(related to Proposition: Set Union is Commutative)

- Suppose that $A$ and $B$ are any sets.
- We want to show that the set union $A\cup B$ is commutative, i.e. $A\cap B=B\cup A.$

- Let $x\in A\cup B.$
- By definition of set union , $x\in A\vee x\in B.$
- By commutativity of disjunction, $x\in B\vee x\in A.$
- It follows $x\in B\cup A.$
- By defintion of subsets, $A\cup B\subseteq B\cup A.$

- The proof is identical to Part 1 if we exchange the denotations of $A$ and $B.$

- It follows from the equality of sets that that $A = B.$∎

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016