 , 1 ), (e i

F _i of a subset

U

defined as

)} ( ), ( ), ( ), ( ), ( { ) ,

(F A  F e₁ Fe₂ F e₃ F e₄ Fe₅ i.e.



(F,A){(c₁,c₂),(c₁,c₃,c₄),(c₃,c₄_,c₅),(c₂,c₃),(c₂,c₄,c₆)}

The soft set (F,A)can also represented as a set ordered pair as follows

))} ( , ( )), ( , ( )), ( , ( )), ( , ( )), ( , {( ) ,

(F A  e₁ Fe₁ e₂ F e₂ e₃ Fe₃ e₄ F e₄ e₅ F e₅



))} , , ( , (

)), , ( , ( )), , ( , ( )), , , ( , ( )), , ( , {( ) , (

6 4 2 5

3 2 4 5 , 4 3 3 4 3 1 2 2 1 1 c c c e

c c e c c c e c c c e c c e A

F 

Notation- (F,A)or (FA)and (FA,E)

By Tabular form (Table1)

U

1

e

Expensive 2

e

Good Design 3

e

Mileage 4

e

Modern 5

e

Space

1

c 1 1 0 0 0

2

c 0 0 0 1 1

3

c 0 1 1 1 0

4

c 1 0 1 0 1

5

c 0 1 1 0 0

6

(2)

represent a soft set in the form of table 1 (corresponding to the soft set in the above example).

2.1 Soft Sub-set

Definition- For two soft set (F,A)and (G,B)over a common universe

U

we say that(F,A) is a soft subset of(G,B)if

(a) ABand

(b) 



A,F(



)andG(



)are identical

approximation, we write

(

,

)

(

,

)

~

B

G

A

F



.

If (F,A)is said to be a soft supper set of(G,B),if (G,B)is

a soft subset of (F,A). We Denote if

(

,

)

(

,

)

~

B

G

A

F



.

2.2 Equality of two Sets

Definition – Two soft set (F,A)and (G,B)over a common universe

U

are said to be soft equal if (F,A) is soft subset of (G,B)and (G,B)is soft subset of(F,A).

Example2.2 - Previous example 2.1

} , , , ,

{e₁e₂ e₃ e₄ e₅

E _{, be the set of parameters with respect to}

U and A{e1,e3,e5}E,B{e1,e2,e3,e5}Eclearly AB

Let (F,A) and (G,B)be two soft sets over the same universe

} , , , , ,

{c₁c₂ c₃ c₄ c₅ c₆

U such that

} , { )

(e₁ c₂ c₄

G  _,G(e2){c1,c3}_,G(e3){c3,c4,c5}_,

} , , { )

(e₄ c₂ c₄ c₆

G 

and } , { )

(e₁ c₂ c₄

F  ,F(e₃){c₃,c₄,c₅},F(e₅){c₂,c₄,c₆}

Therefore ( , ) ( , )

~ B G A

F  ABand _F₍_e₎_G₍_e₎_e_Abut

) , ( ) ,

(G B  F A i.e. (F,A)(G,B)

Remark 2.1 – let soft set (F,A)and (G,B)over a

common universe

U

.( , ) ( , )

~ B G A

F  Does not imply that every

element of (F,A)is an element of (G,B)therefore definition of classical Subset does not hold for soft subset.

Example 2.3 – let U{c₁,c₂,c₃,c₄,c₅,c₆}be universal and }

, , {e₁e₂ e₃

E be the set of parameters with respect to

U

. If }

{e1

A and B{e₁,e₃}

) , ( ) (e₁ c₂ c₄

F  _for

_A

_.

(

F

,

A

)



{

e

₁

,

(

c

₂

,

c

₄

)}

) , , ( ) (

4 3 2

1 c c c

e

G  _,G(e₃)(c₁,c₅)_forB.

))} , ( , ( )), , , ( , {( ) ,

(G B  e₁ c₂ c₃ c₄ e₃ c₁ c₅

Then eA,F(e)G(e)and ABHence( , ) ( , )

~ B G A

F  .

