Suppose students in a class are getting together in groups to do an activity.
The teacher wants to make sure of two things:
 no student is in more than one group (i.e., the groups don't overlap), and
 every student is accounted for (i.e., all the groups, together, make up the entire class).
From a mathematical viewpoint, what has been created is a partition of the class:
DEFINITION
partition
A partition of a set $\,S\,$
is a collection of nonoverlapping subsets of $\,S\,$
that, together,
make up all of $\,S\,$.
More precisely,
‘make up all of $\,S\,$’
means that the
union of the subsets is $\,S\,$.
A partition of a set $\,S\,$
EXAMPLES:
Let $\,S=\{1,2,3,4,5\}\,$.
The sets $\,\{1,2,3\}\,$ and $\,\{4,5\}\,$ form a partition of $\,S\,$:
they don't overlap, and, together, they make up all of $\,S\,$.
The sets
$\,\{1,5\}\,$ and $\,\{2,3,4\}\,$ form a different partition of $\,S\,$.
The sets
$\,\{1\}\,$ and $\,\{3\}\,$ and $\,\{2,4,5\}\,$ form a different partition of $\,S\,$.
The sets
$\,\{1,2\}\,$ and $\,\{3,4\}\,$ do not form a partition of $\,S\,$;
together, they don't make up all of $\,S\,$.
The sets
$\,\{1,2,3\}\,$ and $\,\{3,4,5\}\,$ do not form a partition of $\,S\,$;
they overlap—the number $\,3\,$ is in more than one of the subsets.
The sets
$\,\{1,2\}\,$ and $\,\{2,3\}\,$ and $\,\{4,5\}\,$ do not form a partition of $\,S\,$;
they overlap—the number $\,2\,$ is in more than one of the subsets.
BEING ‘RELATED TO EACH OTHER’ IN A SET
One of the most important ways in math to create a partition of a set
is by using an equivalence relation,
which is the subject of this section.
Given a set $\,S\,$, we first need a way to talk about members of the set being related to each other.
The symbol
‘$\,\sim\,$’
is used for ‘is related to’.
The sentence ‘$\,x\sim y\,$’
is read as
‘$\,x\,$ is related to $\,y\,$’.
The concept is best illustrated with some examples:

Let $\,S\,$ be the set of all people,
and let $\,x\,$ and $\,y\,$ be members of $\,S\,$ .
Define:
$\,x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same sex
Suppose that Carol and Julia are female; Rick and Karl are male.
TRUE SENTENCE 
READ AS: 
WHY TRUE? 
Carol $\,\sim \,$ Julia 
‘Carol is related to Julia’ 
both Carol and Julia are female 
Carol $\,\not\sim\,$ Karl 
‘Carol is not related to Karl’ 
Carol and Karl do not have the same sex 
Rick $\,\sim \,$ Karl 
‘Rick is related to Karl’ 
both Rick and Karl are male 

Let $\,S=\{0,1,2,3,\ldots\}\,$.
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$
TRUE SENTENCE 
READ AS: 
WHY TRUE? 
$\,5\sim 8\,$

‘$\,5\,$ is related to $\,8\,$’ 
When $\,5\,$ is divided by $\,3\,$, the remainder is $\,2\,$:
$5 = 1\cdot 3 + 2$
When $\,8\,$ is divided by $\,3\,$, the remainder is $\,2\,$:
$8 = 2\cdot 3 + 2$
Thus, both have the same remainder ($\,2\,$) when divided by $\,3\,$.

$\,3\sim 12\,$ 
‘$\,3\,$ is related to $\,12\,$’ 
When $\,3\,$ is divided by $\,3\,$, the remainder is $\,0\,$:
$3 = 1\cdot 3 + 0$
When $\,12\,$ is divided by $\,3\,$, the remainder is $\,0\,$:
$12 = 4\cdot 3 + 0$
Thus, both have the same remainder ($\,0\,$) when divided by $\,3\,$.

$\,1\not\sim 11\,$ 
‘$\,1\,$ is not related to $\,11\,$’ 
When $\,1\,$ is divided by $\,3\,$, the remainder is $\,1\,$:
$1 = 0\cdot 3 + 1$
When $\,11\,$ is divided by $\,3\,$, the remainder is $\,2\,$:
$11 = 3\cdot 3 + 2$
Thus, $\,1\,$ and $\,11\,$ have different remainders when divided by $\,3\,$.


