Equivalence Class Definition I am currently reading about the subject given in the title of this thread. The definition they give for equivalence classes in my textbook is a rather ostentatious in its wording, so I just want to make certain that I am understanding it properly. They say to let R be an equivalence relation on a set A, meaning that this this particular relation is reflexive, symmetric, and transitive, right? Essentially the rest of it seems to say that you can partition off the elements that make the relation reflexive, thereby creating a subset of the relation R. Does that seem right? 
I could really use some help, thank you!
 A: 
They say to let R be an equivalence relation on a set A, meaning that this this particular relation is reflexive, symmetric, and transitive, right?

Yes, any relation that satisfies these properties is by definition an equivalence relation. It is called an equivalence relation because it satisfies the fundamental properties of what it means for elements in a set to be "equal".

Essentially the rest of it seems to say that you can partition off the elements that make the relation reflexive, thereby creating a subset of the relation R.

The important thing to understand is that it partitions up the set into disjoint (non-overlapping) subsets.
Let $X$ be a set of people standing in a crowded room, and define an equivalence relation $R$ on $X$ by saying that for any two people $x, y \in X$, $xRy$ if and only if $x$ and $y$ have first names beginning with the same letter. Then you divide up all the people in the room to non-overlapping subsets: $X_{a} \subset X$ people with first names beginning with $a$, $X_{b} \subset X$ beginning with $b$ , and so on.
Then you can write $X$ as the union of partitions defined by the relation $R$:
$$X = X_{a} \cup X_{b} \cup \dots \cup X_{y} \cup X_{z}.$$
A: Yes. For example, consider the integers under the relation of equality modulo 3. This basically divides the integer set into 3 equivalence classes:
$\{3k, 3k+1, 3k+2\}_{k \in \mathbb{Z}}$
A: The point is that equivalence relations are essentially the same as partitions. Indeed, the equivalence classes of an equivalence relation form a partition. Conversely, a partition forms an equivalence relation by relating elements if they lie in the same component of the partition. 
For example, "the same parity" is an equivalence relation on the integers, i.e $\rm\:j \equiv k\iff 2\:|\:j-k.\:$ The equivalence classes of evens $\, {\cal E}\ = 2\,\Bbb Z\,$ and odds $\,{\cal O} = 1+2\,\Bbb Z\,$ form a partition ot the integers $\, {\cal E} \cup\, {\cal O} = \Bbb Z\,$ and $\:{\cal E\cap O} = \emptyset.$ Conversely, given the partition into even and odds, we obtain the parity equivalence relation by defining integers to be equivalent if they lie in the same component, i.e. they are both even, or both odd.