Clearly, (e₁,F(e₁))F(A)and(e₁,F(e₁))G(B)

2.3 Not set of a set of parameters

LetE{e₁,e₂,e₃...,e_n}be the set of Parameters the Not set

of

E

denoted by



E

is defined by

} ., ... , ,

{ e₁ e₂ e₃ e_n

E    

 _wheree_iNot(e_i)i1,2,3,4,...n.

Proposition 2.1

(a)

(A) A

(b)

(AB)AB

(c)

(AB)AB

Remark 2.2 it has been proved that c

A





and the

E

A





and so proposition hold but new assumption that

E

A





and came up with the following proposition

(a)

(AB)AB

(b)

(AB)AB (De Morgan’s law)

Example 2.4 – Consider the above example here



E

{Expensive, Good design, Mileage, Modern, Space capacity} then the Not set of this is





E

{Not Expensive, Not Good design, No Mileage, No Modern, No Space capacity} i.e. E{e₁,e₂,e₃,e₄,e₅}, then Not

set is E{e₁,e₂,e₃,e₄,e₅}.

2.4 Compliment of soft set

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

C

A F, )

( and defined as

), , ( ) ,

(F AC FC A whereFC:AP(U)is a mapping given

by

A F

U

FC()  () .

Example- 2.5 by Example 2.1

Then the soft set (F,A)Cis a parameters family 5

, 4 , 3 , 2 , 1 , ) (e i

F _i C of a subset

U

defined

asF(e₁)CF(e₁){c₁,c₃,c₅,c₆},F(e₂)CF(e₂){c₂,c₄,c₆}

} , , { ) ( )

(e₃ F e₃ c₁c₂c₆

(3)

} , , { ) ( )

(e₅ F e₅ c₁c₃ c₅

F C  

} ) ( , ) ( , ) ( , ) ( , ) ( { ) ,

(F AC Fe₁C Fe₂C Fe₃C Fe₄ C Fe₅C



)} , , ( ), , , , ( ), , , ( ), , , ( ), , , , {( ) ,

(F AC c₁c₃ c₅ c₆ c₂c₄ c₆ c₁c₂c₆ c₁c₄_, c₅c₆ c₁c₃ c₅

The soft set ₍_F_,_A₎C_{can also represented as a set ordered pair}

as follows

)} ) ( , (

), ) ( , ( ), ) ( , ( ), ) ( , ( ), ) ( , {( ) , (

5 5

4 4 3 3 2 2 1 1

C

C C

C

e F e

e F e e F e e F e e F e A F



 





(F,A)C{(e₁,(c₁,c₃,c₅,c₆)),(e₂,(c₂,c₄,c₆)),(e₃,(c₁,c₂,c₆)), ))}

, , ( , ( )), , , , ( ,

(e4 c1c4,c5c6 e5 c1c3c5

2.5Relativecomplement

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

r

A F, )

( and defined as (F,A)r(Fr,A), where Fr:AP(U), is

a mapping given byFr()UF()A.

Example 2.6 –by example 2.1

Then the soft set ₍_F_,_A₎r_{is a parameters family}

5 , 4 , 3 , 2 , 1 , ) (e i

F _i r of a subset

U

defined

asF(e₁)r{c₁,c₃,c₅,c₆},F(e₂)r{c₂,c₄,c₆},F(e₃)r {c₁,c₂,c₆},

} , , , { )

