Union is Associative

Theorem
Set union is associative:


 * $$A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$$

Proof
$$ $$ $$

Therefore, $$x \in A \cup \left({B \cup C}\right)$$ iff $$x \in \left({A \cup B}\right) \cup C$$.

Thus it has been shown that $$A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$$.

Also see

 * Intersection is Associative
 * Set Difference is Not Associative
 * Symmetric Difference is Associative