Answer
$p \rightarrow q$ $\equiv ((p \downarrow q)\downarrow q)\downarrow (p \downarrow q)\downarrow q))$
Work Step by Step
Two propositions are logically equivalent if they have the same truth value for any combination of truth values of the variables.
A conjunction $p\rightarrow q$ is true, if p is false or if both p and q are true.
$p \downarrow q$ is true if and only if both p and q are false.
By exercise 50 part a), $ p \downarrow q \equiv \neg p$
By exercise 50 part b), $p \lor q \equiv (p \downarrow q) \downarrow(p \downarrow q)$
Now, $$p \rightarrow q$$$$\equiv (\neg p \lor q)$$$$\equiv((p \downarrow q) \lor q)$$$$\equiv ((p \downarrow q)\downarrow q)\downarrow (p \downarrow q)\downarrow q))$$
hence, $p \rightarrow q$ $\equiv ((p \downarrow q)\downarrow q)\downarrow (p \downarrow q)\downarrow q))$
This can also be seen through truth table.
$\underline{p \quad q \quad p \rightarrow q \quad p \downarrow q \quad ((p \downarrow q) \lor q)\quad ((p \downarrow q)\downarrow q)\downarrow (p \downarrow q)\downarrow q))}$
$T \quad T \quad\quad T \quad\quad F \quad\quad\quad\quad F\quad\quad\quad\quad\quad\quad\quad\quad\quad T$
$T \quad F \quad\quad F \quad\quad F \quad\quad\quad\quad T\quad\quad\quad\quad\quad\quad\quad\quad\quad F$
$F \quad T \quad\quad T \quad\quad T \quad\quad\quad\quad F\quad\quad\quad\quad\quad\quad\quad\quad\quad T$
$F \quad F \quad\quad T \quad\quad T \quad\quad\quad\quad F\quad\quad\quad\quad\quad\quad\quad\quad\quad T$
Since the two columns of the truth table, $ p \downarrow q$ and $((p \downarrow q)\downarrow q)\downarrow (p \downarrow q)\downarrow q))$ contain the same truth value in every row, so they are logically equivalent.
