Equivalence class

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.

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

Suppose there were two equivalence classes [math][a][/math] and [math][b][/math]. We can write the members of each class as [math][a_n][/math] and [math][b_n][/math].

Suppose the two classes were both nonidentical and nondisjoint. Then there exists [math][a_1] \sim [b_1][/math] and [math][a_2] \nsim [b_2][/math]. However, [math][a_1] \sim [a_2][/math] and [math][b_1] \sim [b_2][/math], thus [math][a_2] \sim [b_2][/math], a contradiction. Therefore the classes must be either identical or disjoint.

This is the motivation for how cosets partition groups.

References

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TODO: Add proofs and whatnot

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