Equivalence Class Equivalent Statements/3 iff 6
Jump to navigation
Jump to search
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 $10.4$