Group Theory and Linear Algebra
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updated: 17 September 2014
Lecture 3: Equivalence relations
A set is a collection of elements.
Let and be sets. The product of and is the set
If then
Let be a set. A relation on is a subset of
is a relation on if there exists
such that
means is in the relation
Let Let
Define a relation on
by
where and
with
and
and
Define to be
where with
and
Let be a set. Let be a relation on Write
if
is in the relation
The relation is reflexive if satisfies:
if then
The relation is symmetric if satisfies:
if and
then
The relation is transitive if satisfies:
if
and and
then
An equivalence relation on is a relation on that is reflexive, symmetric and transitive.
Let be a set. Let be an equivalence relation on Let
The equivalence class of is the set
A partition of is a collection of subsets of such that
(a) |
|
(b) |
If and then
|
Let Then
The equivalence class of is
Recall that
Note that
is a partition of since
(a) |
and
|
(b) |
if and
then
|
Let Then
is an equivalence relation on
|
|
Proof. |
|
To show: |
(a) |
is reflexive.
|
(b) |
is symmetric.
|
(c) |
is transitive.
|
|
To show: |
(a) |
If then
|
(b) |
If and
then
|
(c) |
If and
and
then
|
|
Assume and
and
Let
with
Since
and then
Since is an equivalence relation on
So
and
So is an equivalence relation on
|
|
Notes and References
These are a typed copy of Lecture 3 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on July 29, 2011.
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