# Reducing propositional logic statements

I am having some trouble with reducing some propositional logic statements.

The first one is as follows:

$\neg(P \lor Q) \lor \neg (P \lor \neg Q)$.

I used deMorgan's law to change this to:

$(\neg P \land \neg Q) \lor (\neg P \land Q)$

Is there anything I can do to reduce this further?

The second one is:

$(P \lor Q) \leftrightarrow (\neg P \land Q)$

I have no idea where to even start on this one.

Any tips?

Thanks.

(not(P or Q) or not (P or not Q).

I used deMorgan's law to change this to:

(not P and not Q) or (not P and Q)

\begin{align}\neg (P\lor Q)\lor \neg (P\lor \neg Q) & \equiv (\neg P\land \neg Q)\lor (\neg P\land Q) & \textsf{de'Morgan's Laws} \\ & \equiv \neg P \land (\neg Q\lor Q) & \textsf{Distribution} \end{align}

Can you take it from there?

((P or Q) iff (not P and Q)

First use $A\leftrightarrow B \equiv (A\land B)\lor(\neg A\land \neg B)$

\begin{align}(P\lor Q)\leftrightarrow (\neg P\land Q) & \equiv \big( (P\lor Q)\land(\neg P\land Q) \big) \lor \big(\neg (P\lor Q)\land \neg (\neg P\land Q) \big) \end{align}

Now simplify...

• Ok, so I realize by looking at it that the first part of that ((P∨Q)∧(¬P∧Q)) can be simplified to (¬P∧Q), but what steps do I need to show to get to that? The second part can be simplified to (¬P∧¬Q), but again, I don't know the steps to prove it. – dibdub Jan 24 '16 at 6:25