Statements for Proof by Contradiction I was reading the following notes about Proof by Contradiction

I understood that (as given above) for showing $P\implies Q$ is true $\equiv (\sim P \vee Q)$ is true $\equiv (P \wedge \sim Q)$ is false.
But if I write it in the following manner I get:
$P \implies Q$ is true $\equiv (P \nRightarrow Q)$ is false $\equiv P \implies \sim Q$ is false (I am not sure if this step is right or wrong. I thought one could try it out in this way; and to verify if I get the same statement as above) $\equiv (\sim P \space \vee \sim Q)$ is false $\equiv \space \sim (P \wedge Q)$ is false.
I am not able to figure out specifically what mistake did I commit above, because of which I am getting two different statements. Please help.
Thanks.
 A: Your mistake was to claim $P\not\to Q$ to be equivalent to $P\to\neg Q$.
You can see that this is clearly not the case if you express the first statement as $(P\land\neg Q)$: this says that it is always the case that both $P$ and not $Q$. On the other hand, $P\to\neg Q$ makes the weaker claim that whenever it is the case that $P$, then not $Q$ must follow.
\begin{array}{c|c|c|c}P&Q&P\land\neg Q&P\to\neg Q\\\hline T&T&F&F\\T&F&T&T\\F&T&F&T\\F&F&F&T\end{array}
As you can see, $P\to\neg Q$ is also true when $P$ is false, and is actually equivalent to $\neg(P\land Q)$.
Thus, the following expressions are all equivalent:
$$(P\to Q)\equiv\neg(P\not\to Q)\equiv\neg(\neg(P\to Q))\equiv\neg(P\land\neg Q)\equiv(\neg P\lor Q)$$
A: Your suspicion is right, the second equivalence is incorrect. $P \not\implies Q$ is true $\not\equiv P \implies \sim Q$ is false; that is a strictly stronger claim.
For example, if P is "X is a mammal" and Q is "X is a man" then (given my admittedly limited knowledge of taxonomy), "X is a mammal" implies "X is a man" ($P \implies Q$) is false. This is indeed equivalent to saying "X is a mammal" does not imply "X is a man" ($P \not\implies Q$) is true. However, this is not equivalent to saying that "X is a mammal" implies "X is  not a man" ($P \implies \sim Q$) is true, which is a strictly stronger statement (we can see this is the case because it produces a false inference; I am a mammal and I am also a man).
P.S.: I have stuck to your convention of appending "is true" or "is false" after every statement; however, in longer proofs this is very cumbersome and generally writing $A\equiv B$ is just as good as saying $A$ is true $\equiv B$ is true, and $A\equiv \sim B$ is just as good as saying $A$ is true $\equiv B$ is false.
