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Proof

(related to Proposition: Set Intersection is Commutative)

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

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

• Let $x\in A\cap B.$
• By definition of set intersection, $x\in A\wedge x\in B.$
• By commutativity of conjunction, $x\in B\wedge x\in A.$
• It follows $x\in B\cap A.$
• By defintion of subsets, $A\cap B\subseteq B\cap A.$

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

• 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