LECTURE 17: PROPERTIES OF EQUIVALENCE RELATIONS (
§
8.4)
Mi taku oyasin. (We are all related.) Lakota belief
1.
Equivalence classes
Let
R
be an equivalence relation on
A
.
Let
a
∈
A
.
The set [
a
] =
{
x
∈
A
:
xRa
}
is called the
equivalence class
of
a
. This is the set of all things related to
a
.
Example 1.
Let
A
=
{
1
,
2
,
3
,
4
,
5
,
6
}
, and
R
=
{
((1
,
3)
,
(3
,
4)
,
(5
,
4)
,
(6
,
2)
}
. What do we need to add
to
R
to make it an equivalence relation? Then what are the equivalence classes?
We must add (1
,
1)
,
(2
,
2)
,
(3
,
3)
, . . . ,
(6
,
6) to make the relation reflexive.
We must add (3
,
1)
,
(4
,
3)
,
(4
,
5)
,
(2
,
6) to make the relation symmetric.
We must then add (1
,
4)
,
(4
,
1)
,
(1
,
5)
,
(5
,
1)
,
(3
,
5)
,
(5
,
3) to make the relation transitive.
Then the equivalence classes are [1] =
{
1
,
3
,
4
,
5
}
and [2] =
{
2
,
6
}
.
Example 2.
What are the equivalence classes under the operation = on
Z
? (They are the one element
sets.)
Theorem 3.
Let
R
be an equivalence relation on a nonempty set
A
. Let
a, b
∈
A
. We have
[
a
] = [
b
]
if
and only if
aRb
.
Proof.
(
⇒
) : Suppose [
a
] = [
b
]. Notice that since
R
is reflexive we have
aRa
. Thus
a
∈
[
a
]. This means
a
∈
[
b
]. Hence
aRb
.
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 Spring '11
 Smith
 Equivalence relation, Transitive relation, equivalence class, equivalence classes, Partition of a set, aRb

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