Q:

Concept of an Equivalence class partition

The definition:
Part 1:
The equivalence classes of an equivalence relation on a set S constitute a partition of S. 
Part2:
Conversely, for any partition of S, there is an equivalence relation on S whose equivalence classes are the element of p.
To make sure I'm not blundering over the definition, let's suppose
a and b are integers in the set S.
The equivalence class of $a$ is given as:
$$[a]=\{a+kn|k \in S\}$$
We have that:
$$[1]=\{\dots,(-1-kn),(-1-(k+1)n),\dots,(1+kn),(1+(k+1)n),\dots\}$$
and where a takes on discrete values, we have equivalence classes that forms a partition on a set S.
The partition is: $$[1],[2],[3],....$$
This concludes my understanding of the first part. 
Is my understanding correct? 
Could someone also shed some light on the second part? 

A:

An example. 
Take the integers, and consider this equivalence relation:
A is equivalent to b if they differ by three or a multiple of three. 
So, we have
[1]={..., -5, -2, 1, 4, 7, ...}
[2]={..., -4, -1, 2, 5, 8, ...}
[3]={..., -3, 0, 3, 6, 9, ...}   
These are the only three classes of this relation. 
Every integer appears in exactly one class. 
Any element of a class can be taken as its representative ([3]=[0]). 
Every equivalence relation works like this: it divides a set in "slices" that don't overlap, and cover the entire set.  That is to say, a partition.   
Second part: if you already have a partition, you can find an equivalence relation with that partition.  Just take something equivalent to "x and y are in the same slice"!