An Equivalence Relation is a binary relation $\equiv$ on a set A which has the following properties:

For all a, b and c $\epsilon$ A

• Transitivity: if $a\equiv{b}$ and $b\equiv{c}$ then $a\equiv{c}$
• Symmetry: if $a\equiv{b}$ then $b\equiv{a}$
• Reflexivity: $a\equiv{a}$
Once you have an equivalance relation on a set, you can take one element (say, x) and look at all the elements that are equivalent to it. This is called the equivalence class of x.

• $E(x)=\{y\in{A}:~y{\equiv}x~\}$
It's clear that a finite set can be divided up into a finite number of non-empty, disjoint sets, each of which is an equivalence class (of some element). Most people would agree that a countably infinite set can also be divided up into equivalence classes.

It's less clear that dividing up an uncountably infinite set can always be accomplished. This has a connection with the Axiom of Choice.