Prove that .
Proof. First, let be any element in . By definition of complement, this means that and . Since we have either or which implies, coupled with the fact that is in , means or , respectively. Since is in at least one of these, is in the union . Therefore, .
For the reverse inclusion, let be an arbitrary element of . Then, either or .
If , then and ; hence, . Therefore, is in . On the other hand, if , then and ; hence, . This again implies is in . Therefore, .