Fundamental Theorem on Equivalence Relations

Theorem
Let $\RR \subseteq S \times S$ be an equivalence on a set $S$.

Then the quotient $S / \RR$ of $S$ by $\RR$ forms a partition of $S$.

Proof
To prove that $S / \RR$ is a partition of $S$, we have to prove:


 * $(1): \quad \ds \bigcup {S / \RR} = S$


 * $(2): \quad \eqclass x \RR \ne \eqclass y \RR \iff \eqclass x \RR \cap \eqclass y \RR = \O$


 * $(3): \quad \forall \eqclass x \RR \in S / \RR: \eqclass x \RR \ne \O$

Taking each proposition in turn:

Union of Equivalence Classes is Whole Set
The set of $\RR$-classes constitutes the whole of $S$:

Equivalence Class is not Empty
Thus all conditions for $S / \RR$ to be a partition are fulfilled.

Also see

 * Relation Induced by Partition is Equivalence for the converse.