Union is Commutative

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Theorem

Set union is commutative:

$S \cup T = T \cup S$


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle \paren {S \cup T}\) $\quad$ $\quad$
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x \in S\) \(\lor\) \(\displaystyle x \in T\) $\quad$ Definition of Set Union $\quad$
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x \in T\) \(\lor\) \(\displaystyle x \in S\) $\quad$ Disjunction is Commutative $\quad$
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {T \cup S}\) $\quad$ Definition of Set Union $\quad$

$\blacksquare$


Also see


Sources