(e₄ c₁c₄c₅c₆

F r ,F(e₅)r {c₁,c₃,c₅}

} ) ( , ) ( , ) ( , ) ( , ) ( { ) ,

(F Ar Fe₁r Fe₂ r Fe₃ r Fe₄ r Fe₅ r



)} , , ( ), , , , ( ), , , ( ), , , ( ), , , , {( ) ,

(F Ar  c₁c₃c₅c₆ c₂c₄c₆ c₁c₂c₆ c₁c₄_,c₅c₆ c₁c₃c₅

The soft set ₍_F_,_A₎r_{can also represented as a set ordered pair}

as follows

)} ) ( , ( ), ) ( , ( ), ) ( , ( ), ) ( , ( ), ) ( , {( ) ,

(F Ar e₁Fe₁r e₂ Fe₂r e₃Fe₃ r e₄Fe₄r e₅Fe₅ r



(F,A)r{(e₁,(c₁,c₃,c₅,c₆)),(e₂,(c₂,c₄,c₆)),(e₃,(c₁,c₂,c₆)) ))}

, , ( , ( )), , , , ( ,

(e₄ c₁c₄_,c₅c₆ e₅ c₁c₃c₅ .

Proposition 2.1- let

U

be the universe, (F,A)is soft set over

U

. Then

(a) ((F,A)C)C (F,A)

(b) ((F,A)r)r _(F,A)

(c) r

A A C

A U

U ) ( )

(

~ ~ ~

 



(d) r

A A C

A) U ( )

(

~ ~ ~



 

2.6 Null set

A soft set(F,A)over universal set

U

is said to be a null soft set defined by if F(e)



,eA.

Example – 2.7 suppose that,

U

is the set of wooden houses under the consideration

A

is the set of parameters

Let there be five houses in the universe

U

given by

} , , , ,

{h₁h₂ h₃ h₄ h₅

U and

A

={brick, muddy, steel, stone}.

The soft set(F,A)describe the “construction of the houses”. The soft set (F,A)is

Defined, as

) (brick

F =means the brick built houses

) (muddy

F = means the muddy houses

) (steel

F = means the steel built houses

) (stone

F = means the stone built houses

The soft set (F,A)is the collection of approximations as below:

) ,

(F A = {brick built houses =



, muddy houses= steel

built houses= stone built houses=}.

2.7 Absolute soft set

A soft set (F,A)over universal set

U

is said to be a absolute soft set denoted by

A

~ , ifF(e)U,eA

.Clearly C~

A and

~ C

A

Example 2.8 – Suppose that,

U is the set of wooden houses under the consideration.

Bis the set of parameters.

Let there be five houses in the universe

U

given by

} , , , ,

{h₁h₂ h₃ h₄ h₅

U _And

_B

_{= {not brick, not muddy, not steel,}

not stone}.

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(brick

F =means the houses not built by brick

) (muddy

F = means the houses not by muddy

) (steel

F = means the houses not built by steel

) (stone

F = means the houses not built by stone

The soft set (G,B)is the collection of approximations as below:

) ,

(G B = {not brick built houses ={h₁,h₂,h₃,h₄,h₅}, not

muddy houses={h₁,h₂,h₃,h₄,h₅}, not steel built houses= }

, , , ,

{h₁h₂ h₃ h₄ h₅ , not stone built houses={h₁,h₂,h₃,h₄,h₅}} .

The soft set (G,B)is the absolute soft set

2.8 Relative null soft set

Let

U

be a universe,

E

be a set of parameters and

A



E

.(F,A) is called a relative Null soft set with

respect to

A

denoted by ~_Aif F(e)



,eA.

2.9 Relative whole soft set

Let

U

be a universe,

E

be a set of parameters and

A



E

.(F,A)is called relative whole soft setorA

universal with respect to

A

denoted by

U

A ~

,if

A e U e

F( ) , 

2.10 Relative Absolute soft set

The relative whole soft set with respect to

E

denoted

U

E ~

b is called the relative absolute soft set over

U

.

Example 2.9 – let E{e₁,e₂,e₃,e₄}if

A



{

e

₂

,

e

₃

,

e

₄

}

such

thatF(e2){c2c4}, F(e3),

U e

F( 4) , then soft

set(F,A){(e₂,(c₂,c₄),(e₄,U)}.

ifB{e₁,e₃}such that the soft set (G,B){(e₁,),(e₃,)}then the

soft set (G,B)is a relative null soft set

~ ) ,

(G B 



B .

ifC {e1,e2}such that H(e1)U,H(e2)Uthen the

soft set (H,C)is a relative whole soft set (H,C)U~c .

ifDEsuch that F(e₁)U,e_iE,i1,2,3,4then the soft set

~ ) ,

(F D U_E is an absolute soft set.

