Available online throug
ISSN 2229 – 5046
GENERALIZED FUZZY SOFT SETS: A NEW APPROACH
Manash Jyoti Borah*
Department of Mathematics, Bahona College, Jorhat, Assam, India.
(Received On: 04-07-14; Revised & Accepted On: 22-07-14)
ABSTRACT
I
n this paper, we redefined null, absolute, union, and intersection of generalized fuzzy soft sets and study some of their properties. We also defined disjunctive sum, difference and product of two generalized fuzzy soft sets and their basic properties.Key Words: Fuzzy soft sets, Generalized fuzzy soft set, Disjunctive sum, Difference and Product of fuzzy soft set.
1. INTRODUCTION
Maji et al. [4] 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. In 2011, Neog and Sut [6] put forward some more propositions regarding fuzzy soft set theory. In 2009 Majumder and Samanta [5] initiated another important part, known as Generalized Fuzzy Soft Set Theory. While generalizing the concept of Fuzzy Soft Sets, Majumder and Samanta [5] considered the same set of parameter. But in 2012 Borah, Neog and Sut [3] considering generalized fuzzy soft sets different sets of parameters. In our work, we redefined null, absolute, union, and intersection of generalized fuzzy soft sets and study some of their properties. We also defined disjunctive sum, difference and product of two generalized fuzzy soft sets and their basic properties.
2. PRELIMINARIES
In this section, we recall some basic concepts and definitions regarding fuzzy soft sets and generalized fuzzy soft sets.
Definition 2.1: [4] A pair (F, A) is called a fuzzy soft set over U where F:A→P~(U)is a mapping from A intoP~(U).
Definition 2.2: [4] A soft set (F, A) over U is said to be null fuzzy soft set denoted by ϕ if ∀
ε
∈A,F(ε
) is the null fuzzy set 0 of U where0(x)=0∀x∈U.Definition 2.3: [4] A soft set (F, A) over U is said to be absolute fuzzy soft set denoted by A~ if ∀ε∈A,F(ε) is the null fuzzy set 1 of U where1(x)=1∀x∈U
Definition 2.4: [4] 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) A⊆B
(ii) For all ε∈A,F
( )
ε ⊆G( )
ε and is written as (F , A) ⊆~ (G, B).Definition 2.5: [1] A binary operation *:[0,1]×[0,1]→[0,1] is continuous t-norm if * satisfies the following conditions. (i) * is commutative and associative
(ii) * is continuous (iii) a * 1= a ∀a∈[0,1]
(iv) a*b≤c*d Whenever a≤c,b≤d and a,b,c,d∈[0,1]
Corresponding author: Manash Jyoti Borah*
Definition 2.6: [1] A binary operation ◊:[0,1]×[0,1]→[0,1] is continuous t-conorm if ◊satisfies the following conditions:
(i) ◊ is commutative and associative (ii) ◊ is continuous
(iii) a ◊0 = a ∀a∈[0,1]
(iv) a◊b≤c◊d whenever a≤c,b≤d and a,b,c,d∈[0,1]
Definition 2.7: [5] Let U=
{
x1,x2,x3,...,xn}
be the universal set of elements and E={
e1,e2,e3,...em}
be theuniversal set of parameters. The pair (U, E) will be called a soft universe. Let F:E→IU and
µ
be a fuzzy subset ofE, i.e. µ:E→I=
[ ]
0,1 , where IUis the collection of all fuzzy subsets of U. Let Fµbe the mapping Fµ:E→IU×I be a function defined as follows:)
(
( ), ( ) )(e F e e
Fµ = µ , where F(e)∈IU. Then Fµis called generalized fuzzy soft sets over the soft universe (U,E). Here for each parameter ei, Fµ(ei)=
(
F(ei),µ
(ei))
indicates not only the degree of belongingness of the elements of U in F( )
ei but also the degree of possibility of such belongingness which is represented byµ( )
ei .Definition 2.8: [5] Let Fµ and Gδ be two GFSS over (U, E).
Now, Fµ is said to be a generalized fuzzy soft subset of Gδ if (i) µ is a fuzzy subset of δ
(ii) F(e) is also a fuzzy subset of G(e)∀e∈E.
