Suppose students in a class are getting together in groups to do an activity.
The teacher wants to make sure of two things:
More precisely,
‘make up all of $\,S\,$’
means that the
union of the subsets is $\,S\,$.
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:
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 
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 $\,3\,$. 
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 $\,(2,5)\,$’  $\,1\cdot 5\neq 3\cdot 2\,$ 
Now, we are in a position to define an equivalence relation:
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:
Here is one key reason why equivalence relations are so important:
For Example (1) above, there are only two equivalence classes: males and females.
For Example (2) above, there are three equivalence classes:
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:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
