I am currently working on proving these statements

and

, by means of subsets (I've heard that is is also referred to as, "proof by double inclusion."). I am having trouble with these sort of problems, so I'd certainly appreciate the help.

Here is my work:

Suppose that

; for

to be in the set,

can't be in

. This means that

; however, by this fact,

can be in

,

, or

. Thus, when

,

is also in

, meaning that

I have not commenced the second part, yet, because I feel as though there is something wrong with this, and I feel like I might have left out steps and explanations.