Equivalence Classes are Disjoint

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

Then all $\RR$-classes are pairwise disjoint:


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

Also see

 * Fundamental Theorem on Equivalence Relations


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