some results on generalized fuzzy soft sets - · pdf filesome results on generalized fuzzy...

Some Results on Generalized Fuzzy Soft Sets

Manash Jyoti Borah1, Tridiv Jyoti Neog

2, Dusmanta Kumar Sut

3

1Deptt. of Mathematics, Bahona College, Jorhat, India

E-mail : [email protected] 2 Deptt. of Mathematics, D.K. High School, Jorhat, India

E-mail : [email protected] 3 Deptt. of Mathematics, N. N. Saikia College, Titabor, India

E-mail : [email protected]

Abstract

The purpose of this paper is to put forward some new

notions regarding generalized fuzzy soft set theory. Our

work is an extension of earlier works of Majumder and

Samanta on Generalized Fuzzy Soft Sets.

1. Introduction In many complicated problems arising in the fields

of engineering, social science, economics, medical

science etc involving uncertainties, classical methods

are found to be inadequate in recent times. Molodtsov

[3] pointed out that the important existing theories viz.

Probability Theory, Fuzzy Set Theory, Intuitionistic

Fuzzy Set Theory, Rough Set Theory etc. which can be

considered as mathematical tools for dealing with

uncertainties, have their own difficulties. He further

pointed out that the reason for these difficulties is,

possibly, the inadequacy of the parameterization tool of

the theory. In 1999 he initiated the novel concept of

Soft Set as a new mathematical tool for dealing with

uncertainties. Soft Set Theory, initiated by Molodtsov

[3], is free of the difficulties present in these theories.

In recent times, researches have contributed a lot

towards fuzzification of Soft Set Theory. Maji et al. [5]

introduced the concept of Fuzzy Soft Set and some

properties regarding fuzzy soft union, intersection,

complement of a fuzzy soft set, De Morgan Law etc.

These results were further revised and improved by

Ahmad and Kharal [1]. In 2011, Neog and Sut [8] put

forward some more propositions regarding fuzzy soft

set theory. They studied the notions of fuzzy soft union,

fuzzy soft intersection, complement of a fuzzy soft set

and several other properties of fuzzy soft sets along

with examples and proofs of certain results. In this

paper, we have studied the notion of union and

intersection of two fuzzy soft sets in two fuzzy soft

classes and propose some related properties.

Majumder and Samanta [6] initiated another

important part, known as Generalized Fuzzy Soft Set

Theory. They proposed a way to find the similarity of

two generalized fuzzy soft sets and successfully applied

the same in a medical diagnosis problem. In 2011,

Yang [4] pointed out, with the help of counter

examples, that some results put forward by Majumder

and Samanta [6] are not valid in general. While

generalizing the concept of Fuzzy Soft Sets, Majumder

and Samanta [6] considered the same set of parameter.

In our work, we have put forward the notions of

generalized fuzzy soft sets considering different sets of

parameters.

2. Preliminaries In this section, we recall some basic concepts and

definitions regarding fuzzy soft sets and generalized

fuzzy soft sets.

2.1. Soft Set [3]

A pair (F, E) is called a soft set (over U) if and only

if F is a mapping of E into the set of all subsets of the

set U.

In other words, the soft set is a parameterized family

of subsets of the set U. Every set EF ),( , from

this family may be considered as the set of -

elements of the soft set (F, E), or as the set of -

approximate elements of the soft set.

2.2. Fuzzy Soft Set [5]

A pair (F, A) is called a fuzzy soft set over U where

)(~

: UPAF is a mapping from A into )(~

UP .

2.3. Fuzzy Soft Class [1]

Let U be a universe and E a set of attributes. Then

the pair (U, E) denotes the collection of all fuzzy soft

sets on U with attributes from E and is called a fuzzy

soft class.

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2.4. Null Fuzzy Soft Set [5]

A soft set (F, A) over U is said to be null fuzzy soft

set denoted by if )(, FA is the null fuzzy set

0 of U where Uxx 0)(0 .

