In a course on logic and proofs the professor presented on the following lines to show an example of negation:
$$
\neg (P \Rightarrow Q) \ \ \ \ \Longleftrightarrow \ \ \ \ P \wedge \neg Q
$$
I can't wrap my head around why $\neg (P \Rightarrow Q)$ would be equivalent to the RHS of the above statement. Somehow, we are going from the fact the $P$ does not imply $Q$ to a statement that says that $P$ is true and $Q$ is not, while the LHS statement doesn't say anything about $P$ being true. Is it this line or I that am missing something?
The only logical implication that I can make out of this is that:
$ \exists x. P(x) \Rightarrow \neg Q(x)$
Best Answer
It's because $A\to B$ is equivalent to $(\lnot A)\lor B$ and the negation of that is equivalent to $A\land \lnot B$.