Database Reference
In-Depth Information
1
, id
B
]:
1
where
[⊥
A
‡
B
→
B,
[
id
A
,
⊥
]:
A
‡
B
→
A
. In fact, in the proof of
Lemma
9
, we have shown that
i
1
:
→
A
B
B
A
is an isomorphism. Let us
1
1
, id
B
]:
A
‡
B
→
A
B
is equal to the isomorphism
show that
[
id
A
,
⊥
,
⊥
1
1
,id
B
]
B
. In fact,
[
id
A
,
⊥
,
⊥
id
A
id
B
id
A
id
B
:
A
‡
B
→
A
={
id
A
, id
B
}=
and
1
1
, id
B
]=
hence, from Definition
23
,
[
id
A
,
⊥
,
⊥
id
A
id
B
. Consequently, we
can define the composition
i
1
◦
id
A
,
1
,
⊥
1
, id
B
:
(id
B
◦
⊥
→
→
→
id
A
)
A
‡
B
A
B
B
A
B
‡
A,
and to show that it is equal to
i
1
as follows:
i
1
◦
id
A
,
1
,
⊥
1
, id
B
(id
B
id
A
)
◦
⊥
◦
⊥
1
, id
B
,
id
A
,
1
◦
id
A
,
1
,
⊥
1
, id
B
=
(id
B
id
A
)
⊥
⊥
=
id
B
◦
⊥
1
, id
B
, id
A
◦
id
A
,
1
◦
id
A
,
1
,
⊥
1
, id
B
⊥
⊥
=
⊥
1
, id
B
,
id
A
,
1
◦
id
A
,
1
,
⊥
1
, id
B
⊥
⊥
(
from case 1 in Example
19
)
=
⊥
1
, id
B
◦
id
A
,
⊥
1
,
⊥
1
, id
B
◦
⊥
1
, id
B
,
id
A
,
1
◦
id
A
,
1
,
id
A
,
1
◦
⊥
1
, id
B
⊥
⊥
⊥
=
⊥
id
B
,
,
⊥
1
1
,
id
A
=
⊥
1
, id
B
,
id
A
,
1
=
⊥
i
1
.
Claim 3. (Associativity)
A
‡
(B
‡
C)
A
(B
‡
C)
A
(B
C) (
from Lemma
12
)
A
B
C
=
(A,B,C) (
from Lemma
9
)
‡
(A,B,C)
=
A
‡
B
‡
C
from Lemma
12
)
1
1
, id
C
]:
Similarly to the proof above,
id
A
[
id
B
,
⊥
]
,
[⊥
A
‡
(B
‡
C)
→
A
‡
B
‡
C
represents this isomorphism.
3.4
Equivalence Relations in
DB
Category
We can introduce a number of different equivalence relations for instance-databases:
•
Identity
relation—two instance-databases (sets of relations)
A
and
B
are identical
when the set identity
A
=
B
holds.
•
Isomorphism
relation '
'in
DB
.
•
Behavioral equivalence
relation '
' in Definition
19
—two instance-databases
A
and
B
are behaviorally equivalent when each view obtained from a database
A
can also be obtained from a database
B
, and viceversa.
≈
