Upload
duongdiep
View
224
Download
1
Embed Size (px)
Citation preview
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.
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
583
ISSN:2229-6093
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 .]
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
584
ISSN:2229-6093
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
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
585
ISSN:2229-6093
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.
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
586
ISSN:2229-6093
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
GF , be a GFSS over (U, E), then
(i)
CBACBCAGFGF )
~(~)(
~)(
(ii)
CBCACBAGFGF )(
~)(~)
~(
Proof:
We have,
))()(~
)(( eGF CBCA
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
587
ISSN:2229-6093
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
GF , be a GFSS over (U, E), then
(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
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
588
ISSN:2229-6093
),(~
),( 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.
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
589
ISSN:2229-6093
[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
Tridiv Jyoti Neog et al ,Int.J.Computer Technology & Applications,Vol 3 (2), 583-591
590
ISSN:2229-6093