In this case, we write Fµ ⊆~ Gδ
Definition 2.9: [5] Let Fµ be a GFSS over (U, E). Then the complement of Fµ, denoted by Fµc and is defined by
δ µ G
F c= , where
δ
(e)=µ
c(e)andG(e)=Fc(e),∀e∈E.Definition 2.10: [5] The union of two GFSS Fλand Gµover (U, E) is denoted by Fλ ∪~Gµand defined by GFSS
I I E
Hδ : → U× such that for each e∈E
)) ( ) ( ), ( ) ( ( )
(e F e Ge e e Hδ = ◊ λ ◊µ
Definition 2.11: [5] The intersection of two GFSS Fλand Gµover (U, E) is denoted by Fλ ∩~Gµand defined by GFSS Hδ :E→IU ×I such that for each e∈E
)) ( * ) ( ), ( * ) ( ( )
(e F e G e e e
Hδ = λ µ
Definition 2.12: [5] A GFSS is said to be a generalized null fuzzy soft set, denoted byθφ, if
θ
φ:E→IU ×I such that(
( ), ( ))
)
(e F e φ e
θφ = Where F(e)=0,φ(e)=0,∀e∈E
Definition 2.13: [5] A GFSS is said to be a generalized absolute fuzzy soft set, denoted byΑ~λ, if A~λ:E→IU ×I
such that
(
(
),
(
)
)
)
(
e
F
e
e
A
φ=
λ
Where F(e)=1,λ(e)=1,∀e∈EDefinition 2.14: [3] Let X ={x1,x2,x3,...,xn} be the universal set of elements and E={e1,e2,e3,...,em}be
the set of parameters. Let A⊆Eand F:A→IUand λbe a fuzzy subset of A i.e.
λ
:A→I=[0,1], where IUis the collection of all fuzzy subsets of U. Let F:A→IU×I be a function defined as follows:(
( ), ( ))
)(e F e e
FλA = λ , whereF(e)∈IU.Then FλAis called a generalized fuzzy soft set (GFSS) over (U, E). Here for each parameter , ( i)
A i F e
e λ indicates not only degree of belongingness of the elements of U in F(ei)but also
Example 2.1: Let U =
{
S1,S2,S3}
, E={
e1,e2, e3, e4} and A={
e1,e3,e4}
⊆E.We define FλAas follows:
{
/0.3, /0.5, /0.7}
,0.4) ()
(e1 S1 S2 S3
FλA = ,
{
/0.6, /0.1, /0.3}
,0.5) ()
(e3 S1 S2 S3
FλA = ,FλA(e4)=(
{
S1/0.3,S2/0.9,S3/0.7}
,0.8) is the generalized fuzzy soft set.Definition 2.15: [3] Let FλAand GµBbe two generalized fuzzy soft set over (U, E). Now FλAis called a generalized fuzzy soft subset of GµBif
(i) A⊆B,
(ii) λ is a fuzzy subset of µ,
(iii) ∀λ∈A,F(e)is a fuzzy subset of G(e) i.e.F(e)⊆G(e),∀e∈E
We write FλA ⊆~ GµB
Example 2.2: We consider the GFSS FλAgiven in Example 2.1 and let B=
{
e1,e2,e3,e4}
⊆E We define GµBas follows:{
/0.6, /0.9, /0.8}
,0.4) ()
(e1 S1 S2 S3
GµB = ,
{
/0.6, /0.1, /0.3}
,0.3) ()
(e2 S1 S2 S3
GµB = ,
{
/0.8, /0.7, /0.5}
,0.7) ()
(e3 S1 S2 S3
GµB = ,G (e4) (
{
S1/0.6,S2/0.9,S3/0.8}
,0.9)B =
µ
ThenFλA ⊆~ GµB.
Definition 2.16: [3] Let FµA be a GFSS over (U, E). Then the complement ofFµA, denoted by FµAc and is defined
byFµAc =GδA, where δ(e)=µc(e)andGA(e)=FAc(e),∀e∈A.
