Q:

Help with simplifying p,q, logic and proof

How do I simplify the following:
$ (p \wedge q) \vee q$
$ p \to (p \to q)$
$\neg (\neg p \wedge \neg q)$

A:

Take a look at this page: Boolean algebra 
As regards the first one, by Absorption 2:
$(p \wedge q) \vee q\equiv q$.
As regards the second one, note that $p\to q\equiv \neg p \vee q$, hence
$$p \to (p \to q)\equiv \neg p \vee (p \to q)\equiv \neg p \vee (\neg p \vee q)\equiv (\neg p \vee \neg p) \vee q\equiv\neg p \vee q\equiv p \to q.$$
As regards the third one, use the De Morgan's Laws. Thus
$$\neg (\neg p \wedge \neg q)\equiv p \vee q.$$

