Theorem
Union is associative
$ A\cup (B\cup C) = (A\cup B)\cup C $
where $ A $, $ B $ and $ C $ are sets.
Proof
$ \begin{align} A\cup (B\cup C)&= \{x\in\mathcal S:\;x\in A\;\mbox{or}\; x\in (B\cup C)\}\\ &= \{x\in\mathcal S:\;x\in A\;\mbox{or}\; x\in B\;\mbox{or}\; x\in C\}\\ &= \{x\in\mathcal S:\;x\in (A\cup B)\;\mbox{or}\; x\in C)\}\\ &= (A\cup B)\cup C \\ \blacksquare \end{align} $
Because of this property, A ∪ (B ∪ C) or (A ∪ B) ∪ C is written simply as A ∪ B ∪ C.