3. GENARALIZED 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 null, absolute, union, intersection, complement etc. of generalized fuzzy soft sets in the following manner:
Definition 3.1: A generalized fuzzysoft sets (GFSS) is said to be a generalized null fuzzy soft set denoted byθφA, if
I I A U A → ×
:
φ
θ
such that θφA(e)=(
F(e),φ(e))
whereF
(
e
)
=
0
,
φ
(
e
)
=
1
,
∀
e
∈
A
⊆
E
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.
Definition 3.2: A GFSS is said to be a generalized absolute fuzzy soft set, denoted byΑ~λ, if ~Aλ:A→IU×I such that
(
(
),
(
)
)
)
(
~
e
e
F
e
A
φ=
λ
WhereF
(
e
)
=
1
,
λ
(
e
)
=
0
,
∀
e
∈
A
⊆
E
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.
Definition 3.3: The union of two GFSS FλAand GµBover (U, E) is denoted by FλA∪~GµBand defined by GFSS
I I B A
HδA∪B: ∪ → U× such that for each e∈A∪B andA,B⊆E
where
∩
∈
∗
◊
−
∈
−
∈
=
∪
B
A
e
e
e
e
G
e
F
A
B
e
e
e
G
B
A
e
e
e
F
e
H
A Bif
)),
(
)
(
),
(
)
(
(
if
)),
(
),
(
(
if
)),
(
),
(
(
)
(
µ
λ
µ
λ
Example 3.1: From Example 2.1and 2.2
{
/
0
.
72
,
/
0
.
95
,
/
0
.
94
}
,
0
.
16
)
(
)
(
e
1S
1S
2S
3H
δAB=
,H
δAB(
e
2)
=
(
{
S
1/
0
.
60
,
S
2/
0
.
10
,
S
3/
0
.
30
}
,
0
.
30
)
,{
/
0
.
92
,
/
0
.
73
,
/
0
.
65
}
,
0
.
35
)
(
)
(
e
3S
1S
2S
3H
δAB=
,H
(
e
4)
(
{
S
1/
0
.
72
,
S
2/
0
.
99
,
S
3/
0
.
94
}
,
0
.
72
)
B A
=
δ
Remark 3.1: If we consider F(e)◊G(e)=max{F(e),G(e)},
λ
(
e
)
*
µ
(
e
)
=
min{
λ
(
e
),
µ
(
e
)}.
Then
{
/
0
.
6
,
/
0
.
9
,
/
0
.
8
}
,
0
.
4
)
(
)
(
e
1S
1S
2S
3H
δA∪B=
,H (e2) ({
S1/0.6,S2/0.1,S3/0.3}
,0.3)B A =
δ ,
{
/
0
.
8
,
/
0
.
7
,
/
0
.
5
}
,
0
.
5
)
(
)
(
e
3S
1S
2S
3H
δAB=
,H
δAB(
e
4)
=
(
{
S
1/
0
.
6
,
S
2/
0
.
9
,
S
3/
0
.
8
}
,
0
.
8
)
Definition 3.4: The intersection of two GFSS FλAand GµBover (U, E) is denoted by FλA∩~GµBand defined by GFSS KδA∩B:A∩B→IU ×Isuch that for each e∈A∩B andA,B⊆E KδA∩B(e)=
(
K(e),δ(e)),
)
(
)
(
)
(
),
(
*
)
(
)
(
Where
K
e
=
F
e
G
e
δ
e
=
λ
e
◊
µ
e
. In order to avoid degenerate case, we assume here thatϕ
≠ ∩BA .
Example 3.2: From Example 2.1 and 2.2
{
/
0
.
18
,
/
0
.
45
,
/
0
.
56
}
,
0
.
64
)
(
)
(
e
1S
1S
2S
3K
δA∩B=
,K
(
e
3)
(
{
S
1/
0
.
48
,
S
2/
0
.
07
,
S
3/
0
.
15
}
,
0
.
85
)
B A∩
=
δ ,
{
/
0
.
18
,
/
0
.
81
,
/
0
.
56
}
,
0
.
98
)
(
)
(
e
4S
1S
2S
3K
δA∩B=
Remark 3.2: Let us define KδA∩B(e)=
(
K(e),δ(e)),
where K(e)=min{F(e),G(e)},
δ(e)
=
max{
λ
(
e
),
µ
(
e
)}.
