r
)
−
5
R
−
S
((
5
R
−
S
(
r
)
×
s
)
−
5
R
−
S
,
S
(
r
))
To see that this expression is true, we observe that
5
R
−
S
(
r
) gives us
all tuples
t
that satisfy the Frst condition of the deFnition of division.
The expression on the right side of the set difference operator
5
R
−
S
((
5
R
−
S
(
r
)
×
s
)
−
5
R
−
S
,
S
(
r
))
serves to eliminate those tuples that fail to satisfy the second condition
of the deFnition of division. Let us see how it does so. Consider
5
R
−
S
(
r
)
×
s
. This relation is on schema
R
, and pairs every tuple
in
5
R
−
S
(
r
) with every tuple in
s
. The expression
5
R
−
S
,
S
(
r
) merely
reorders the attributes of
r
.
Thus, (
5
R
−
S
(
r
)
×
s
)
−
5
R
−
S
,
S
(
r
) gives us those pairs of tuples from
5
R
−
S
(
r
)and
s
that do not appear in
r
.Ifatuple
t
j
is in
5
R
−
S
((
5
R
−
S
(
r
)
×
s
)
−
5
R
−
S
,
S
(
r
))
then there is some tuple
t
s
in
s
that does not combine with tuple
t
j
to form a tuple in
r
.Thus
,
t
j
holds a value for attributes
R
−
S
that
does not appear in
r
÷
s
. It is these values that we eliminate from
5
R
−
S
(
r
).
6.5
Let the following relation schemas be given:
R
=
(
A
,
B
,
C
)
S
=
(
D
,
E
,
F
)
Let relations
r
(
R
)and
s
(
S
) be given. Give an expression in the tuple rela
tional calculus that is equivalent to each of the following:
a.
5
A
(
r
)
b.
s
B
=
17
(
r
)
c.
r
×
s
d.
5
A
,
F
(
s
C
=
D
(
r
×
s
))
Answer:
a.
{
t
∃
q
∈
r
(
q
[
A
]
=
t
[
A
])
}
b.
{
t

t
∈
r
∧
t
[
B
]
=
17
}
c.
{
t
∃
p
∈
r
∃
q
∈
s
(
t
[
A
]
=
p
[
A
]
∧
t
[
B
]
=
p
[
B
]
∧
t
[
C
]
=
p
[
C
]
∧
t
[
D
]
=
q
[
D
]
∧
t
[
E
]
=
q
[
E
]
∧
t
[
F
]
=
q
[
F
])
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View Full DocumentPractice Exercises
5
d.
{
t
∃
p
∈
r
∃
q
∈
s
(
t
[
A
]
=
p
[
A
]
∧
t
[
F
]
=
q
[
F
]
∧
p
[
C
]
=
q
[
D
]
}
6.6
Let
R
=
(
A
,
B
,
C
), and let
r
1
and
r
2
both be relations on schema
R
.Give
an expression in the domain relational calculus that is equivalent to each
of the following:
a.
5
A
(
r
1
)
b.
s
B
=
17
(
r
1
)
c.
r
1
∪
r
2
d.
r
1
∩
r
2
e.
r
1
−
r
2
f.
5
A
,
B
(
r
1
)
1
5
B
,
C
(
r
2
)
Answer:
a.
{
<
t
>
∃
p
,
q
(
<
t
,
p
,
q
>
∈
r
1
)
}
b.
{
<
a
,
b
,
c
>

<
a
,
b
,
c
>
∈
r
1
∧
b
=
17
}
c.
{
<
a
,
b
,
c
>

<
a
,
b
,
c
>
∈
r
1
∨
<
a
,
b
,
c
>
∈
r
2
}
d.
{
<
a
,
b
,
c
>

<
a
,
b
,
c
>
∈
r
1
∧
<
a
,
b
,
c
>
∈
r
2
}
e.
{
<
a
,
b
,
c
>

<
a
,
b
,
c
>
∈
r
1
∧
<
a
,
b
,
c
>
6∈
r
2
}
f.
{
<
a
,
b
,
c
>
∃
p
,
q
(
<
a
,
b
,
p
>
∈
r
1
∧
<
q
,
b
,
c
>
∈
r
2
)
}
6.7
Let
R
=
(
A
,
B
)and
S
=
(
A
,
C
), and let
r
(
R
)and
s
(
S
)bere
la
t
ions
.W
r
i
te
expressions in relational algebra for each of the following queries:
a.
{
<
a
>
∃
b
(
<
a
,
b
>
∈
r
∧
b
=
7)
}
b.
{
<
a
,
b
,
c
>

<
a
,
b
>
∈
r
∧
<
a
,
c
>
∈
s
}
c.
{
<
a
>
∃
c
(
<
a
,
c
>
∈
s
∧∃
b
1
,
b
2
(
<
a
,
b
1
>
∈
r
∧
<
c
,
b
2
>
∈
r
∧
b
1
>
b
2
))
}
Answer:
a.
5
A
(
s
B
=
17
(
r
))
b.
r
1
s
c.
5
A
(
s
1
(
5
r
.
A
(
s
r
.
b
>
d
.
b
(
r
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 Spring '13
 Dr.Khansari
 Relational Database, Relational model, Formal Relational Query

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