Equivalence Class Equivalent Statements/3 iff 5

Theorem

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

Let $x, y \in S$.

The following statements are equivalent:

$x \mathrel \RR y$

$y \in \eqclass x \RR$

Proof

This follows through dint of the symmetry of $\RR$ and the definition of Equivalence Class.

$\blacksquare$

Sources

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