## Thinking Mathematically (6th Edition)

$\begin{array}{llllll} p & q & r & \sim q & \sim q\vee r & p\wedge(\sim q\vee r)\\ \hline T & T & T & F & T & T\\ T & T & F & F & F & F\\ T & F & T & T & T & T\\ T & F & F & T & T & T\\ F & T & T & F & T & F\\ F & T & F & F & F & F\\ F & F & T & T & T & F\\ F & F & F & T & T & F \end{array}$
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 column: $\sim q$, negation of column 2: $\begin{array}{lllll} p & q & r & \sim q & \\ \hline T & T & T & F & \\ T & T & F & F & \\ T & F & T & T & \\ T & F & F & T & \\ F & T & T & F & \\ F & T & F & F & \\ F & F & T & T & \\ F & F & F & T & \end{array}$ Next column: $\sim q\vee r,$ disjunction, inputs from columns 3 and 4: $\begin{array}{lllllll} p & q & r & \sim q & \sim q\vee r & & \\ \hline T & T & T & F & T & & \\ T & T & F & F & F & & \\ T & F & T & T & T & & \\ T & F & F & T & T & & \\ F & T & T & F & T & & \\ F & T & F & F & F & & \\ F & F & T & T & T & & \\ F & F & F & T & T & & \end{array}$ Final column: $p\wedge(\sim q\vee r),$ conjunction, inputs from columns 1 and 5: $\begin{array}{llllll} p & q & r & \sim q & \sim q\vee r & p\wedge(\sim q\vee r)\\ \hline T & T & T & F & T & T\\ T & T & F & F & F & F\\ T & F & T & T & T & T\\ T & F & F & T & T & T\\ F & T & T & F & T & F\\ F & T & F & F & F & F\\ F & F & T & T & T & F\\ F & F & F & T & T & F \end{array}$