Simplify $(p \vee q) \wedge(p \vee \neg q) $ to obtain $ (p \vee(q \wedge \neg q))$

Question

I am unable to simplify the equation $$ (p \vee q) \wedge(p \vee \neg q) $$ to $$ (p \vee(q \wedge \neg q)) $$.

Answer 1

Use absorption in a chain.

$$(p\lor q)\land (p\lor \lnot q)$$

$$=(p\land p)\lor (p\land q)\lor(p\land \lnot q) \lor (q \land\lnot q)\quad(\text{distribution)}$$

$$=p\lor (p\land q)\lor(p\land \lnot q) \lor (q \land\lnot q)\quad(\text{idempotence)}$$

$$=p\lor(p\land \lnot q) \lor (q \land\lnot q)\quad(\text{absorption)}$$

$$=p \lor (q \land\lnot q)\quad(\text{absorption)}$$

Answer 2

$q \wedge \neg q$ is always false, so $(p \vee(q \wedge \neg q))=p$.

If $p$ is true, $(p \vee q) \wedge(p \vee \neg q)$ will be "true and true", it therefore is true. Conversely, if $p$ is false, $(p \vee q) \wedge(p \vee \neg q)$ will be "false and false", it therefore is false. Thus, $(p \vee q) \wedge(p \vee \neg q)=p=(p \vee(q \wedge \neg q))$.

Answer 3

I'm Transforming this to Binary Notations as it makes it slightly simpler to handle with $\displaystyle ( p+q) *( p+\overline{q}) \ =\ pp+p\overline{q} \ +\ qp\ +\ q\overline{q} \ =\ p+p\overline{q} +qp\ =\ p( 1+\overline{q}) +qp$
$\displaystyle p+pq\ =\ p( 1+q) \ =\ p$
Now the second statement is $\displaystyle p+( q*\overline{q}) \ =\ p$
Since both statements are equating to the same value, both statements have to be equal to each other.