## Thinking Mathematically (6th Edition)

$\begin{array}{lllllll} p & q & r & \sim p & \sim q & r\vee\sim p & (r\vee\sim p)\wedge\sim q\\ \hline T & T & T & F & F & T & F\\ T & T & F & F & F & F & F\\ T & F & T & F & T & T & T\\ T & F & F & F & T & F & F\\ F & T & T & T & F & T & F\\ F & T & F & T & F & T & F\\ F & F & T & T & T & T & T\\ F & F & F & T & T & T & T \end{array}$
Set up a truth table for three inputs, p,q and r: $\begin{array}{lllll} p & q & r & ... & \\ \hline T & T & T & & \\ T & T & F & & \\ T & F & T & & \\ T & F & F & & \\ F & T & T & & \\ F & T & F & & \\ F & F & T & & \\ F & F & F & & \end{array}$ Next two columns: $\sim p$ and $\sim q$, negation of columns 1 and 2: $\begin{array}{lllllll} p & q & r & \sim p & \sim q & & \\ \hline T & T & T & F & F & & \\ T & T & F & F & F & & \\ T & F & T & F & T & & \\ T & F & F & F & T & & \\ F & T & T & T & F & & \\ F & T & F & T & F & & \\ F & F & T & T & T & & \\ F & F & F & T & T & & \end{array}$ Next column: $r\vee\sim p$, disjunction of columns 3 and 4: $\begin{array}{lllllll} p & q & r & \sim p & \sim q & r\vee\sim p & \\ \hline T & T & T & F & F & T & \\ T & T & F & F & F & F & \\ T & F & T & F & T & T & \\ T & F & F & F & T & F & \\ F & T & T & T & F & T & \\ F & T & F & T & F & T & \\ F & F & T & T & T & T & \\ F & F & F & T & T & T & \end{array}$ Final column: $(r\vee\sim p)\wedge\sim q$, conjunction of columns 6 and 5: $\begin{array}{lllllll} p & q & r & \sim p & \sim q & r\vee\sim p & (r\vee\sim p)\wedge\sim q\\ \hline T & T & T & F & F & T & F\\ T & T & F & F & F & F & F\\ T & F & T & F & T & T & T\\ T & F & F & F & T & F & F\\ F & T & T & T & F & T & F\\ F & T & F & T & F & T & F\\ F & F & T & T & T & T & T\\ F & F & F & T & T & T & T \end{array}$