So you've got

1. $\displaystyle p\to q$

2. $\displaystyle \neg p\to x$

3. $\displaystyle y\to z$

4. $\displaystyle a\to \neg x$

5. $\displaystyle \neg y\to \neg q$.

You're asked to prove or disprove $\displaystyle z\to a$.

Since all of your assumptions are implications, the only inference rules you need are modus ponens and modus tollens. In this case, you'd have to assume $\displaystyle z$, and try to show $\displaystyle a$. For modus ponens, you'd need to start somewhere with a $\displaystyle z$ on the LHS of an implication. For modus tollens, you'd need to start somewhere with a $\displaystyle \neg z$ on the RHS of an implication. That does not occur anywhere in your assumptions. Therefore, I deduce that you're not going to be able to prove $\displaystyle z\to a$. But how do you disprove it? You have to be able to assign truth values to all of the propositions such that all your assumptions are true, and yet $\displaystyle z\to a$ is false. In order for $\displaystyle z\to a$ to be false, $\displaystyle z=\text{TRUE}$ and $\displaystyle a=\text{FALSE}.$ So now see if you can assign the rest of the truth values of your propositions such that all the assumptions are satisfied. Make sense?