Prove the following.

- .
- .
- if and only if .

*Proof.*To show two sets are equal, we want to show that each is a subset of the other (i.e., we want to show that and ).

First, since the only element of is , and we have .

Second, the only element of is , and we have . Hence, . Thus, . ∎*Proof.*Again, we want to show and .

First, since the elements of are and , and .

Second, the elements of are and . Since we have every element of is in ; thus, .

Therefore, . ∎*Proof.*Assume . Since , we must have ; hence, every element of must be contained in . This means that both and are in . Since is the only element of , we must have .

Conversely, assume . Then and from part (a) we know ; hence .∎