Originally Posted by

**jvignacio** Rewrite: $\displaystyle q \wedge ( \sim (((p \vee (p \wedge q)) \wedge (p \vee q)) \vee q))$ and determine if its tautology, contradiction or neither. Using truth table I got contradiction.

My working out:

$\displaystyle q \wedge ( \sim (((p \vee (p \wedge q)) \wedge (p \vee q)) \vee q))$

$\displaystyle \equiv$ $\displaystyle q \wedge ( \sim ((p \wedge (p \vee q)) \vee q))$

$\displaystyle \equiv$ $\displaystyle q \wedge ( \sim (p \vee q))$

$\displaystyle \equiv$ $\displaystyle q \wedge ( \sim p \wedge \sim q)$

$\displaystyle \equiv$ $\displaystyle q \wedge ( \sim q \wedge \sim p)$

$\displaystyle \equiv$ $\displaystyle {\color{red}(q \wedge \sim q)} \wedge \sim p$