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

- 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, $A\cap B\subseteq B\cap A.$

- 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