Union is Idempotent

From ProofWiki
Jump to: navigation, search

Theorem

Set union is idempotent:

$S \cup S = S$


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle S \cup S\) $\quad$ $\quad$
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x \in S\) \(\lor\) \(\displaystyle x \in S\) $\quad$ Definition of Set Union $\quad$
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S\) $\quad$ Rule of Idempotence: Disjunction $\quad$

$\blacksquare$


Also see


Sources