Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 3 - Logic - 3.3 Truth Tables for Negation, Conjunction and Disjunction - Exercise Set 3.3: 38

Answer

$\begin{array}{llllll} p & q & r & \sim q & \sim q\wedge r & p\vee(\sim q\wedge r)\\ \hline T & T & T & F & F & T\\ T & T & F & F & F & T\\ T & F & T & T & T & T\\ T & F & F & T & F & T\\ F & T & T & F & F & F\\ F & T & F & F & F & F\\ F & F & T & T & T & T\\ F & F & F & T & F & F \end{array}$

Work Step by Step

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