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

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 a * b and b * c then a * c

* 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}

Reflexivity: a * 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.

Symmetry: if a * b then b * a

* 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.