2.5. Absolute Fuzzy Soft Set [5] A soft set (F, A) over U is said to be absolute fuzzy

soft set denoted by A~

if )(, FA is the null fuzzy

set 1 of U where Uxx 1)(1

2.6. Fuzzy Soft Sub Set [5] For two fuzzy soft sets (F, A) and (G, B) in a fuzzy

soft class (U, E), we say that (F, A) is a fuzzy soft

subset of ( G, B), if

(i) BA

(ii) For all A , GF and is written as

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

2.7. Union of Fuzzy Soft Sets [5]

Union of two fuzzy soft sets (F, A) and (G, B) in a

soft class (U, E) is a fuzzy soft set (H, C) where

BAC and C ,

BAGF

ABG

BAF

H

if ),()(

if ),(

if ),(

)(

And is written as CHBGAF ,,~

, .

2.8. Intersection of Fuzzy Soft Sets [5] Intersection of two fuzzy soft sets (F, A) and (G, B)

in a soft class (U, E) is a fuzzy soft set (H, C) where

BAC and C , )(or )()( GFH (as

both are same fuzzy set) and is written as

CHBGAF ,,~

, .

Ahmad and Kharal [1] pointed out that generally

)(F or )(G may not be identical. Moreover in order

to avoid the degenerate case, he proposed that

BA must be non-empty and thus revised the above

definition as follows -

2.9. Intersection of Fuzzy Soft Sets Redefined [1] Let (F, A) and (G, B) be two fuzzy soft sets in a soft

class (U, E) with BA .Then Intersection of two

fuzzy soft sets (F, A) and (G, B) in a soft class (U, E) is

a fuzzy soft set (H,C) where BAC and C ,

)()()( GFH .

We write CHBGAF ,,~

, .

2.10. Complement of a Fuzzy Soft Set [7]

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

by (F, A)c and is defined by (F, A)

c = (F

c, A) where

)(~

: UPAF c is a mapping given

by cc FF )()( , A .

2.11. T – norm [2]

A binary operation ]1,0[]1,0[]1,0[:* is

continuous t - norm if * satisfies the following

conditions.

(i) * is commutative and associative

(ii) * is continuous

(iii) a * 1= a ]1,0[a

(iv)

dcba ** whenever dbca , and

]1,0[,,, dcba

An example of continuous t - norm is a * b = ab.

2.12. T – conorm [2]

A binary operation ]1,0[]1,0[]1,0[: is

continuous t -conorm if satisfies the following

conditions:

(i) is commutative and associative

(ii) is continuous

(iii) a 0 = a ]1,0[a

(iv) dcba whenever dbca , and

]1,0[,,, dcba

An example of continuous t - conorm is

a * b = a + b – ab.

2.13. Generalized Fuzzy Soft Set [6]

Let

nxxxxU ....,,.........,, 321 be the universal set

of elements and meeeeE ,......,, 321 be the universal

set of parameters. The pair (U, E) will be called a soft

universe. Let UIEF : and be a fuzzy subset of

E, i.e. 1,0: IE , where UI is the collection of

all fuzzy subsets of U. Let F be the mapping

IIEF U : be a function defined as follows:

)(),()( eeFeF , where UIeF )( . Then F is

called generalized fuzzy soft sets over the soft universe

(U,E). Here for each parameter ,ie

)(),()( iii eeFeF indicates not only the degree of

belongingness of the elements of U in ieF but also

the degree of possibility of such belongingness which is

represented by ie .]


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2.14. Generalized Fuzzy Soft Subset [6]

Let F and

G be two GFSS over (U, E). Now

F is said to be a generalized fuzzy soft subset of G

if

(i) is a fuzzy subset of

(ii) )(eF is also a fuzzy subset of

. )( EeeG

In this case, we write

~F G

2.15. Complement of a Generalized Fuzzy Soft Set

[6]

