Visualizing relations

So I am having some trouble visualizing relations. This is my analogy to

what a relation is currently.

-A relation R on A is the subset of the power set of (A X A)

Therefore the (A X A) has a R in its folder. Or R is contained in the A X A ven-diagram you know what i mean ? or is the relation an element? or is it both?

Lets give an R an arbitrary number as a relation since more than n

relations can exist

- R1 is the relation in which it sorts elements of the Cartesian product of A X A n-tuples in this

case 2-tuples, into a specific relationship

Therefore within the R1 folder exist ordered pairs which are elements of R1

Are my conclusions valid ? or are there misconceptions ?

Equivalence classes

Defining an Equivalence relations

R is an equivalence relation it is transitive reflexive and

symmetric

So let R be an equivalence relation on A

Therefore R is in A x A folder R is a subset of A x A

I don't quite understand what an Equivalence class is

my current understanding is

The Equivalence Class is in the folder of the Equivalence relation R

The Equivalence relation has to do with only either the element (a) from

the ordered pair of the relation of R or either (a) or (b)

is says in my textbook that [a]R = {s|(a,s) E R} saying that the

Equivalence class is a subset or element ?