What does it mean to prove if $p$ then $q$? Does that mean to prove $p\rightarrow q$ is a true statement? Then since when $p$ is false, $p\rightarrow q$ is vacuously true, do I only have to prove $q$ is true when $p$ is true?
 A: That is the gist of it, yeah.
There are, in practice, several ways to do this, and here is a short summary. A direct proof uses intermediate, already-known implications chained together like this.
$$
p\to p_1\\
p_1\to p_2\\
\vdots\\
p_n\to q
$$
A contrapositive proof is a direct proof of the statement
$$
\text{not }q\to\text{not }p
$$ and a proof by contradiction is a direct proof of the statement
$$
(p\text{ and not }q)\to \text{contradiction / absurdity}
$$
A: In classical logic, $p \implies q$ means only that it is false that both $p$ is true and $q$ is false. 

$p \implies q \space \space \equiv \space\space \neg (p \land \neg q)$

We can prove $p\implies q$ by either:


*

*Assuming $p$ is true, and then proving that $q$ must also be true.

*Assuming $q$ is false, and then proving that $p$ must also be false.

*Assuming $p$ is true and $q$ is false, and then obtainimg a contradiction of the forms $r\land \neg r$ or $r \iff \neg r$ 

*Proving $p$ is false. (Then there is no need to prove anything about $q$.)

*Proving $q$ is true. (Then there is no need to prove anything about $p$.)
