Database Reference
In-Depth Information
This projection is given by the following logical equivalence
r(x
i
1
,...,x
i
k
)
⇔∃
x
j
1
...x
j
n
r
1
(x
1
,...,x
ar(r
1
)
),
where
n
=
ar(r
1
)
−
k
and for all 1
≤
m
≤
n
,
j
m
/
∈{
i
1
,...,i
k
}
, so that
r
=
∃
.
4. Selection is a unary operation written as _ WHERE
C
, where a condition
C
is a finite-l
e
ngth logical formula that consists of atoms '
(name
i
θ nam
e
j
)
'or
'
(name
i
θ d)
', with built-in predicates
θ
∈
Σ
θ
⊇{
x
j
1
...x
j
n
r
1
(x
1
,...,x
ar(r
1
)
)
.
=
,>,<
}
, a constant
d
, and
the logical operators
(NOT), such that for a relation
r
1
and
name
i
,
name
j
the names of its columns, we define the relation
r
by
r
1
WHERE
C,
∧
(AND),
∨
(OR) and
¬
=
as the relation with
atr(r)
atr(r
1
)
and the function
nr
r
equal to
nr
r
1
, where
r
for which
C
is satisfied.
This selection is given by the following logical equivalence:
is composed by the tuples in
r
1
r
1
(x
1
,...,x
ar(r
1
)
)
C(
x
)
,
r(x
i
1
,...,x
i
k
)
⇔
∧
where
C(
x
)
is obtained by substitution of each
name
i
=
nr
r
1
(j)
(of the
j
th col-
umn of
r
1
) in the formula
C
by the variable
x
j
, so that
r
=
r
1
(x
1
,...,x
ar(r
1
)
)
∧
C(
x
)
.
4.1. We assume as an
iden
ti
ty
un
ary operation
, the operation _ WHERE
C
when
C
is the atomic condition 1
.
1 (i.e., a tautology).
5. Union is a binary operation written as _ UNION _ such that for two union-
compatible relations
r
1
and
r
2
, we define the relation
r
by:
r
1
UNION
r
2
, where
=
r
r
1
∪
r
2
, with
atr(r)
=
atr(r
1
)
, and the functions
atr
r
=
atr
r
1
, and
nr
r
=
nr
r
1
. This union is given by the following logical equivalence:
r(x
1
,...,x
n
)
⇔
r
1
(x
1
,...,x
n
)
r
2
(x
1
,...,x
n
)
,
∨
where
n
=
ar(r)
=
ar(r
1
)
=
ar(r
2
)
, so that
r
2
(x
1
,...,x
n
)
.
6. Set difference is a binary operation written as _ MINUS_ such that for two union-
compatible relations
r
1
and
r
2
, we define the relation
r
by
r
1
MINUS
r
2
, where
=
r
1
(x
1
,...,x
n
)
r
∨
r
{
t
|
t
∈
r
1
such that
t
/
∈
r
2
}
, with
atr(r)
=
atr(r
1
)
, and the functions
atr
r
=
nr
r
1
.
Let
r
1
and
r
2
be the predicates (relational symbols) for these two relations.
Then their difference is given by the following logical equivalence:
r(x
1
,...,x
n
)
⇔
r
1
(x
1
,...,x
n
)
∧¬
r
2
(x
1
,...,x
n
)
,
atr
r
1
, and
nr
r
=
=
=
=
where
n
ar(r)
ar(r
1
)
ar(r
2
)
, and hence
r
1
(x
1
,...,x
n
)
r
2
(x
1
,...,x
n
)
.
r
=
∧¬
