Equivalence Classes are Disjoint

Theorem
Let $\mathcal R$ be an equivalence relation on a set $S$.

Then all $\mathcal R$-classes are pairwise disjoint:


 * $\tuple {x, y} \notin \mathcal R \iff \eqclass x {\mathcal R} \cap \eqclass y {\mathcal R} = \O$

Also see

 * Fundamental Theorem on Equivalence Relations


 * Union of Equivalence Classes is Whole Set
 * Equivalence Class is not Empty