Union is Idempotent

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Theorem

Set union is idempotent:

$S \cup S = S$


Proof

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

$\blacksquare$


Also see


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