1. RELATIONS AND THEIR PROPERTIES
205
Often the relations in our examples do have special properties, but be careful not to
assume that a given relation must have any of these properties.
1.2. Examples.
Example
1.2.1
.
Let
A
=
{
a, b, c
}
and
B
=
{
1
,
2
,
3
,
4
}
, and let
R
1
=
{
(
a,
1)
,
(
a,
2)
,
(
c,
4)
}
.
Example
1.2.2
.
Let
R
2
⊂
N
×
N
be defined by
(
m, n
)
∈
R
2
if and only if
m

n
.
Example
1.2.3
.
Let
A
be the set of all FSU students, and
B
the set of all courses
offered at FSU. Define
R
3
as a relation from
A
to
B
by
(
s, c
)
∈
R
3
if and only if
s
is
enrolled in
c
this term.
Discussion
There are many different types of examples of relations. The previous examples
give three very different types of examples. Let’s look a little more closely at these
examples.
Example 1.2.1.
This is a completely abstract relation. There is no obvious reason
for
a
to be related to 1 and 2. It just is. This kind of relation, while not having any
obvious application, is often useful to demonstrate properties of relations.
Example 1.2.2.
This relation is one you will see more frequently. The set
R
2
is
an infinite set, so it is impossible to list all the elements of
R
2
, but here are some
elements of
R
2
:
(2
,
6)
,
(4
,
8)
,
(5
,
5)
,
(5
,
0)
,
(6
,
0)
,
(6
,
18)
,
(2
,
18)
.
Equivalently, we could also write
2
R
2
6
,
4
R
2
8
,
5
R
2
5
,
5
R
2
0
,
6
R
2
0
,
6
R
2
18
,
2
R
2
18
.
Here are some elements of
N
×
N
that are
not
elements of
R
2
:
(6
,
2)
,
(8
,
4)
,
(2
,
5)
,
(0
,
5)
,
(0
,
6)
,
(18
,
6)
,
(6
,
8)
,
(8
,
6)
.
Example 1.2.3.
Here is an element of
R
3
: (you, MAD2104).
Example
1.2.4
.
Let
A
and
B
be sets and let
f
:
A
→
B
be a function. The graph
of
f
, defined by
graph
(
f
) =
{
(
x, f
(
x
))

x
∈
A
}
, is a relation from
A
to
B
.
Notice the previous example illustrates that any function has a relation that is
associated with it. However, not all relations have functions associated with them.