As for the first question, since and are independent, we have

Let denote the complement of . Then they state that

This is correct - for instance, try checking it with a Venn diagram.

Thus:

which means that and are independent.

For the other problem: What if ? Then as is a subset of , we have also , which equals .

What if ? Then the event definitely occurs, no matter what is. See if you can write it down formally.