# Commutativity of intersection

(Redirected from Intersection is commutative)

$A\cap B=B\cap A$

## Note

This is somewhere between a theorem and a definition because at some point you have to accept "and" is commutative or something.

## Proof

### $\implies$

$x\in A\cap B\implies x\in A\text{ and }x\in B\implies x\in B\text{ and }x\in A\implies x\in B\cap A$, thus by the implies and subset relation we see $A\cap B\subset B\cap A$

### $\impliedby$

By the exact same procedure we see $B\cap A\subset A\cap B$

Thus we conclude $A\cap B=B\cap A$