Prove that and that .

*Proof (Associativity of unions).*First, let be any element in . This means that or . If then ; hence, . On the other hand, if , then . This means or . If , then . If , then . Hence, .

For the reverse inclusion, let be any element of . Then, or . If , we know or . If , then . If , then ; hence, . On the other hand, if , then , and so, . Therefore, .

Thus, .∎

* Proof (Associativity of intersections). * To expedite matters, will prove both inclusions at once: is in if and only if and , further, if and only if and . Similarly, if and only if and and . Thus, and have exactly the same elements (since if and only if and and if and only if ).

Thus, .∎