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 $10.4$
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