Logic Equivalences I am having trouble finding the logical equivalence to $\neg P \Rightarrow \neg Q$. I have tried factoring out the negation and going from their but it is not working out. Any ideas on how to g about doing this?  

eIt depends on the rules, but $A \Rightarrow B$ is normally defined as $\neg A \lor B$. Then
$$
\neg P \Rightarrow \neg Q \equiv (\neg \neg P) \lor \neg Q \equiv 
P \lor \neg Q \equiv Q \Rightarrow P
$$
All those espressions are logically equivalent.
If you want to prove that $P \Rightarrow Q$ and $\neg P \Rightarrow \neg Q$ are not equivalent you can just look at a counterexample, e.g. if $P$ is false and $Q$ is true then $P \Rightarrow Q$ is true, but $\neg P \Rightarrow \neg Q$ is false.
