Equivalence class

Definition

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

$$[a]=\{b|a\sim b\}$$

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.

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

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

Suppose the two classes were both nonidentical and nondisjoint. Then there exists $$[a_1] \sim [b_1]$$ and $$[a_2] \nsim [b_2]$$ . However, $$[a_1] \sim [a_2]$$ and $$[b_1] \sim [b_2]$$ , thus $$[a_2] \sim [b_2]$$ , 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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