Proposition 2.2- let

U

be the universe,

E

a set of parameters, A,B,CE.if(F,A),(G,B) and

) ,

(H C are soft set over

U

. Then

(e) F A UA

~ ~ ) ,

( 

(f) ( , )

~ ~

A F 



(g) ( , ) ( , ) ~

A F A F 

(h) ( , ) ( , ) ~

B G A

F  and ( , ) ( , ) ~

C H B

G  implies

) , ( ) , (

~ C H A F 

) , ( ) ,

(F A  G B and(G,B)(H,C)implies )

, ( ) ,

(F A  H C

3 SOFT SET OPERATIONS

3.1 Union

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

U

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

) , ( ) , (

~ B G A

F  by is a soft set(H,C), where

B

A

C





and



e



C

    

    

  

 

  

B A e e G e F

A B e e G

B A e e F e H

), ( ) (

), (

), ( ) (

Example 3.1 – Consider the soft set(F,A) which describes the “cost of the houses” and the soft set (G,B)which describes the “attractiveness of the houses”.

Suppose thatU{h₁,h₂,h₃,h₄,h₅,h₆,h₇,h₈,h₉,h₁₀}_,



A

{Very costly, Costly, Cheap} And



B

{Cheap, Beautiful, in the green surroundings}



A{e₁,e₂,e₃}And A{e3,e₄,e5}respectively.

LetF(e₁){h₂,h₄,h₇,h₈},F(e₂){h₁,h₃,h₅},F(e₃){h₆,h₉,h₁₀}, And

} , , { )

(e₃ h₆ h₉ h₁₀

G  G(e₄){h₂,h₃,h₇},G(e₅){h₅,h₆,h₈}, then

) , ( ) , (

~ B G A

F  =(H,C)  CAB, where

} , , , { )

(e₁ h₂h₄ h₇ h₈

H  ,H(e₂){h₁,h₃,h₅},

) (e₃

H {h₆,h₉,h₁₀}_,

} , , { )

(e₄ h₂ h₃ h₇

(5)

U

.then the intersection of (F,A)And(G,B), denoted ( , ) ( , )

~ B G A

F  by is a soft set (H,C)where CAB

C

e





,H(e)F(e)orG(e)(as both are same set).

Example 3.2 by example 3.1



( , ) ( , ) ~

B G A

F  =(H,C)



C



A



B

, Where )

(e1

H 



,H(e2)



,H(e3) {h6,h9,h10},

) (e4

H 



,H(e₅) 



Proposition 3.1 -let

U

be the Universe, (F,A)is soft set over

U

. Then

(a) ( , ) ( , ) ( , ) ~

A F A F A

F  

(b) ( , ) ( , ) ( , ) ~

A F A F A

F  

(c) (F,A)~







,where



is a null set

(d) (F,A)~







where



is a null set

(e) F A AA ~ ~ ) ,

( ,where

~

A

is absolute set

(f) ( , ) ( , )

~ ~

A F A A F  

Proposition 3.2

(a) ₍₍_F_,_A₎_~₍_G_,_B₎₎C_₍_F_,_A₎C_~₍_G_,_B₎C

(b) ₍₍_F_,_A₎_~₍_G_,_B₎₎C_₍_F_,_A₎C_~₍_G_,_B₎C

Proof-

(a) Suppose that ( , ) ( , ) ( , ) ~

B A H B G A

F    ,where

. ),

( ) (

), (

, ),

( ) (

B A if G F

A B if G

B A if F

H

  



  

  

 

  

Therefore, ₍₍_F_,_A₎_~₍_G_,_B₎₎C _₍_H_,_A__B₎C

) ,

(HC AB



Now,HC()UH(),AB

Therefore,

. ),

( ) (

), (

, ),

( ) (

B A if

G F

A B if

G

B A if

F H

C C

C C C

        

     



     

 

  

 



Again, ( , ) ( , ) ( , ) ( , ), ~

~

B G A F B

G A

F C C  C   C 

) ,

(K AB

 (Say)

Where,

. ),

( ) (

), (

, ),

( ) (

B A if G F

A B if G

B A if F

K

C C

        

     



     

 

  

 



C

H

 and

K

are same function . Hence proved.

