Equivalence Class is not Empty

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Then no $\mathcal R$-class is empty.


Proof

\(\, \displaystyle \forall \eqclass x {\mathcal R} \subseteq S: \exists x \in S: \, \) \(\displaystyle x\) \(\in\) \(\displaystyle \eqclass x {\mathcal R}\) Definition of Equivalence Class
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass x {\mathcal R}\) \(\ne\) \(\displaystyle \O\) Definition of Empty Set


$\blacksquare$


Also see


Sources