## Thinking Mathematically (6th Edition)

$\begin{array}{llllll} p & q & \sim q & p\wedge\sim q & p\wedge q & (p\wedge\sim q)\vee(p\wedge q)\\ \hline T & T & F & F & T & T\\ T & F & T & T & F & T\\ F & T & F & F & F & F\\ F & F & T & F & F & F \end{array}$
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 the next column (3rd) , use the negation table for $\sim q$ $\begin{array}{lllll} p & q & \sim q & & \\ \hline T & T & F & & \\ T & F & T & & \\ F & T & F & & \\ F & F & T & & \end{array}$ Next column: $p\wedge\sim q$ (conjunction, inputs: columns 1 and 3) $\begin{array}{lllll} p & q & \sim q & p\wedge\sim q & \\ \hline T & T & F & F & \\ T & F & T & T & \\ F & T & F & F & \\ F & F & T & F & \end{array}$ Next column: $p\wedge q$ (conjunction, inputs: columns 1 and 2) $\begin{array}{llllll} p & q & \sim q & p\wedge\sim q & p\wedge q & \\ \hline T & T & F & F & T & \\ T & F & T & T & F & \\ F & T & F & F & F & \\ F & F & T & F & F & \end{array}$ Final column: $(p\wedge\sim q)\vee(p\wedge q)$ (disjunction, inputs: columns $4$ and $5$) $\begin{array}{llllll} p & q & \sim q & p\wedge\sim q & p\wedge q & (p\wedge\sim q)\vee(p\wedge q)\\ \hline T & T & F & F & T & T\\ T & F & T & T & F & T\\ F & T & F & F & F & F\\ F & F & T & F & F & F \end{array}$