.
.
Fall 2007
CPE/CSC 365:
Introduction to Database Systems
Alexander Dekhtyar
.
.
Relational Algebra.
Part 1. Definitions.
Relational Algebra
Notation
R, T, S, . . .
– relations.
t, t
1
, t
2
, . . .
– tuples of relations.
t
(
n
)
– tuple with
n
attributes.
t
[1]
, t
[2]
, . . ., t
[
n
] – 1st, 2nd,. . . nth attribute of tuple
t
.
t.name
– attribute
name
of tuple
t
(attribute names and numbers are
inter
changable
).
R.name, R
[1]
, . . .
– attribute
name
or attribute number 1 (etc.) of relation
R
.
R
(
X
1
:
V
1
, . . . , X
n
:
V
n
) – relational schema specifying relation
R
with
n
attributes
X
1
, . . . , X
n
of types
V
1
, . . . , V
n
. We let
V
i
specify both the
type
and
the set of possible values in it.
Base Operations
Union
Definition 1
Let
R
=
{
t
}
and
S
=
{
t
′
}
be two relations over
the same
rela
tional schema. Then, the union of
R
and
S
, denoted
R
∪
S
is a relation
T
, such
that:
T
=
{
t
′′

t
′′
∈
R
or
t
′′
∈
S
}
.
Union
of two relations combines in one relation all their tuples.
Note:
it is important to note that for union to be applicable to two relations
R
and
S
, they
must have the
same
schema
.
1
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Difference
Definition 2
Let
R
and
S
be two relations over
the same
relational schema.
Then, the difference of
R
and
S
, denoted
R
−
S
is a relation
T
such that:
T
=
{
t

t
∈
R
and
t
negationslash∈
S
}
.
Difference
of two relations
R
and
S
is a relation that contains all tuples from
R
that are
not contained
in
S
.
Cartesian Product
Definition 3
Let
R
and
S
be two relations. The cartesian product of
R
and
S
,
denoted
R
×
S
is a relation
T
such that:
T
=
{
t

t
= (
a
1
, . . . , a
n
, b
1
, . . . , b
m
)
∧
(
a
1
, . . . , a
n
)
∈
R
∧
(
b
1
, . . . , b
m
)
∈
S
}
.
Cartesian product
“glues” together all tuples of one relation with all tuples
of the other one.
Selection
Definition 4
Let database
D
consist of relations
R
1
, . . . R
n
.
Let
X, Y
be at
tributes of one of the relations
R
1
, . . . R
n
. An atomic selection condition is an
expression of one of the forms:
X
op
α
;
X
op
Y,
where
α
∈
V
and
V
is
X
’s domain;
op
∈ {
=
,
negationslash
=
, <, >,
≤
,
≥}
and
X
and
Y
have
comparable types.
An atomic selection condition is a selection condition.
Let
C
1
and
C
2
be two selection conditions. Then
C
1
∧
C
2
,
C
1
∨
C
2
and
¬
C
1
are
selection conditions.
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 Spring '08
 dekhtyar
 Databases, Tuple, C1 C2, S. Attributes

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