How to negate an implication in English? How to negate this proposition:
"If $xy$ is irrational then either $x$ is irrational or $y$ is irrational. "
Because the negation of $p\Rightarrow q$ is $p \wedge \text{not } q$. If I translate this sentence into English, it would be "$xy$ is irrational and $x$ and $y$ are rational." It is so strange. 

$P$ in this case is "$xy$ is irrational" and $Q$ is "either $x$ is irrational or $y$ is irrational". You wrote that $\lnot(P \implies Q)$ is $(P \wedge \lnot Q)$. So, colloquially, I would suggest that $\lnot Q$ is "neither $x$ nor $y$ are irrational". This colloquially gives us:
$P \wedge \lnot Q$: "$xy$ is irrational and neither $x$ nor $y$ are irrational."
