compound proposition logically equivalent I can not solve this question 
Find a compound proposition logically equivalent to $p \to q$ using only the logical operator $\downarrow$.

It's quite simple; let start with :

$\lnot p \equiv p \downarrow  p$.

Then :


$p \rightarrow q \equiv ((p \downarrow p) \downarrow q) \downarrow ((p \downarrow p) \downarrow q)$.


In order to check the definition, we have to use the truth-table for $\downarrow$ : it is true only when both $p$ and $q$ are false. 
This fact justify the definition of $\lnot p$ as $p \downarrow p$. 
For the conditional, we will work by steps: 

$((p \downarrow p) \downarrow q)$ is $\lnot p \downarrow q$ [see the definition of $\lnot$ above]. 

Thus, the complete formula is simply : $\lnot ( \lnot p \downarrow q)$.
Now we may check that the only case when it is false (i.e.$0$) is when $p=1$ and $q=0$. 
Note. You can see this paper on Adeqaute set of connectives for a general overview of the topic.
