Equivalence Classes are Disjoint

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

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


 * $\left({x, y}\right) \notin \mathcal R \iff \left[\!\left[{x}\right]\!\right]_\mathcal R \cap \left[\!\left[{y}\right]\!\right]_\mathcal R = \varnothing$

Also see

 * Fundamental Theorem on Equivalence Relations


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