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: 41

Answer

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

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 two columns: $\sim r$ and $(p\vee q)$, negation of column 3, disjunction of columns 1 and 2: $\begin{array}{lllllll} p & q & r & \sim r & p\vee q & & \\ \hline T & T & T & F & T & & \\ T & T & F & T & T & & \\ T & F & T & F & T & & \\ T & F & F & T & T & & \\ F & T & T & F & T & & \\ F & T & F & T & T & & \\ F & F & T & F & F & & \\ F & F & F & T & F & & \end{array}$ Next column: $\sim(p\vee q)$, negation of column 5: $\begin{array}{lllllll} p & q & r & \sim r & p\vee q & \sim(p\vee q) & \\ \hline T & T & T & F & T & F & \\ T & T & F & T & T & F & \\ T & F & T & F & T & F & \\ T & F & F & T & T & F & \\ F & T & T & F & T & F & \\ F & T & F & T & T & F & \\ F & F & T & F & F & T & \\ F & F & F & T & F & T & \end{array}$ Next column: $\sim(p\vee q)\wedge\sim r$, conjunction of columns 6 and 4: $\begin{array}{lllllll} p & q & r & \sim r & p\vee q & \sim(p\vee q) & \sim(p\vee q)\wedge\sim r\\ \hline T & T & T & F & T & F & F\\ T & T & F & T & T & F & F\\ T & F & T & F & T & F & F\\ T & F & F & T & T & F & F\\ F & T & T & F & T & F & F\\ F & T & F & T & T & F & F\\ F & F & T & F & F & T & F\\ F & F & F & T & F & T & T \end{array}$
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