## Thinking Mathematically (6th Edition)

$\begin{array}{llllll} p & q & \sim p & \sim q & p\vee\sim q & \sim p\wedge( p\vee\sim q)\\ \hline T & T & F & F & T & F\\ T & F & F & T & T & F\\ F & T & T & F & F & F\\ F & F & T & T & T & T \end{array}$
Set up a truth table for two inputs, p and q: $\begin{array}{llll} p & q & ... & ...\\ \hline T & T & & \\ T & F & & \\ F & T & & \\ F & F & & \end{array}$ In columns 3 and 4, use the negation table for $\sim p$ and $\sim q$ $\begin{array}{llllll} p & q & \sim p & \sim q & & \\ \hline T & T & F & F & & \\ T & F & F & T & & \\ F & T & T & F & & \\ F & F & T & T & & \end{array}$ In the next column (5th) use inputs from columns 1 and 4 for $p\vee\sim q$ and the disjunction truth table, (true if any of the inputs are true) $\begin{array}{llllll} p & q & \sim p & \sim q & p\vee\sim q & \\ \hline T & T & F & F & T & \\ T & F & F & T & T & \\ F & T & T & F & F & \\ F & F & T & T & T & \end{array}$ Finally, for $\sim p\wedge( p\vee\sim q)$, use conjunction truth table on inputs from columns 3 and 5 (true only if both inputs are true) $\begin{array}{llllll} p & q & \sim p & \sim q & p\vee\sim q & \sim p\wedge( p\vee\sim q)\\ \hline T & T & F & F & T & F\\ T & F & F & T & T & F\\ F & T & T & F & F & F\\ F & F & T & T & T & T \end{array}$