Let F be a GFSS over (U, E). Then the

complement of F , denoted by c

F and is defined

by GFc , where ),()( and )()( eFeGee cc

.Ee

2.16. Union of Generalized Fuzzy Soft Sets [6]

The union of two GFSS F and G over (U, E) is

denoted by GF ~ and defined by GFSS

IIEH U : such that for each Ee

))()(),()(()( eeeGeFeH

2.17. Intersection of Generalized Fuzzy Soft Sets [6]

The intersection of two GFSS F and G over (U,

E) is denoted by GF ~ and defined by GFSS

IIEH U : such that for each Ee

))(*)(),(*)(()( eeeGeFeH

2.18. Generalized Null Fuzzy Soft Set [6] A GFSS is said to be a generalized null fuzzy soft

set, denoted by , if IIE U : such that

)(),()( eeFe where EeeeF ,0)(,0)(

2.19. Generalized Absolute Fuzzy Soft Set [6]

A GFSS is said to be a generalized absolute fuzzy

soft set, denoted by ~

, if IIEA U :~ such that

)(),()( eeFe where

EeeeF ,1)(,1)(

3. Generalized Fuzzy Soft Set Redefined

In this section, we put forward the notion of

generalized fuzzy soft sets considering different sets of

parameters and accordingly redefine the notions of

union, intersection, complement etc. of generalized

fuzzy soft sets in the following manner:

3.1. Generalized Fuzzy Soft Set

Let }..,,.........,,{ 321 nxxxxX be the universal set

of elements and },........,,,{ 321 meeeeE be the set of

parameters. Let EA and UIAF : and be a

fuzzy subset of A i.e. ]1,0[: IA , where UI is the

collection of all fuzzy subsets of U. Let

IIAF U : be a function defined as follows:

)(),()( eeFeFA

, where UIeF )( .Then A

F is

called a generalized fuzzy soft set (GFSS) over (U, E).

Here for each parameter )(, iA

i eFe indicates not only

degree of belongingness of the elements of U in

)( ieF but also degree of preference of such

belongingness which is represented by )( ie .

3.2. Example

Let U = 321 ,, SSS be the set of students under

consideration and

E = 1e (expertise in English), 2e (expertise in

mathematics), 3e (expertise in chemistry), 4e (expertise

in computer science)} be the set of parameters and

EeeeA 431 ,, . Let ]1,0[: IA be given as

follows:

8.0)(,5.0)(,4.0)( 431 eee . We define A

F as

follows:

)4.0,7.0/,5.0/,3.0/()( 3211 SSSeFA

,

)5.0,3.0/,1.0/,6.0/()( 3213 SSSeFA

,

)8.0,7.0/,9.0/,3.0/()( 3214 SSSeFA

is the

generalized fuzzy soft set representing overall

expertness of the students.

In tabular form this can be expressed as

AF

0.4 0.5 0.8

1e

3e 4e

1S

0.3 0.6 0.3

2S

0.5 0.1 0.9

3S

0.7 0.3 0.7


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3.2. Generalized Fuzzy Soft Sub Set

Let A

F and B

G be two generalized fuzzy soft set

over (U, E). Now A

F is called a generalized fuzzy soft

subset of B

G if

(i) ,BA

(ii) is a fuzzy subset of ,

(iii) )(, eFA is a fuzzy subset of )(eG

i.e. EeeGeF ),()(

We write A

F ~B

G

3.3. Example

We consider the GFSS A

F given in Example 3.2

and let EeeeeB 4321 ,,, Let ]1,0[: IA be

given as follows:

,3.0)(,4.0)( 21 ee 9.0)(,7.0)( 43 ee

We define B

G as follows:

)4.0,8.0/,9.0/,6.0/()( 3211 SSSeGB

,

)3.0,3.0/,1.0/,6.0/()( 3212 SSSeGB

,

)7.0,5.0/,7.0/,8.0/()( 3213 SSSeGB

,

)9.0,8.0/,9.0/,6.0/()( 3214 SSSeGB

.

ThenA

F ~B

G

3.4. Intersection of Generalized Fuzzy Soft Sets

The intersection of two GFSS A

F and B

G over

(U, E) is denoted by BA

GF ~ and defined by GFSS

IIBAK UBA

: such that for each

EBABAe , and

,)(),()( eeKeKBA

)(*)()(),(*)()( Where eeeeGeFeK . In

order to avoid degenerate case, we assume here that

BA .

