Difference between revisions of "Equivalence class"

Latest revision as of 20:00, 14 November 2015

Definition

Given an Equivalence relation $\sim$ the equivalence class of $a$ is denoted as follows:

Notations

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

• $\hat{a}$, where again $a$ 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 $[a]_{\sim_1}$ for a class in $\sim_1$ and $[f]_{\sim_2}$ for $\sim_2$

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

1. Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici

TODO: Add proofs and whatnot