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 [ilmath]\pi:G\rightarrow G/N[/ilmath] is a surjection we can say that "as we have an equivalence relation, we get a surjective map with the property that:

• [ilmath]x\sim y\implies \pi(x)=\pi(y)[/ilmath]

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

Claims

Let [ilmath]X[/ilmath] be a set and let [ilmath]\sim[/ilmath] be an equivalence relation, then:

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

Proof of claims

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