Then
{
/
0
.
3
,
/
0
.
5
,
/
0
.
7
}
,
0
.
4
)
(
)
(
e
1S
1S
2S
3K
δA∩B=
,{
/
0
.
6
,
/
0
.
1
,
/
0
.
3
}
,
0
.
5
)
(
)
(
e
3S
1S
2S
3K
δA∩B=
,K
δA∩B(
e
4)
=
(
{
S
1/
0
.
3
,
S
2/
0
.
9
,
S
3/
0
.
7
}
,
0
.
8
)
Proposition 3.1: If FλA be a GFSS over (U, E), then (i) FλA ~θφA=FλA
(ii) FλA ~θφA =θφA
(iii) FλA~Α~α =Α~α
(iv) FλA~Α~α =FλA
Proposition 3.2: If FλA,GµB,HδCbe a GFSS over (U, E), then (i) FλA~GµB =GµB~FλA
(ii) FλA~GµB =GµB~FλA
(iii) FλA~(GµB~HδC)=(FλA~GµB)~HδC
C B A C
B A
H G F H
G
Fλ ~( µ ~ δ )=( λ ~ µ )~ δ
Proof: Since the t - norm function and t - co norm functions are commutative and associative, therefore the theorem follows.
Proposition 3.3: If FλA,GµAbe a GFSS over (U, E), then (i) (FλA~GµA)C =(GµA)C~(FλA)C
(ii) (FλA~GµA)C =(GµA)C~(FλA)C
Proposition 3.4: The following results are valid if we take max and min operations. If FλA,GµB,HδCbe a GFSS over (U, E), then
(i) FλA ~(GµB ~HδC) =(FλA~GµB)~(FλA~HδC) (ii)FλA ~(GµB ~HδC) =(FλA~GµB)~(FλA~HδC)
Definition 3.5: The equal of two GFSS FλAand GµBover (U, E) is denoted by
F
λA=
G
µB,)
(
)
(
),
(
)
(
Where
F
e
=
G
e
λ
e
=
µ
e
Proposition 3.5: If FλA,GµB,HδCbe a GFSS over (U, E), then
F
λA=
G
µB,
G
µB=
H
δC⇔
F
λA=
H
δCProof: Since
C B
B A
H
G
G
F
λ=
µ,
µ=
δTherefore
{
F
(
e
)
=
G
(
E
),
λ
(
e
)
=
µ
(
e
)
,{
G
(
e
)
=
H
(
E
),
µ
(
e
)
=
δ
(
e
)
Therefore
{
F
(
e
)
=
H
(
E
),
λ
(
e
)
=
δ
(
e
)
Hence
C A
H
F
λ=
δ.
4. SOME NEW OPERATIONS OF GENERALIZED FUZZY SOFT SETS
Definition 4.1: Let FλAand GµBbe two GFSS over (U, E).We define the disjunctive sum of FλAand GµBbe as the
generalized fuzzy soft set
K
δCover (U, E). Written as'
F
λA⊕
~
G
µB=
K
δC'
where
C
=
A
∩
B
≠
φ
,
∀
e
∈
C
and
A
,
B
⊆
E
K
(
e
)
(
K
(
e
),
(
e
)
)
,
C
δ
δ
=
))
(
)
(
),
(
)
(
max(
)
(
Where
K
e
=
F
e
∗
µ
e
G
e
∗
λ
e
δ
(
e
)
=
min(min(
G
(
e
)
◊
λ
(
e
),
F
(
e
)
◊
µ
(
e
))
Example 4.1: Let U =
{
S1,S2,S3}
and E={
e1,e2,e3, e4} be the set of parameters and A={
e1,e3,e4}
⊆E,{
e
e
e
}
E
B
=
1,
2,
3⊆
We define FλAand
G
µBas follows:{
/
0
.
3
,
/
0
.
5
,
/
0
.
7
}
,
0
.
