Prove that and that .

*Proof.*If , then or by definition of union. Since (by definition of ), we must have . Hence, .

On the other hand, if , then by definition of union, so .

Therefore, .∎

* Proof. * If , then by definition of intersection and . But, by definition of , there is no such that . Hence, must be empty. By uniqueness of the empty set then, we have .∎