3.5. Example

From Example 3.2 and 3.3

)16.0,56.0/,45.0/,18.0/()( 3211 SSSeKBA

,

)35.0,15.0/,07.0/,48.0/()( 3213 SSSeKBA

,

)72.0,56.0/,81.0/,18.0/()( 3214 SSSeKBA

3.6. Remark

Let us define ,)(),()( eeKeKBA

where

)(eK )}(),(min{ eGeF

)(e )}.(),(min{ ee

Then

)4.0,7.0/,5.0/,3.0/()( 3211 SSSeKBA

,

)5.0,3.0/,1.0/,6.0/()( 3213 SSSeKBA

,

)8.0,7.0/,9.0/,3.0/()( 3214 SSSeKBA

3.7. Union of Generalized Fuzzy Soft Sets

The union of two GFSS A

F and B

G over (U, E) is

denoted by BA

GF ~ and defined by GFSS

IIBAH UBA

:

such that for each

EBABAe , and

Where

BAeeeeGeF

ABeeeG

BAeeeF

eHBA

if )),()(),()((

if )),(),((

if )),(),((

)(

3.8. Example From Example 3.2and 3.3

)64.0,94.0/,95.0/,72.0/()( 3211 SSSeHBA

,

)30.0,30.0/,10.0/,60.0/()( 3212 SSSeHBA

,

)85.0,65.0/,73.0/,92.0/()( 3213 SSSeHBA

,

)98.0,94.0/,99.0/,72.0/()( 3214 SSSeHBA

3.9. Remark

If we consider )},(),(max{)()( eGeFeGeF

)}.(),(max{)()( eeee

Then

)4.0,8.0/,9.0/,6.0/()( 3211 SSSeHBA

,

)3.0,3.0/,1.0/,6.0/()( 3212 SSSeHBA

,

)7.0,5.0/,7.0/,8.0/()( 3213 SSSeHBA

,

)9.0,8.0/,9.0/,6.0/()( 3214 SSSeHBA

3.10. Generalized Null Fuzzy Soft Set A GFSS is said to be a generalized null fuzzy soft

set, denoted byA

, if IIA UA: such that

)(),()( eeFeA

where

EAeeeF ,0)(,0)(

It is clear from our definition that the generalized

fuzzy soft null set is not unique in our way, it would

depend upon the set of parameters under consideration.


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3.11. Generalized Absolute Fuzzy Soft Set A GFSS is said to be a generalized absolute fuzzy

soft set, denoted by ~

, if IIAA U :~ such that

)(),()( eeFe Where

EAeeeF ,1)(,1)(

It is clear from our definition that the generalized

fuzzy soft absolute set is also not unique in our way, it

would depend upon the set of parameters under

consideration.

3.12. Complement of a Generalized Fuzzy Soft Set

Let

AF be a GFSS over (U, E). Then the

complement of A

F , denoted by cA

F and is defined

byAcA

GF , where

),()( and )()( eFeGeecAAc .Ae

3.13. Proposition

If A

F be a GFSS over (U, E), then

(i) AAA

FF ~

(ii) AAA

F ~

(iii) ~~~

AF

(iv) AAFF

~~

(v) AAAFFF

~

(vi) AAA

FFF ~

3.14. Remark The results (v) and (vi) above take the following forms

if we take max and min operations.

(vii) AAA

FFF ~

(viii) AAA

FFF ~

3.15. Proposition

If CBA

HGF ,, be a GFSS over (U, E), then

(i) ABBA

FGGF ~~

(ii) ABBA

FGGF ~~

(iii) CBACBA

HGFHGF ~

)~

()~

(~

(iv) CBACBA

HGFHGF ~

)~

()~

(~

Proof: Since the t - norm function and t - conorm

functions are commutative and associative, therefore

the theorem follows.

