I'm trying to understand the logic behind "proof by contradiction" and hoping that I can clear up a few things in this post.
First of all, suppose I have a proposition $P$ and from this I can imply some statement $Q$, i.e. $P\Rightarrow Q$, then to prove this by contradiction I assume the negation of this, $\neg \left(P\Rightarrow Q \right)$, which is equivalent to assuming $P\wedge \neg Q$, i.e. $\neg \left(P\Rightarrow Q \right)=P\wedge \neg Q$. Is there any way to prove this algebraically (apologies, I'm fairly new to logic)? I have shown this is true via a truth table, but not sure whether this counts as a proof or not?!
Next, having assumed $P\wedge \neg Q$, following logically from this if we find some statement $R$ such that a contradiction arises, $R\wedge \neg R$ then it follows that $\neg \left(P\Rightarrow Q \right)\Rightarrow R\wedge \neg R$. Now, my understanding of this is that, as $R\wedge \neg R$ is always false, then the only way this implication can be true is if $\neg \left(P\Rightarrow Q \right)$ is false, which implies that $P\Rightarrow Q $ is true, hence proving think intial claim. Is this the correct reasoning?
Thanks for your time.