Lecture Notes, Concepts of Mathematics (21127)
Lecture 1, Recitation A–D, Spring 2008
3
Relations
3.3
Relations, Equivalence Relations, and Partitions
Recall the definition of the Cartesian product
X
×
Y
of two sets
X
and
Y
. From basic set
theory, we have:
Theorem 3.1
If
A
,
B
,
C
, and
D
are sets, then
(a)
A
×
(
B
∪
C
) = (
A
×
B
)
∪
(
A
×
C
)
(b)
A
×
(
B
∩
C
) = (
A
×
B
)
∩
(
A
×
C
)
(c)
A
× ∅
=
∅
(d)
(
A
×
B
)
∩
(
C
×
D
) = (
A
∩
C
)
×
(
B
∩
D
)
(e)
(
A
×
B
)
∪
(
C
×
D
)
⊆
(
A
∪
C
)
×
(
B
∪
D
)
Proof:
(Part (a)) The ordered pair (
x, y
)
∈
A
×
(
B
∪
C
)
iff
x
∈
A
and
y
∈
B
∪
C
iff
x
∈
A
and (
y
∈
B
or
y
∈
C
)
iff (
x
∈
A
and
x
∈
B
) or (
x
∈
A
and
y
∈
C
)
iff (
x, y
)
∈
A
×
B
or (
x, y
)
∈
A
×
C
iff (
x, y
)
∈
(
A
×
B
)
∪
(
A
×
C
). Therefore
A
×
(
B
∪
C
) = (
A
×
B
)
∪
(
A
×
C
).
The other parts of the proof are left as an exercise.
Definition 3.1
A relation
R
from a set
X
to a set
Y
is a subset of
X
×
Y
:
R
=
{
(
x, y
)
∈
X
×
Y

xRy
}
.
Subsets of
X
×
X
are called relations on
X
.
Definition 3.2
The domain
of the relation
R
from
X
to
Y
is the set
Dom
(
R
) =
{
x
∈
X

there exists
y
∈
Y
such that
xRy
}
.
The range
of the relation
R
from
X
to
Y
is the set
Rng
(
R
) =
{
y
∈
Y

there exists
x
∈
X
such that
xRy
}
.
1
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A relation
R
can be defined by listing the elements of
R
or by describing the property that
relates the two elements.
Example 3.1
Let
A
=
{
1
,
2
,
3
}
and
B
=
{
a,
5
,
{
b
}
, c
}
and
R
be given by
R
=
{
(1
, a
)
,
(1
,
{
b
}
)
,
(2
, a
)
,
(2
, c
)
}
.
Then
•
we have
1
Ra
,
1
R
{
b
}
,
2
Ra
,
2
Rc
,
3
R
5
•
Dom
(
R
) =
{
1
,
2
} ⊆
A
•
Rng
(
R
) =
{
a,
{
b
}
, c
} ⊆
B
Example 3.2
Let
P
be the set of all people. Let
L
=
{
(
a, b
)
∈
P
×
P

a
and
b
have the same last name
}
.
Then
L
is a relation on
P
.
Example 3.3
Let
S
be a relation on the set
N
×
N
defined by
(
m, n
)
S
(
k, j
)
iff
m
+
n
=
k
+
j
.
Then
(3
,
17)
S
(12
,
8)
but
(5
,
4)
S
(6
,
15)
. Note that we can write
S
as
S
=
{
((
m, n
)
,
(
k, j
))
∈
N
2
×
N
2

m
+
n
=
k
+
j
}
.
What are the domain and range of
S
?
Relations that are defined on familiar sets can be expressed visually by using a graph
. For
example, let
S
=
{
(
x, y
)
∈
R
×
R

x
2
+
y
2
≤
1
}
and consider its graph (draw graph here). Now consider the set
A
=
{
3
,
6
,
9
,
12
}
and let
R
on
A
be defined by, for
x, y
∈
A
,
xRy
iff
x
divides
y
. Not only can this be represented with
a graph, it can also be represented with a directed graph
or digraph
. We think of objects in
A
as vertices and think of
R
as telling us which vertices are connected by edges. The edges
are directed from one vertex to another using arrows. There is an edge from
x
to
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 Spring '08
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 Math, Set Theory, Sets, Equivalence relation, Binary relation, Transitive relation, Symmetric relation, relation

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