Source: p 46, How to Prove It, by Daniel Velleman
Please beware that although the author writes the original apodosis as 'You’ll fail the course',
I shorten it to 'You'll fail', for convenience.
Let $P$ be the statement “You will neglect your homework” and $Q$ be “You’ll fail.”
Then “You won’t neglect your homework, or you’ll fail.” = $\lnot P \vee Q$.
But what message is the teacher trying to convey with this statement?
Clearly the intended message is
“If you neglect your homework, then you’ll fail the course,” or in other words $P \rightarrow Q.$ Thus, in this example, the statements $\lnot P \vee Q$ and $ P \rightarrow Q $ seem to mean the same thing.
But why can the bolded NOT be symbolised as:
$P \vee \lnot Q $ = "You neglect your homework or you won't fail"?
How would this motivate the Conditional Law?