Let $\,S\,$ be the set of all ordered pairs of positive integers.
For $\,(a,b)\,$ and $\,(c,d)\,$ in $\,S\,$, define:
$\,(a,b) \sim (c,d) \,$
if and only if
$\,ad=bc\,$
TRUE SENTENCE 
READ AS: 
WHY TRUE? 
$\,(1,3)\sim (2,6)\,$ 
‘$\,(1,3)\,$ is related to $\,(2,6)\,$’ 
$\,1\cdot 6 = 3\cdot 2\,$

$\,(2,5)\sim (4,10)\,$ 
‘$\,(2,5)\,$ is related to $\,(4,10)\,$’ 
$\,2\cdot 10 = 5\cdot 4\,$

$\,(1,3)\not\sim (2,5)\,$ 
‘$\,(1,3)\,$ is not related to $\,(2,5)\,$’ 
$\,1\cdot 5\neq 3\cdot 2\,$

(This concept is very familiar to you; it's just being cast into an unfamiliar setting.
Hint—think about fractions!)
EQUIVALENCE RELATIONS
Now, we are in a position to define an equivalence relation:
DEFINITION
equivalence relation
Let $\,S\,$ be a set.
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$
if and only if
the following properties are
satisfied for all members $\,x\,$, $\,y\,$ and $\,z\,$ in $\,S\,$:

REFLEXIVE property:
$\,x\sim x\,$
(every member is related to itself)

SYMMETRY property:
if $\,x\sim y\,$,
then $\,y\sim x\,$
(if one member is related to another,
then the other is related to the one)

TRANSITIVE property:
if $\,x\sim y\,$ and $\,y\sim z\,$,
then $\,x\sim z\,$
(if one is related to another,
and this other is related to a third,
then the first is related to the third)
EXAMPLE:
an equivalence relation
on the set of all people
(1)
Let $\,S\,$ be the set of all people,
and let $\,x\,$ and $\,y\,$ be members of $\,S\,$ .
Define:
$\,x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same sex
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$, as follows:

reflexivity:
every person has the same sex as his/her self

symmetry:
if a person has the same sex as another person,
then the other person has the same sex as the first

transitivity:
if person #1 has the same sex as person #2,
and person #2 has the same sex as person #3,
then person #1 has the same sex as person #3
EXAMPLE:
an equivalence relation
on the set of all nonnegative integers
(2)
Let $\,S=\{0,1,2,3,\ldots\}\,$.
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$\,x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$, as follows:

reflexivity:
every number has the same remainder upon division by $\,3\,$ as itself

symmetry:
if a number has the same remainder upon division by $\,3\,$ as another,
then the other has the same remainder upon division by $\,3\,$ as the first

transitivity:
if a number has the same remainder upon division by $\,3\,$ as another,
and this other has the same remainder upon division by $\,3\,$ as a third,
then the first number has the same remainder upon division by $\,3\,$ as the third
EXAMPLE:
NOT an equivalence relation
(3)
Let $\,S=\{0,1,2,3,\ldots\}\,$.
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$\,x\sim y\,$
if and only if
$\,x\lt y\,$
Then, $\,\sim \,$ is not an equivalence relation on $\,S\,$.
It fails both reflexivity
(since $\,x\,$ is not less than $\,x\,$)
and symmetry
(if $\,x\,$ is less than $\,y\,$,
then $\,y\,$ is not less than $\,x\,$).
EQUIVALENCE CLASSES
Once you select a member of a set which has an equivalence relation on it,
you often want to study all the members which are related to it.
This leads us to:
DEFINITION
equivalence class
Let $\,\sim \,$ be an equivalence relation on a set $\,S\,$.
Let $\,x\,$ be a member of $\,S\,$.
The equivalence class determined by $\,x\,$
is the set of all members of $\,S\,$ that are related to
$\,x\,$.
Here is one key reason why equivalence relations are so important:
The equivalence classes of a set always form a PARTITION of the set.
For Example (1) above, there are only two equivalence classes:
males and females.
For Example (2) above, there are three equivalence classes:

$\,\{0,3,6,\ldots\}\,$
(numbers that are divisible by $\,3\,$;
that is, the remainder is $\,0\,$)

$\,\{1,4,7,\ldots\}\,$
(numbers that leave a remainder of $\,1\,$ upon division by $\,3\,$)

$\,\{2,5,8,\ldots\}\,$
(numbers that leave a remainder of $\,2\,$ upon division by $\,3\,$)
As a final example,
congruence is an equivalence relation on the set of all geometric figures.
Let $\,G_1\,$, $\,G_2\,$, and $\,G_3\,$ be geometric figures.
Then:

Reflexivity:
$\,G_1 \cong G_1$
(every geometric figure is congruent to itself)

Symmetry:
if $\,G_1\cong G_2\,$ then $\,G_2\cong G_1\,$

Transitivity:
if $\,G_1\cong G_2\,$ and $\,G_2\cong G_3\,$,
then $\,G_1\cong G_3\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Triangle Congruence