4
)
(
)
(
e
1S
1S
2S
3F
λA=
,{
/0.6, /0.1, /0.3}
,0.5) ()
(e3 S1 S2 S3
FλA = ,FλA(e4)=(
{
S1/0.3,S2/0.9,S3/0.7}
,0.8){
/0.6, /0.9, /0.8}
,0.4) ()
(e1 S1 S2 S3
GµB = ,GµB(e2)=(
{
S1/0.6,S2/0.1,S3/0.3}
,0.3),{
/0.8, /0.7, /0.5}
,0.7) ()
(e3 S1 S2 S3
GµB = ,
Then
'
F
λA⊕
~
G
µB=
K
δC'
WhereC
=
A
∩
B
=
{e
1,
e
3}
{
=
C
K
δ{
}
1 1 2 3
( )
(
/ max{0.12, 0.24},
/ max{0.20, 0.36},
/ max{0.28, 0.32 ;
min{min(0.76, 0.58), min(0.94, 0.70), min(0.88, 0.82)})
C
{
}
3 1 2 3
( )
(
/ max{0.42, 0.40},
/ max{0.07, 0.35},
/ max{0.21, 0.25 ;
min{min(0.90, 0.88), min(0.85, 0.73), min(0.75, 0.79)})
C
K
δe
=
S
S
S
i.e
.
K
(
e
1)
(
{
S
1/
0
.
24
,
S
2/
0
.
36
,
S
3/
0
.
32
}
,
0
.
58
)
C
=
δ
{
/
0
.
42
,
/
0
.
35
,
/
0
.
25
}
,
0
.
73
)
(
)
(
e
3S
1S
2S
3K
δC=
Proposition 4.1:
Let FλAand GµBbe two GFSS over (U, E).Then the following results hold.
(i)
F
λA⊕
~
G
µB=
G
µB⊕
~
F
λA(ii)
F
λA⊕
~
(
G
µB⊕
~
H
δC)
=
(
F
λA⊕
~
G
µB)
⊕
~
H
δCProof: Since
(
( ), ( ))
)(e F e e
FλA = λ , where
e
∈
E
,
A
⊆
E
,
G
(
e
)
(
G
(
e
),
(
e
)
)
B
µ
µ
=
, wheree
∈
E
,
B
⊆
E
Now
(
F
λA⊕
~
G
µB)(
e
)
=
K
δC(
e
)
whereC
=
A
∩
B
,
K
(
e
)
(
K
(
e
),
(
e
)
)
C
δ
δ
=
))
(
)
(
),
(
)
(
max(
)
(
Where
K
e
=
F
e
∗
µ
e
G
e
∗
λ
e
,δ
(
e
)
=
min(min(
G
(
e
)
◊
λ
(
e
),
F
(
e
)
◊
µ
(
e
))
Let
(
G
µB⊕
~
F
λA)(
e
)
=
H
ζC(
e
)
whereC
=
A
∩
B
,
H
(
e
)
(
H
(
e
),
(
e
)
)
C
ζ
ζ
=
))
(
)
(
),
(
)
(
max(
)
(
Where
H
e
=
G
e
∗
λ
e
F
e
∗
µ
e
=
max(
F
(
e
)
∗
µ
(
e
),
G
(
e
)
∗
λ
(
e
))
=
K
(
e
)
ζ
(
e
)
=
min(min(
F
(
e
)
◊
µ
(
e
),
G
(
e
)
◊
λ
(
e
))
=
min(min(
G
(
e
)
◊
λ
(
e
),
F
(
e
)
◊
µ
(
e
))
=
δ
(
e
)
Therefore)
(
)
(
e
H
e
K
δC=
ζCHence
A B B A
F
G
G
F
λ⊕
~
µ=
µ⊕
~
λ .Proof of (ii) can be done in a similar way.
Proposition 4.2: A A A
F
F
λ⊕
~
θ
φ=
λProof:
(
( ), ( ))
)(e F e e
FλA = λ , where
e
∈
E
,
A
⊆
E
,
(
e
)
=
( )
0
,
1
A
φ
θ
, wheree
∈
E
,
B
⊆
E
Now
(
F
λA⊕
~
θ
φA)(
e
)
=
K
δA(
e
)
(
(
),
(
)
)
)
(
e
K
e
e
K
δA=
δ
)
(
)
0
),
(
max(
))
(
0
,
1
)
(
max(
)
(
Where
K
e
=
F
e
∗
∗
λ
e
=
F
e
=
F
e
δ
(
e
)
=
min(min(
0
◊
λ
(
e
),
F
(
e
)
◊
1
)
=
min{min(
λ
(
e
),
1
)}
=
λ
(
e
)
K
(
e
)
(
K
(
e
),
(
e
)
)
(
F
(
e
),
(
e
))
F
(
e
)
A A
λ
Hence
A A A
F
F
λ⊕
~
θ
φ=
λ.
