I have the following venn diagram with the yellow that show with regions are True
My task is to:
Write a propositional statement using $\to$ and $\neg$ (if necessary) equivalent to regions 3, 5 and 6.
How exactly do I do this?
$$ 3 - [P \land Q \land \neg R] $$ $$ 5 - [P \land \neg Q \land R] $$ $$ 6 - [P \land Q \land R] $$
How do I proceed from here? By the way, the answer is: $$ (P \to \neg Q) \to \neg (P \to \neg R) $$
You have the correct definitions for 3, 5, 6 however since you want all three of them you can describe it more easily as $(P \land Q) \lor (P \land R)$.
However since we want to write it only using $\to$ and $\neg$ we need to get it in a nicer form. First let's remember that $A \to B = (\neg A \lor B)$. However since we want to go the other way we need to remember that $\neg \neg A = A$. So $(A \lor B) = (\neg A \to B)$.
Going back to our original statement we need to switch the $\land$s to $\lor$s. So we use the rule $A\land B=\neg(\neg A \lor \neg B)$.
So rewriting the original we get $\neg(\neg P \lor \neg Q) \lor \neg(\neg P \lor \neg R)$.
Now substituting the identity we had earlier for the $\lor$ statements we get $\neg (P \to \neg Q) \lor \neg (P \to \neg Q)$.
And substituting one last time to get rid of the final $\lor$ we get $ (P \to \neg Q) \to \neg (P \to \neg R)$.
I hope this helps.