Proving "$p\wedge q$ imples $r$" problems by contraposition I am trying to prove a particular problem of the form "p^q $\Rightarrow$ r" by contrapositive, so I have to show that " $\neg$ r $\Rightarrow$ $\neg$ (p^q)". However, this is also equivalent to "$\neg$ r $\Rightarrow$ $\neg$ p v $\neg$q. Is this also equivalent to proving that "$\neg$ r ^ p $\Rightarrow$$\neg$ q"? That is, if I prove "$\neg$ r ^ p $\Rightarrow$$\neg$ q", have I proven "p^q $\Rightarrow$ r"?
 A: They are the same, but I'll try and prove it using a practical example, which you can translates into predicate language.
P="There are clouds in the sky".
Q="They are filled with water".
R="It's going to rain".
$P \wedge Q \implies R = $ "If there are clouds in the sky and they are filled with water then it's going to rain."
$\neg R \implies \neg (P \wedge Q) \iff \neg R \implies \neg P \vee \neg Q)$ = "If it doesn't rain then either there are no clouds in the sky or they are not filled with water".
$\neg R \wedge P \implies \neg Q$ = "If it hasn't rained, and there are clouds on top, then surely they will not contain water".
It is clear that the first statement implies the second, namely $(P \wedge Q \implies R) \implies (\neg R \wedge P \implies \neg Q)$. But you need to study implication the other way, because you are starting with the second statement and trying to prove the first.
So start with the second statement. Suppose it is true that "If it hasn't rained, and there are clouds in the sky, then surely they will not contain water"(Call this statement (*)). Then if we are trying to prove the first statement, we must assume that "there are clouds in the sky and they contain water"(call this statement (+)), and try to prove that it did rain. We do this by contradiction, namely assuming both (*) and (+) are true and showing it can't possibly have not rained. 
Suppose it didn't rain. By (+) we know it's true that there are clouds in the sky. By (*), we know that  these clouds don't contain water. But by the other part of (+), the clouds contain water. The clouds can't possibly both contain and not contain water, hence there is contradiction, so it must have rained.
This is a non-trivial implication, I thought I could make it clear with this example (forgive me for my climate-scientific errors if any). Hence you are correct with your assumption. 