3.16. Proposition

If BA

GF , be a GFSS over (U, E), then

(i)

CBCACBAGFGF )(

~)(~)

~(

(ii)

CBACBCAGFGF )

~(~)(

~)(

Proof: (i)

))(~

( eGFBA

BAeeeEGeF

ABeeeG

BAeeeF

if )),()(),()((

if )),(),((

if )),(),((

Therefore,

)()~

( eGF CBA

BAeeeEGeF

ABeeeG

BAeeeF

CC

CC

CC

if),))()((,))()(((

if )),(),((

if )),(),((

BAeeeEGeF

ABeeeG

BAeeeF

CCCC

CC

CC

if),))((*))((,))((*))(((

if )),(),((

if )),(),((

Again

))()(~

)(( eGF CBCA

BAeeeEGeF

ABeeeG

BAeeeF

CCCC

CC

CC

if ),))(())((,))(())(((

if )),(),((

if )),(),((

But )()(~)()( eGeFeGeF

It follows that, CBCACBA

GFGF )(~

)(~)~

(

(ii) This proof similarly follows.

3.17. Proposition

If BA


(i)

CBACBCAGFGF )

~(~)(

~)(

(ii)

CBCACBAGFGF )(

~)(~)

~(

Proof:

We have,

))()(~

)(( eGF CBCA


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BAeeeEGeF CCCC )))((*))((,))((*))(((

And

))(~

( eGFBA

BAeeeEGeF

ABeeeG

BAeeeF

if )),()(),()((

if )),(),((

if )),(),((

Therefore, BAe

)()

~( eGF CBA

BAeeeEGeF

ABeeeG

BAeeeF

CC

CC

CC

if),))()((,))()(((

if )),(),((

if )),(),((

BAeeeEGeF

ABeeeG

BAeeeF

CCCC

CC

CC

if),))((*))((,))((*))(((

if )),(),((

if )),(),((

follows.result the, Since BABA

(ii) This proof is similar to (i) above.

3.18. Proposition

If AA


(i)

CACACAAFGGF )(

~)()

~(

(ii)

CACACAAFGGF )(

~)()

~(

Proof: The proof is straight forward and follows from

definition.

3.19. Proposition The following results are valid if we take max and

min operations.

If CBA

HGF ,, be a GFSS over (U, E), then

(i) )~

(~ CBA

HGF

)~

(~

)~

(CABA

HFGF

(ii) )~

(~ CBA

HGF

)~

(~

)~

(CABA

HFGF

4. Relation on generalized Fuzzy Soft Sets

4.1. Generalized Fuzzy Soft Relation

Let BA

GF , be a GFSS over (U, E). Then

generalized fuzzy soft relation (in short GFSR) R from

AF to

BG is a function IIBAR U : defined

by

.),(),(~

)(),( BAeeeGeFeeR babB

aA

ba

4.2. Inverse of a Generalized Fuzzy Soft Relation

If R is a GFSR from A

F to B

G then 1R is

defined as BAeeeeReeR baabba ),(),,(),(1

4.3. Remark

If R is a GFSR from A

F to B

G then 1R is a

GFSR from B

G to A

F .

4.4. Proposition

If R and S be two fuzzy soft relations from A

F to

BG then

(i) RR 11)(

(ii) 11 SRSR

Proof:

(i) ),(),(),()( 111baabba eeReeReeR

Hence RR 11)(

(ii) Same as (i).

4.5. Composition of Generalized Fuzzy Soft Relation

Let R and S be two generalized fuzzy soft relations

from A

F to B

G andB

G to C

H respectively.

Then the composition of R and S is defined by

),(~

),(),)(( cbbaca eeSeeReeSR

4.6. Theorem Let R and S be two generalized fuzzy soft relations

from A

F to B

G andB

G to C

H respectively.