(b) Suppose that,( , ) ( , ) ( , )

~

B A H B G A

F   

Therefore ₍₍_F_,_A₎_~₍_G_,_B₎₎C _₍_H_,_A__B₎C )

,

(HC AB



Again

), , ( ) , ( ) , ( ) , (

~ ~

B G A F B

G A

F C C  C   C 

) ,

(K AB

 (Say)

Where,



AB, we have

) ( )

( FC 

K orGC()

=F(



)orG(



) , Where



AB

=H(



) =HC()

C

H

 And Kare same function. Hence proved.

Proposition 3.3 if(F,A),(G,B) and (H,C)are three soft sets over

U

. Then

(a) (( , ) ( , )) ( , ) ( , ) (( , ) ( , ))

~ ~ ~

~

C H B G A F C H B G A

F     

(b) (( , ) ( , )) ( , ) ( , ) (( , ) ( , ))

~ ~ ~

~

C H B G A F C H B G A

F     

(c) (( , ) ( , )) ( , ) (( , ) ( , )) (( , ) ( , ))

~ ~ ~ ~

~

C H A F B G A F C H B G A

F      

(d) (( , ) ( , )) ( , ) (( , ) ( , )) (( , ) ( , ))

~ ~ ~ ~

~

F      

3.3 AND

U

.then the AND operation of (F,A)and(G,B), denoted by (F,A)AND(G,B)or(F,A)(G,B)by is a soft set defined by (F,A)(G,B)=(H,AB), where

B A b a b G a F b a

H( , ) ( ) ( )( , ) 

Example 3.3 –by example 3.1

) , ( ) ,

(F A  G B =(H,AB), where,H(e₁,e₃) 



_,

) , (e1 e4

(6)

, (e2 e4

H {h₃},H(e₂,e₅) {h₅},H(e₃,e₃) {h₆,h₉,h₁₀} )

, (e₃ e₄

H 



H(e₃,e₅) {h₆}

3.4 OR

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

U

.then the AND operation of (F,A)and(G,B), denoted by(F,A))OR(G,B), therefore

) , ( ) ,

(F A  G B by is a soft set defined by

) , ( ) ,

(F A  G B =(H,AB),where

B A b a b G a F b a

H( , ) ( ) ( )( , )  .

Then,(F,A)(G,B)=(H,AB), where

) , (e₁e₃

H ={h₂,h₄,h₆,h₇,h₈,h₉,h₁₀}, ₍ _, ₎

4 1 e

e H }

, , , , ,

{h₂ h₃ h₄ h₅ h₇ h₈

 ,H(e₁,e₅) {h₂,h₄,h₅,h₆,h₇,h₈},H(e₂,e₃)

} , , , , ,

{h₁h₃ h₅ h₆ h₉ h₁₀

 _, ₍ _, ₎

4 2 e

e

H {h₁,h₂,h₃,h₅,h₇,h₈},H(e₂,e₅) }

, , , ,

{h₁h₃ h₅ h₆ h₈

 _,H(e₃,e₃) {h₆,h₉,h₁₀}_,H(e₃,e₄)

} , , , ,

{h₂ h₃ h₆ h₇ h₉

 ,H(e₃,e₅) {h₅,h₆,h₈,h₉,h₁₀}.

Remark 3.1 We also use the tabular form for AND, OR solving the examples.

