Math 100
Relations
Martin H. Weissman
1.
Relations on Sets
1.1.
Examples of Relations.
Let
S
be a set. A
relation
on
S
is a subset of
S
2
. In other words, a relation
is a set of ordered pairs, whose entries are elements of
S
.
If
R
is a relation on
S
, and (
a,b
)
∈
S
2
, then we say that “
a
is related to
b
” if (
a,b
)
∈
R
.
Let’s look at a simple example:
•
Let
S
=
{
a,b
}
.
•
Let
R
=
{
(
a,a
)
,
(
b,b
)
}
. Then
R
is a relation on
S
.
In this case, the following two sentences are true:
•
a
is related to
a
.
•
b
is related to
b
.
There are no other relations among the elements of
S
(for example,
a
is
not
related to
b
). The relation
R
is
called
equality
, because one element of
S
is related to another element of
S
if and only if they are equal to
each other.
Here’s another example:
•
Let
S
=
{
a,b
}
.
•
Let
R
=
{
(
a,b
)
,
(
b,a
)
}
. Then
R
is a relation on
S
.
In this case, the following two sentences are true:
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 Fall '08
 Weissman
 Sets, Equivalence relation, Binary relation, Transitive relation, Symmetric relation, relation

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