have no idea about this question
I don't know what proof techniques you discussed in class, but one can prove (a), for example. The thing is that every formula that one wants to prove has some form: A and B; A or B; A implies B; for every x, A(x); S is a subset of S', etc. There is a way to prove formulas of each of those forms. E.g., to prove "for every x, A(x)" you fix an arbitrary x and prove that A(x) is true for this x.
The claim you need to prove in (a) has the form "for every $\displaystyle f$, $\displaystyle {O}(f)$ is a subset of $\displaystyle \Gamma(f)$. If you now what it means to prove a formula of each form, you can gradually rewrite the claim and then see what you can do about it. Start with: "Fix some arbitrary $\displaystyle f\in\mathcal{F}$. Need to prove: $\displaystyle {O}(f)\subseteq\Gamma(f)$."