Most recent change of EquivalenceRelation

Edit made on July 18, 2009 by ColinWright at 18:57:59

Deleted text in red / Inserted text in green

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 EQN:a\equiv{b} and b * c EQN:b\equiv{c} then a * c EQN:a\equiv{c}

Reflexivity: a
* a

Symmetry: if a * b EQN:a\equiv{b} then b 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.