# Equivalence classes are either equal or disjoint

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
I'm sure I've already done this SOMEWHERE - find it!

## Statement

Let [ilmath]X[/ilmath] be a set, let [ilmath]\sim\subseteq X\times X[/ilmath] be an equivalence relation on [ilmath]X[/ilmath], let [ilmath]\frac{X}{\sim} [/ilmath] denote the quotient of [ilmath]X[/ilmath] by [ilmath]\sim[/ilmath][Note 1] and lastly let [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath] given by [ilmath]\pi:x\mapsto[x][/ilmath] be the canonical projection of the equivalence relation. Then:

• We claim that [ilmath]\frac{X}{\sim} [/ilmath] is a partition of [ilmath]X[/ilmath]. That is:
1. [ilmath]\forall x\in X\exists y\in\frac{X}{\sim}[x\in y][/ilmath] - all elements of [ilmath]x[/ilmath] belong to an element of the partition
2. [ilmath]\forall u,v\in\frac{X}{\sim}[u\cap v\neq\emptyset\implies u\eq v][/ilmath] - if [ilmath]u[/ilmath] and [ilmath]v[/ilmath] are not disjoint, they are equal
• Equivalently (by contrapositive[Note 2]): [ilmath]\forall u,v\in\frac{X}{\sim}[u\neq v\implies u\cap v\eq\emptyset][/ilmath]