SHEN’S CLASS NOTES-191
1
Chapter Three
Relations
3.1 Relations
Relations generalize the notion of functions.
Definition 3.1
A (binary) relation
R
from a set
X
to a set
Y
is
a subset of the Cartesian product
X
×
Y
. If (
x
,
y
)
∈
R
, we say
that
x
is related to
y
and write
x
R
y
. If
X
=
Y
, we call
R
a
(binary) relation on
X
.
Example 1 (3.1.3)
Let
X
= {2, 3, 4} and
Y
= {3, 4, 5, 6, 7}.
We define a relation
R
as follows.
R
= {(
x
,
y
) |
x
∈
X
,
y
∈
Y
,
x
divides
y
}.
Then,
R
= {(2, 4), (2, 6), (3, 3), (3, 6), (4, 4)}.
A binary relation can also be represented by a table or an arrow
diagram as shown in Fig. 1. From the figure, we can see that a
relation does not require that (
x
,
y
) must be unique for each
x
. A
function is a special case of a relation.

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