Proposition 3.4

(c) C C C

B G A F B

G A

F, ) ( , )) ( , ) ( , ) ((

~

 



(d) C C C

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((   

Proof-

(c) Suppose that (F,A)(G,B)(H,AB),

Therefore,

C C

B A H B

G A

F, ) ( , )) ( , )

((   

)) (

,

(HC  AB

 , now

), , ( ) , ( ) , ( ) ,

(F A C  G B C  FC A  GC B

) ,

(K AB

 , where K(x,y)FC(x)GC(y),

)) (

,

(K  AB



Now take (,)(AB))

Therefore,

) , (

), ( ) (

), ( ) ( [

), , ( ) , (

 

  



  

   

  

   

K

G F

G F U

H U K

C C

C

H

 And Kare same function. Hence proved.

(d) Suppose that (F,A)(G,B)(H,AB),

Therefore, C C

B A H B

G A

F, ) ( , )) ( , ) ((

~

 



)) (

,

(HC _ A_B



Now, (F,A)C _(G,B)C _(FC,_A)_(GC,_B),

) ,

(K AB

 , where

), ( ) ( ) ,

(x y F x G y

K _ C _ C

)) (

,

(K  AB



Now take (,)(AB))

Therefore,

) , (

), ( ) (

)], ( [ )] ( [

), ( ) ( [

), , ( ) , (

 

  



  

   

   

  

   

K

G F

G U F U

G F U

H U K

C C

C

H

 And

K

are same function. Hence proved

3.5 Extended intersection

U

.then the extended intersection of

) ,

(F A and(G,B), denoted by (F,A)_ (G,B) is a

soft set(H,C), where

C



A



B

and



e



C

    

    

  

 

  

B A e e G e F

A B e e G

B A e e F e H

), ( ) (

), (

), ( ) (

Example 3.5- Consider the soft set(F,A) which describes the “cost of the houses” and the soft set(G,B)which describes the “attractiveness of the houses”.

Suppose thatU{h₁,h₂,h₃,h₄,h₅,h₆}, E{e₁,e₂,e₃,e₃,e₄,e₅}

respectively.



E

{Very costly, Costly, Cheap, beautiful, in the green surroundings}



A

{Very costly, Costly, Cheap},

B



{cheap, beautiful, in the green surroundings}



A{e₁,e₂,e₃}E,B{e3,e₄,e5}Erespectively.

LetF(e₁){h₂,h₄},F(e₂){h₁,h₃,h₅}_,F(e₃){h₃,h₄,h₅}_and

} , , { )

(e₃ h₁h₂ h₃

(7)

) , ( ) ,

(F A  G B =(H,C), where CAB



e



C

    

    

  

 

  

B A e e G e F

A B e e G

B A e e F e H

), ( ) (

), (

), ( )

( .

} , { )

(e1 h2 h4

H  ,H(e₂){h₁,h₃,h₅},

) (e₃

H {h₃},H(e₄){h₂,h₃,h₆},H(e₅){h₂,h₃,h₄}

3.6 Restricted Intersection

U

.then the restricted intersection of

) ,

(F A and(G,B), denoted by(F,A)R(G,B) is a soft

set (H,C)where

C



A



B

C

e





,H(e)F(e)G(e)

Then, (F,A)R(G,B)=(H,C), where

C



A



B

C

e





, H(e)F(e)G(e).

) (e1

H 



,H(e2)



,H(e3) {h3},

) (e₄

H 



,H(e₅) 



3.7 Restricted Union

U

.then the restricted union of

) ,

(F A and(G,B), denoted by (F,A)R (G,B) is a soft set (H,C)where

C



A



B

C

e





,H(e)F(e)G(e)

Example3.7 – by example 3.5

Then,(F,A)R (G,B)=(H,C), where

B

A

C







e



C

,H(e)F(e)G(e). )

(e1

H 



,H(e2)



,H(e3) {h1,h2,h3,h4,h5},

) (e4

H 



,H(e₅) 



