Notes:Equivalence relations

Overview

I'm brushing on some old abstract algebra pages, specifically involving the quotient group, and it occurs to me, a lot of this work can actually be done "higher up" the abstractional chain, that is, rather than showing $\pi:G\rightarrow G/N$ is a surjection we can say that "as we have an equivalence relation, we get a surjective map with the property that:

• $x\sim y\implies \pi(x)=\pi(y)$

This is just some notes to get what I've done on paper into the system

Claims

Let $X$ be a set and let $\sim$ be an equivalence relation, then:

1. $X/\sim$ is a partition of $X$
2. $\pi:X\rightarrow X/~$ given by $\pi:x\mapsto [x]$ is a surjective function satisfying the property:
   • If $x\sim y$ then $\pi(x)=\pi(y)$
3. $\pi$ is the unique function that does this. Warning:Just throwing that in there, I have no source for this, I suspect that $\pi$ satisfies some sort of universal property though

Proof of claims

Here $X$ is a set and $\sim$ an equivalence relation, $X/\sim$ denotes the set of equivalence classes of $\sim$.