Database Reference
In-Depth Information
(
( .
t A t A
,
. )
( . ) max(
t A
( . ),
t A
(
t A
. ))
( .
t s
t s
.
))
µ
α µ
∧
=
µ
µ
β
∧
=
≠ ∗ ⇔
EQ
T
Z
Z
T T
T
(
µ
( .
t A t A
,
. )
α µ
∧
( . ) max(
t A
=
µ
( . ),
t A
µ
(
t A
. ))
β
∧
( .
t
s
=
t s
.
≠ ∗ ∨
))
EQ
R
Z
Z
R R
R
(
( .
t A t A
,
. )
( . ) max(
t A
( . ),
t A
( . ))
t A
( .
t s
t s
.
))
µ
α µ
∧
=
µ
µ
β
∧
=
≠ ∗ ∨
EQ
S
Z
Z
S
S
S
(
µ
( .
t A t A
,
. )
α µ
∧
( . ) max(
t A
=
µ
( . ),
t A
µ
(
t A
. ))
β
∧
( .
t s
=
t s
.
≠ ∗ ⇔
))
EQ
T
Z
Z
T
T
T
(
( .
t A t A
,
. )
( . ) max(
t A
( .
t
A
),
(
t A
. ))
( .
t s
t s
.
))
µ
α µ
∧
=
µ
µ
β
∧
=
≠ ∗ ∨
EQ
R
Z
Z
R R
R
(
µ
( .
t A t A
,
. )
α µ
∧
( . ) max
t A
=
(
µ
( . ),
t A
µ
(
t A
. ))
β
∧
( .
t s
=
t s
.
≠ ∗ ⇔
))
EQ
Y
Z
Z
Y Y
Y
f
s
sameDB
( .
t A
,
µ
( . ), . )
t A t s
∈
∪
(
R
Y
)
⇔
Z
α β
,
fs
sameDB
fs
same
( .
t A
,
µ
( . ), . )
t A t s
∈
(
R
(
S
DB
T
))
∪
∪
Z
α β
,
α β
,
fs
anyDB
α β
,
is associative. i.e.
Z
fs
anyDB
fs
anyDB
fs
anyDB
fs
anyDB
Theorem:
È
= ∪
(
R
S
)
∪
T
= ∪
R
(
S
∪
T
)
α β
,
α β
,
α β
,
α β
,
fs
anyDB
α β
,
can have either non-`*' or `*' as its source value.
Case 1: Resultant tuple with its source value non-`*' (say `s'). It can be shown by using the procedure
similar to that of previous proof that:
Proof:
Tuples in the result of È
fs
anyDB
fs
anyDB
fs
anyDB
fs
anyDB
( .
t A
,
µ
( . ), . )
t A t s
∈
(
R
∪
S
)
∪
T
⇔
( .
t A
,
µ
( . )
t A
, . )
t s
∈ ∪
R
(
S
∪
T
)
α β
,
α β
,
α β
,
α β
,
Z
Z
Case 2: Resultant tuple with `*' value of its source attribute.
Proof: Let S, R and T be FTS relations of same arity and domain of
i
th
attribute of S,R,T is the same.
In case of
anyDB
option, result must have non-`*' source attribute value.
fs
anyDB
= ∪
fs
anyDB
= ∪
α β
Let FTS relation Let FTS relation
X
(
R
S
)
and FTS relation
Y
S
T
)
. Hence,
α β
,
( .
t A
,
µ
µ
( . ), )
t A
∗ ∈ ⇔
∗ ∈
Z
Z
fs
sameDB
( .
t A
,
( . ), )
t A
(
X
T
)
⇔
∪
α β
Z
,
f
f
f
t A
.
∈
(
X
T
)
∧
µ
( . )
t A
β
∧
( .
t s
= ∗ ∨
t s
.
≠
t s
. ))
⇔
π
A
α β
π
∪
,
α β
,
,
A
,
α β
,
Z
X
T
f
f
f
f
f
t A
.
∈
(
R
S
T
)
∧
µ
( . )
t A
β
∧
( .
t s
= ∗ ∨
t s
.
≠
t
.
s
≠
t s
. ))
⇔
π
α β
π
α β
π
∪
∪
A
,
α β
,
,
A
,
α β
,
,
A
,
α β
,
Z
R
S
T
f
f
f
t A
.
∈
(
R
Y
)
∧
µ
( . )
t A
β
∧
( .
t s
= ∗ ∨
t
.
s
≠
t s
. ))
⇔
π
α β
π
∪
A
,
α β
,
,
A
,
α β
,
Z
R
Y
fs
sameDB
( .
t A
,
µ
µ
( . ), )
t A
∗ ∈
(
R
Y
)
⇔
∪
Z
α β
,
fs
sameDB
fs
sameDB
( .
t A
,
( . ),
t A
∗ ∈
∪
)
(
R
(
S
T
)).
∪
Z
α β
,
α β
,
fs
sameDB
α β
fs
anyDB
α β
,
and are associative, however, they are mutually non-associative.
Although È
and È
,
fs
sameDB
fs
anyDB
fs
sameDB
fs
anyDB
Theorem:
(
R
∪
S
)
∪
T
≠ ∪
R
(
S
∪
T
)
α β α β α β α β
Proof:
This can be proved using a counter example given in Figure 7.
Similarly properties related with other FTS relational operations can also be proved.
,
,
,
,