Then SR is GFSR from A

F to C

H

Proof:

)(~

)(),( bB

aA

ba eGeFeeR

.),( ))(*)()},(*)(({ BAeeeeeGeF bababa

)(~

)(),( cC

bB

cb eHeGeeS

.),( ))(*)()},(*)(({ CBeeeeeHeG cbcbcb

),)(( ca eeSR


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),(~

),( cbba eeSeeR

)},(*)(*)(*)(({ cbba eHeGeGeF

))(*)(*)(*)( cbba eeee

CBAeee cba ),,(

Now, )(*)()}(*)(*)(*)({ cacbba eHeFeHeGeGeF

And also

)(*)())(*)(*)(*)(( cacbba eeeeee

Hence

),)(( ca eeSR

),(~

),( cbba eeSeeR

)(~

)( cC

aA

eHeF

Thus SR is GFSR from A

F to C

H

4.7. Proposition

111)( RSSR where R and S be two fuzzy soft

relations from A

F to B

G andB

G to C

H

respectively.

Proof:

Let CeBeAe cba ,,

),()( 1ac eeSR

),)(( ca eeSR

),(~

),( cbba eeSeeR

),(~

),( bacb eeReeS

),(~

),( 11abbc eeReeS

),)(( 11ac eeRS

Hence 111)( RSSR .

4.8. Union and Intersection of Generalized Fuzzy

Soft Relations

Let R and S be two generalized fuzzy soft relations

from A

F to B

G . Then SRSR , are defined as

follows

),)(( ba eeSR

)},(),,(max{ baba eeSeeR

),)(( ba eeSR

BAeeeeSeeR bababa ),()},,(),,(min{

4.9. Proposition

Let R is a GFSR from A

F to B

G and S, T are

GFSR B

G to C

H . Then

(i) )()()( TRSRTSR

(ii) )()()( TRSRTSR

Proof:

(i) Let CeBeAe cba ,,

),(),().( cbcbba eeTeeSeeR

),(),(~

),( cbcbba eeTeeSeeR

)},(),,(min{~

),( cbcbba eeTeeSeeR

)},(~

),(),,(~

),(min{ cbbacbba eeTeeReeSeeR

)},)((),,)(min{( caca eeTReeSR

),))(()(( ca eeTRSR

Hence )()()( TRSRTSR .

(ii) Same as (i)

4.10. Proposition

Let R and S are GFSR from A

F to B

G , then

(i) 111)( SRSR

(ii) 111)( SRSR

Proof:

(i) ),()( 1ba eeSR

),)(( ab eeSR

)},(),,(min{ abab eeSeeR

)},(),,(min{ 11baba eeSeeR

),)(( 11ba eeSR

Hence 111)( SRSR

(ii) Same as (i)

5. Conclusion

We have put forward some new notions regarding

generalized fuzzy soft set theory. We have also given

some results and examples on generalized fuzzy soft

relation based on our new notion. Future work in this

regard would be required to study whether the notions

put forward in this paper yield a fruitful result.

6. References

[1] B. Ahmad and A. Kharal, “On Fuzzy Soft Sets”,

Advances in Fuzzy Systems, Volume 2009.

[2] B. Schweirer, A. Sklar, “Statistical metric space”,

Pacific Journal of Mathematics 10(1960), 314-334.


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[3] D. A. Molodtsov, “Soft Set Theory - First Result”,

Computers and Mathematics with Applications, Vol.

37, pp. 19-31, 1999

[4] H. L. Yang, “Notes On Generalized Fuzzy Soft

Sets”, Journal of Mathematical Research and

Exposition, Vol 31, No. 3, May - 2011, pp.567-570

[5] P. K. Maji, R. Biswas and A.R. Roy, “Fuzzy Soft

Sets”, Journal of Fuzzy Mathematics, Vol 9 , no.3,pp.-

589-602,2001

[6] P. Majumdar, S. K. Samanta, “Generalized Fuzzy

Soft Sets”, Computers and Mathematics with

Applications,59(2010), pp.1425-1432

[7] T. J. Neog, D. K. Sut, “On Fuzzy Soft Complement

and Related Properties” , Accepted for publication in

International Journal of Energy, Information and

communications (IJEIC), Japan.

[8] T. J. Neog, D. K. Sut, “On Union and Intersection

of Fuzzy Soft Sets”, Int.J. Comp. Tech. Appl., Vol 2 (5),

1160-1176


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