Definition 4.2: Let FλAand GµBbe two GFSS over (U, E).We define the difference of FλAand GµBbe as the
generalized fuzzy soft set
H
δCover (U, E). Written as'
F
λAΘ
~
G
µB=
H
δC'
whereC
=
A
∩
B
≠
φ
,
∀
e
∈
C
and
A
,
B
⊆
E
H
(
e
)
(
H
(
e
),
(
e
)
)
,
C
δ
δ
=
)
(
)
(
)
(
Where
H
e
=
F
e
∗
µ
e
,δ
(
e
)
=
max(
G
(
e
)
◊
λ
(
e
))
Example 4.2: From Example 4.1
{
/
0
.
12
,
/
0
.
20
,
/
0
.
28
}
,
max{
0
.
76
,
0
.
94
,
0
.
88
})
(
)
(
e
1S
1S
2S
3K
δC=
{
/
0
.
42
,
/
0
.
07
,
/
0
.
21
}
,
max{
0
.
90
,
0
.
85
,
0
.
75
})
(
)
(
e
3S
1S
2S
3K
δC=
i.e.
{
/
0
.
12
,
/
0
.
20
,
/
0
.
28
}
,
0
.
94
)
(
)
(
e
1S
1S
2S
3K
δC=
{
/
0
.
42
,
/
0
.
07
,
/
0
.
21
}
,
0
.
90
)
(
)
(
e
3S
1S
2S
3K
δC=
Proposition 4.3 A A A
F
F
λΘ
~
θ
φ=
λProof:
(
( ), ( ))
)(e F e e
FλA = λ , where
e
∈
E
,
A
⊆
E
,
(
e
)
=
( )
0
,
1
A
φ
θ
, wheree
∈
E
,
B
⊆
E
Now
(
F
λAΘ
~
θ
φA)(
e
)
=
K
δA(
e
)
(
(
),
(
)
)
)
(
e
K
e
e
K
δA=
δ
)
(
1
)
(
)
(
Where
K
e
=
F
e
∗
=
F
e
,δ
(
e
)
=
max(
0
◊
λ
(
e
))
=
λ
(
e
)
Therefore
(
(
),
(
)
)
(
(
),
(
))
(
)
)
(
e
K
e
e
F
e
e
F
e
K
δA=
δ
=
λ
=
λAHence
A A A
F
F
λΘ
~
θ
φ=
λProposition 4.4: A A
A
F
λΘ
~
~
µ=
θ
φProof:
(
( ), ( ))
)(e F e e
FλA = λ , where
e
∈
E
,
A
⊆
E
,
(
)
( )
1
,
0
~
=
e
A
µNow
(
F
λAΘ
~
A
~
µ)(
e
)
=
K
δA(
e
)
(
(
),
(
)
)
)
(
e
K
e
e
K
δA=
δ
0
0
)
(
)
(
)
(
)
(
Where
K
e
=
F
e
∗
µ
e
=
F
e
∗
=
,δ
(
e
)
=
max(
1
◊
λ
(
e
))
=
1
Therefore
(
(
),
(
)
)
(
~
0
,
1
)
(
)
)
(
e
K
e
e
e
K
δA=
δ
=
=
θ
φAHence
A A
A
F
λΘ
~
~
µ=
θ
φ5. PROUCT OF GENERALIZED FUZZY SOFT SETS
Definition 5.1: The
∧
-Product of two GFSS FλAand GµBover (U, E) is denoted byF
λA∧
G
µBand defined byGFSS
H
δA∧B:
A
×
B
→
I
U×
I
such that for each(
α
,
β
)
∈
A
×
B
and
A
,
B
⊆
E
(
(
,
),
(
,
)
)
,
)
,
(
α
β
α
β
δ
α
β
δ
H
H
A∧B=
)
(
)
(
)
,
(
),
(
)
(
)
,
(
Where
H
α
β
=
F
α
∗
G
β
δ
α
β
=
λ
α
∗
µ
β
Example 5.1:
We define FλAandGµBas follows:
{
/
0
.
