Database Reference
In-Depth Information
first performing a selection on the export fuzzy relation from
DB
i
followed by making the fuzzy tuples
global using the merge operation.
fs
f
s
fs
Lemma 1 2
.
:
∀
L
L G merge L
,
=
(
L
),
G
=
merge
(
L
,
,
L
).
σ
σ
σ
p
α β
,
p
α β
,
p
α β
,
1
j
,
,
nj
1
j
,
,
nj
j
1
j
nj
Proof:
g
fs
fs
fs
fs
{
}
∈
G
⇔ ∃ ∈
i
1
,
,
n g A
,
.
∈
L where A Attr
=
(
G
)
=
Attr L
(
)
⇔ ∈
g merge
(
L
,
,
L
).
σ
σ
σ
σ
p
α β
,
p
α β
,
p
α β
,
p
α β
,
j
ij
j
ij
1
j
nj
Lemma 1 3
.
∀
L
, ...,
L G
,
=
merge L
(
, ...,
L
),
1
j
nj
j
1
j
nj
fs
fs
fs
G
=
merge
(
L
, ...,
L
)
σ
σ
σ
p
α β
,
j
p
α β
,
1
j
p
α β
,
nj
fs
Proof:
g
G
⇔
σ
p
α β
,
j
fs
∃ ∈
i
{ , ..., },
1
n
g A
.
∈
L where A Attr G
=
(
)
=
Attr L
(
)
⇔
σ
p
α β
,
ij
j
ij
fs
fs
, ...,
L
).
g merge
∈
(
L
σ
σ
p
α β
,
1
j
p
α β
,
nj
A global projection operation with
sameDB
option is equivalent to projecting the required attributes
from the export relations first followed by integrating the projected export relations. Hence the global
FTS
project
operation is consistent as given below.
Lemma 1 4
.
∀
L
, ...,
L G
,
=
merge L
(
, ...,
L
),
1
j
nj
j
1
j
nj
fs
fs
fs
sameDB
sameDB
sameDB
G
=
merge
(
L
, ...,
L
)
π
π
π
A
,
α β
,
j
A
,
α β
,
1
j
A
,
α β
,
nj
fs
fs
Proof: Let
G
=
(
sameDB
G
)
and
me i
L
=
(
L
).
π
π
A
,
α β
,
j
A
,
α β
,
ij
g G and g s DB for so
∈
.
=
∈
{ , ..., }
1
n
⇔
i
fs
sameDB
g
∈
G and g s DB for some i
.
=
∈
{ , ...
1
, }
n
⇔
π
A
,
α β
,
j
i
f
g A
.
∈
L for some i
∈
{ , ..., }
1
n
⇔
π
A
,
α β
µ
,
ij
g
=
( .
g A
,
( . ),
g A DB
)
∧
µ
( . )
g A
β
for some i
∈
{ , ..., }
1
n
⇔
L
i
L
g L
g L
g merge
∈ ⇔
∈
f
(
)
⇔
π
A
,
α β
,
ij
f
f
(
L
, ...,
L
).
∈
π
π
A
,
α β
,
1
j
A
,
α β
,
nj
Using similar proof techniques, we have proved that global FTS
join
,
union, intersection
, and
minus
operations are also consistent with the respective fuzzy relational operations on the export fuzzy rela-
tions as given below:
