What is the inverse of $"P \mbox{ implies } (Q \mbox{ or } R)”$? What is the inverse of "$P \mbox{ implies} (Q \mbox{ or } R)$". 
I remember it to if not (Q or R) then not P.
Does this mean that that both Q and R have to be false for P to be false?

What do you mean by "inverse" ?

The contrapositive of $\;P\to (Q\vee R)\;$ is $\;(\neg Q\wedge \neg R)\to \neg P\;$.
If $P$ is true, then either $Q$ or $R$ are true.   If both $Q$ and $R$ are false, then $P$ is false.   These statements are equivalent.

The negation of $\;P\to (Q\vee R)\;$ is $\;P\wedge (\neg Q\wedge \neg R)$.
If $P$ is true, then either $Q$ or $R$ are true.   $P$ is true and both $Q$ and $R$ are false.   These statements contradict each other.
$$\begin{array}{c:c:c|c:c|c} P & Q & R & P\to(Q\vee R) & (\neg Q \wedge \neg R)\to P & P \wedge (\neg Q\wedge \neg R) \\ \hline T & T & T & T & T & \color{navy}{F} \\ T & T & \color{navy}{F} & T & T & \color{navy}{F} \\ T & \color{navy}{F} & T & T & T & \color{navy}{F}\\[1ex] T & \color{navy}{F} & \color{navy}{F} & \color{navy}{F} & \color{navy}{F} & T \\[1ex] \color{navy}{F} & T & T & T & T & \color{navy}{F} \\ \color{navy}{F} & T & \color{navy}{F} & T & T & \color{navy}{F} \\ \color{navy}{F} & \color{navy}{F} & T & T & T & \color{navy}{F} \\ \color{navy}{F} & \color{navy}{F} & \color{navy}{F} & T & T & \color{navy}{F}\end{array}$$