4
,
/
0
.
6
,
/
0
.
5
}
,
0
.
3
)
(
)
(
e
1S
1S
2S
3F
λA=
,F
(
e
2)
(
{
S
1/
0
.
2
,
S
2/
0
.
9
,
S
3/
1
.
0
}
,
0
.
7
)
A
=
λ
and
{
/
0
.
8
,
/
0
.
2
,
/
1
.
0
}
,
0
.
4
)
(
)
(
e
2S
1S
2S
3G
µB=
,G
µB(
e
3)
=
(
{
S
1/
0
.
2
,
S
2/
0
.
3
,
S
3/
0
.
6
}
,
0
.
2
)
Then
{
/
0
.
32
,
/
0
.
12
,
/
0
.
50
}
,
0
.
12
)
(
)
,
(
e
1e
2S
1S
2S
3H
δA∧B=
{
/
0
.
08
,
/
0
.
18
,
/
0
.
30
}
,
0
.
06
)
(
)
,
(
e
1e
3S
1S
2S
3H
δA∧B=
{
/
0
.
16
,
/
0
.
18
,
/
1
.
00
}
,
0
.
28
)
(
)
,
(
e
2e
2S
1S
2S
3H
δA∧B=
{
/
0
.
04
,
/
0
.
27
,
/
0
.
06
}
,
0
.
14
)
(
)
,
(
e
2e
3S
1S
2S
3H
δA∧B=
Definition 5.2: The V-product of two GFSS FλAand GµBover (U, E) is denoted by
F
λA∨
G
µBand defined by GFSSI
I
B
A
K
A∨B×
→
U×
:
δ such that for each
(
α
,
β
)
∈
A
×
B
and
A
,
B
⊆
E
(
(
,
),
(
,
)
)
,
)
,
(
α
β
α
β
δ
α
β
δ
K
K
A∨B=
)
(
)
(
)
,
(
),
(
)
(
)
,
(
Where
K
α
β
=
F
α
◊
G
β
δ
α
β
=
λ
α
◊
µ
β
Example 5.2: From Example 5.1
{
/
0
.
88
,
/
0
.
68
,
/
1
.
00
}
,
0
.
58
)
(
)
,
(
e
1e
2S
1S
2S
3K
δA∨B=
{
/
0
.
52
,
/
0
.
72
,
/
0
.
80
}
,
0
.
44
)
(
)
,
(
e
1e
3S
1S
2S
3K
δA∨B=
{
/
0
.
84
,
/
0
.
92
,
/
1
.
00
}
,
0
.
82
)
(
)
,
(
e
2e
2S
1S
2S
3K
δA∨B=
Definition 5.3: The
∧
-product of two GFSS FλAand GµBover (U, E) is denoted byF
λA∧
G
µBand defined byGFSS
H
δA∧B:
A
×
B
→
I
U×
I
such that for each(
α
,
β
)
∈
A
×
B
and
A
,
B
⊆
E
(
(
,
),
(
,
)
)
,
)
,
(
α
β
α
β
δ
α
β
δ
H
H
A∧B=
Where
H
(
α
,
β
)
=
F
(
α
)
∗
G
C(
β
),
δ
(
α
,
β
)
=
λ
(
α
)
∗
µ
C(
β
)
Example 5.3: From Example 5.1
{
/
0
.
08
,
/
0
.
48
,
/
0
.
00
}
,
0
.
18
)
(
)
,
(
e
1e
2S
1S
2S
3H
δA∧B=
{
/
0
.
32
,
/
0
.
42
,
/
0
.
20
}
,
0
.
24
)
(
)
,
(
e
1e
3S
1S
2S
3H
δA∧B=
{
/
0
.
04
,
/
0
.
72
,
/
0
.
00
}
,
0
.
42
)
(
)
,
(
e
2e
2S
1S
2S
3H
δA∧B=
{
/
0
.
