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Proof

(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. $A\cap B=B\cup A.$

Part 1: $A\cup B\subseteq B\cup A$

• 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, $A\cup B\subseteq B\cup A.$

Part 2: $B\cup A\subseteq A\cup B$

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

Conclusion

• It follows from the equality of sets that that $A = B.$
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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016