3.8 Restricted difference

U

.then the restricted difference of

) ,

(F A and(G,B), denoted by(F,A)R(G,B) is a

soft set (H,C)where

C



A



B

C

e





,H(e)F(e)G(e) Example 3.8 –by example 3.5

Then, (F,A)R(G,B)=(H,C), where

B

A

C







e



C

,H(e)F(e)G(e). )

(e1

H 



,H(e2)



,H(e3) {h4,h5},

) (e4

H 



,H(e₅) 



3.9 Restricted symmetric difference

U

.then the restricted symmetric difference of

) ,

(F A and(G,B), denoted by( , ) ( , ) ~

B G A

F  is a soft

set (H,C)where

C



A



B

C

e





,H(e) F(e)G(e)

In other words

U

.then the restricted symmetric difference of

) ,

(F A and(G,B), denoted ( , ) ( , ) ~

B G A

F  is a soft set

defined by

)) , ( ) , (( )) , ( ) , (( ) , ( ) , (

~ ~

B G A F B

G A F B G A

F   R _R 

)) , ( ) , (( )) , ( ) , (( ) , ( ) , (

~

A F B G B

G A F B G A

F   R R R

Then,

)) , ( ) , (( )) , ( ) , (( ) , ( ) , (

~

A F B G B

G A F B G A

F   R R R .

now, (F,A)R(G,B)=(H,C), where

C



A



B

C

e





, H(e)F(e)G(e).

) (e₁

H 



,H(e₂)



,H(e₃) {h₄,h₅}_,

) (e₄

H 



,H(e₅) 



) , ( ) ,

(G B R F A =(K,D), where DAB



e



D

,

) ( ) ( )

(e G e H e

K  

) (e₁

K 



,K(e₂) 



,

) (e₃

K _{ _, _}

2 1 h

h

 ,K(e₄) 



,K(e₅) 



_{, now the value of}

)) , ( ) , (( )) , ( ) , (( ) , ( ) , (

~

A F B G B

G A F B G A

(8)



e



Z

,

) ( ) ( )

(e H e K e

J   ,

) (e1

J 



,J(e2) 



,J(e3) {h1,h2,h4,h5},J(e4) 



,

) (e₅

J 



4. PROPERTIES OF SOFT SET OPERATION 4.1 Idempotent properties

(a) ( , ) ( , ) ( , ) ( , ) ( , ) ~

A F A

F A F A F A

F    _R

(b) ( , ) ( , ) ( , ) ( , ) ( , )

~

A F A

F A F A F A

F    _

4.2Identity properties

(a)(F,A)~



~ (F,A)(F,A)_R



~ (b)(F,A)~U~ (F,A)(F,A)_U~

(a) (F,A)_R



~ (F,A)(F,A)~



~

(b) ( , ) ( , ) ( , ) ( , )

~ ~

A F A F A

F A

F R 



 

4.3 Domination properties

(a) (F,A)~U~ U~ (F,A)_RU~

(b) (F,A)~~ ~ (F,A)_ ~

4.4 Complementation properties

(a) C U r

~ ~ ~

   

(b) UC Ur

~ ~ ~

 

4.5 Double Complementation (involution) property

(a) C C r r

A F A F A

F, ) ) ( , ) (( , ) )

((  

4.6 Exclusion property

(a) r

R r

A F A

F U A F A

F, ) ( , ) ( , ) ( , ) (

~ ~

 

 

4.7 Contradiction property

(a) r r

A F A

F A

F, ) ( , ) ( , ) ( , )

(

~ ~

    



Remark4.1– Exclusion and contradiction properties do not hold with respect to complement in definition 2.6.