16
,
/
0
.
63
,
/
0
.
40
}
,
0
.
56
)
(
)
,
(
e
2e
3S
1S
2S
3H
δA∧B=
Definition 5.4: The
∨
−
-product of two GFSS FλAand GµBover (U, E) is denoted by
F
λA∨
G
µB−
and defined by
GFSS
K
δA∨
−B:
A
×
B
→
I
U×
I
such that for each(
α
,
β
)
∈
A
×
B
and
A
,
B
⊆
E
(
(
,
),
(
,
)
)
,
)
,
(
α
β
α
β
δ
α
β
δ
K
K
A∨
−B=
Where
(
α
,
β
)
(
α
)
(
β
),
δ
(
α
,
β
)
λ
(
α
)
µ
(
β
)
C C
G
F
Example 5.4: From Example 5.1
{
/
0
.
52
,
/
0
.
92
,
/
0
.
50
}
,
0
.
72
)
(
)
,
(
e
1e
2S
1S
2S
3K
δA∨
−B=
{
/
0
.
88
,
/
0
.
88
,
/
0
.
70
}
,
0
.
86
)
(
)
,
(
e
1e
3S
1S
2S
3K
δA∨
−B=
{
/
0
.
36
,
/
0
.
98
,
/
1
.
00
}
,
0
.
88
)
(
)
,
(
e
2e
2S
1S
2S
3K
δA∨
−B=
{
/
0
.
84
,
/
0
.
97
,
/
1
.
00
}
,
0
.
94
)
(
)
,
(
e
2e
3S
1S
2S
3K
δA∨
−B=
Proposition 5.1:
If FλA,GµAbe a GFSS over (U, E), then (i)
F
λA∧
F
λA⊆
F
λA(ii)
F
λA∨
F
λA⊇
F
λA(iii)
F
λA∧
F
λA⊆
F
λA(iv)
F
λA∨
F
λA⊇
F
λA−
(v)
(
F
λA∨
G
µA)
C=
(
G
µA)
C∧
(
F
λA)
C(vi)
(
F
λA∧
G
µA)
C=
(
G
µA)
C∨
(
F
λA)
C(vii)
(
F
λA∨
G
µA)
C=
(
G
µA)
C∧
(
F
λA)
C −(viii)
(
F
λAG
µA)
C(
G
µA)
C∨
(
F
λA)
C −=
∧
Proof: The proof is straight forward and follows from definition.
6. CONCLUSION
In this paper, we redefined null, absolute, union, and intersection of generalized fuzzy soft sets and study some of their properties. Also we have put forward some new idea such as disjunctive sum, difference and product of two generalized fuzzy soft sets and their basic properties. It is hoped that our findings will help enhancing this study on generalized fuzzy soft sets.
ACKNOWLEDGEMENT
The author would like to thank the fund for university Grant Commission (UGC), India, for funding the research elaborated on in this paper. [Grant No. F.5-60/2013-14/MRP/NERO/17424]
7. REFERENCES
[1] B. Schweirer, A. Sklar, “Statistical metric space”, Pacific Journal of Mathematics 10(1960), 314-334.
[2] H. L.Yang, “Notes On Generalized Fuzzy Soft Sets”, Journal of Mathematical Research and Exposition, Vol. 31, No. 3, May - 2011, pp.567-570.
[3] M.J.Borah, T.J.Neog and D.K.Sut, “Some Results on Generalized Fuzzy Soft Sets”, International Journal of Computer Technology and Application, Vol. 3(2), 583-591.
[4] P. K. Maji, R. Biswas and A.R. Roy, “Fuzzy Soft Sets”, Journal of Fuzzy Mathematics, Vol. 9, no.3, pp.-589-602,2001.
[5] P. Majumdar, S. K. Samanta, “Generalized Fuzzy Soft Sets”, Computers and Mathematics with Applications, 59(2010), pp.1425-1432.
[6] T. J. Neog, D. K. Sut, “On Union and Intersection of Fuzzy Soft Sets”, Int.J. Comp. Tech. Appl., Vol. 2 (5), 1160-11.
[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.
Source of support: University Grant Commission (UGC), India, Conflict of interest: None Declared
