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 .∎