Prove that and .

*Proof.*First, it is clear that (see Exercise 12 of Section I.2.5, or simply note that implies since is in ).

For the reverse inclusion, if is any element of then or . But implies (and ). So, in either case ; hence, .

Therefore, .∎

* Proof. * From Exericse 12 (Section I.2.5) we know that for any set ; hence, .

For the reverse inclusion, if is any element in , then and ; hence, . Therefore, .

Hence, .∎