4.8De Morgan’s properties

(a) C C C

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((

~

  



(b) C C C

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((

~  



(c) r r r

R G B F A G B

A

F, ) ( , )) ( , ) ( , ) ((

~

 



(d) r

R r r

B G A

F B

G A

F, ) ( , )) ( , ) ( , ) ((

~

 



(e) r r r

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((

~

  



(f) r r r

B G A F B

G A

F, ) ( , )) ( , ) ( , )

((

~  

_

(g) C C C

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((   

(h) C C C

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((   

(i) r r r

B G A F B

G A

F, ) ( , )) ( , ) ( , )

((   

(j) r r r

B G A

F B

G A

F, ) ( , )) ( , ) ( , )

((   

Remark 4.2 De Morgan’s property does not hold for restricted union and restricted intersection with respect to complement in definition that means

(a) C C C

R G B F A G B

A

F, ) ( , )) ( , ) ( , ) ((

~

 



(b) C

R C C

B G A

F B

G A

F, ) ( , )) ( , ) ( , ) ((

~

 



4.9 Absorption properties

(a)( , ) (( , ) ( , )) ( , )

~ ~

A F B G A F A

F   

(b) ( , ) (( , ) ( , )) ( , )

~ ~

A F B G A F A

F   

(c) (F,A)R((F,A) (G,B))(F,A) (d) (F,A) ((F,A)R (G,B))(F,A)

Remark 4.3

(a) ~



And



do not absorb over each other

(b) ~



And _Rdo not absorb over each other

4.10 Commutative properties

(a) ( , ) ( , ) ( , ) ( , )

~ ~

A F B G B G A

F   

(b) (F,A)R(G,B)(G,B)R(F,A)

(c) ( , ) ( , ) ( , ) ( , ) ~ ~

A F B G B G A

F   

(d) (F,A) (G,B)(G,B) (F,A)

(e) ( , ) ( , ) ( , ) ( , ) ~ ~

A F B G B G A

(9)

(a) And do not commute over each other

4.11 Associative properties

(a) ( , ) (( , ) ( , )) (( , ) ( , )) ( , )

~ ~ ~ ~ C H B G A F C H B G A

F     

(b) ( , ) (( , ) ( , )) (( , ) ( , )) ( , )

~ ~ ~ ~ C H B G A F C H B G A

F     

(c) ) , ( )) , ( ) , (( )) , ( ) , (( ) ,

(F A R G B R H C  F A R G B R H C

(d) (F,A)_ ((G,B)_(H,C))((F,A)_(G,B))_ (H,C)

(e) (F,A)((G,B)(H,C))((F,A)(G,B))(H,C)

(f) (F,A)((G,B)(H,C))((F,A)(G,B))(H,C)

4.12 Complementation properties

(a) ( , ) (( , ) ( , )) ( , ) ( , ) ( , ) ( , )

~ ~ ~ ~ ~ C H A F B G A F C H B G A

F      

(b) ( , ) (( , ) ( , )) ( , ) ( , ) ( , ) ( , )

~ ~ ~ ~ ~ C H A F B G A F C H B G A

F      

(c) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) ,

(F A R G B  H C  F A R G B  F A R H C

(d) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) ,

(F A  G B R H C  F A  G B R F A  H C (e) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) , ( ~ ~ C H A F B G A F C H B G A

F R   R  R

(f) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) , ( ~ ~ ~ C H A F B G A F C H B G A

F  R   R 

(g) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) , ( ~ ~ C H A F B G A F C H B G A

F R   R  R

(h) (i) (j) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) , ( ~ ~ (k) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) , ( ~ ~ ~ C H A F B G A F C H B G A

F  R   R 

(l) ) , ( ) , ( ) , ( ) , ( )) , ( ) , (( ) , ( ~ ~ ~ ~ ~ C H A F B G A F C H B G A

F      

Remark 4.5 ~



And



_do not distribute over each other

And do not distribute over each other ~

, Rand  do not distribute over R ~

~ , ,

,  

 R _ AndRdo not distribute overand 

R

 Distribute over ~ but converse is false

~



Distribute over but the converse is false

5. CONCLUSIONS

In this paper, we have discussed in detail the fundamentals of soft set theory such as equality of soft set, subset, Complement with examples. Its properties with solved examples. It was observe that some properties does not hold some classic result and soft set operation. In this paper we discussed the De Morgan’s law. And all operation union, intersection, AND, OR, and alsorelative operations and its examples. Same as we proof the all properties in future.

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(10)

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