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An Equivalence Relation is a binary relation EQN:\equiv on a set A which has the following properties: For all a, b and c EQN:\epsilon A * Transitivity: if EQN:a\equiv{b} and EQN:b\equiv{c} then EQN:a\equiv{c} * Symmetry: if EQN:a\equiv{b} then EQN:b\equiv{a} * Reflexivity: EQN: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. * EQN: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.