# Equivalence class

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## Contents

## Definition

Given an Equivalence relation [ilmath]\sim[/ilmath] the equivalence class of [ilmath]a[/ilmath] is denoted as follows:

[math][a]=\{b|a\sim b\}[/math]

## Notations

An equivalence class may be denoted by [ilmath][a][/ilmath] where [ilmath]a[/ilmath] is the *representative* of it. There is an alternative representation:

- [ilmath]\hat{a} [/ilmath], where again [ilmath]a[/ilmath] is the representative of the class.
^{[1]}

I quite like the hat notation, however I recommend one *avoids* using it when there are multiple Equivalence relations at play.

If there are multiple ones, then we can write for example [ilmath][a]_{\sim_1} [/ilmath] for a class in [ilmath]\sim_1[/ilmath] and [ilmath][f]_{\sim_2} [/ilmath] for [ilmath]\sim_2[/ilmath]

## Equivalence relations partition sets

An equivalence relation is a partition

## Equivalence classes are either the same or disjoint

This is the motivation for how cosets partition groups.

## References

- ↑ Functional Analysis - George Bachman and Lawrence Narici

TODO: Add proofs and whatnot