Equivalence Class Equivalent Statements/3 iff 6

Theorem

Let $\RR$ be an equivalence relation on $S$.

Let $x, y \in S$.

The following statements are equivalent:

$x \mathrel \RR y$

$\eqclass x \RR \cap \eqclass y \RR \ne \O$

Proof

Follows directly from Equivalence Classes are Disjoint.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations: Theorem $10.4$