Boolean simplification expression

Question

In words my problem is

NOT(p AND q) AND (NOT p OR q) AND (p OR p).

I have rewritten it in symbols

¬(p ∧ q) ∧ (¬p ∨ q) ∧ (p ∨ p)

I got this far:

(¬p ∨ ¬q) ∧(¬p ∨ q) ∧ p

Any help please?

Answer 1

Hint: Using De Morgan's law, your last step is: [(~p v (~q and q) ) and p].

Answer 2

You can use De Morgan's law to start off with

$$\neg( (p \wedge q) \vee (p \wedge \neg q) \vee \neg p)$$

Then you can combine the first and second terms as $(p \wedge q) \vee (p \wedge \neg q) = p$. See below for the intuition beind this: $(p \wedge q)$ in red, and $ (p \wedge \neg q) = p$ in green.

You should be good to go from there on in.

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Alternatively, without the need for any intutition, it's possible to simply follow the rules of Boolean algebra. First of all you multiply out the brackets using the distributive law.

$$\begin{align} (\neg p \vee \neg q) \wedge (\neg p \vee q) \wedge p &= (\neg p \vee \neg q) \wedge (\neg p \wedge p \;\vee\; q \wedge p) \\ &= (\neg p \vee \neg q) \wedge (q \wedge p) \\ &= \neg p \wedge q \wedge p \;\;\vee\;\; \neg q \wedge q \wedge p \end{align}$$

Second you use the associative law to rearrange the logical AND on the left to $\neg p \wedge p \wedge q$.