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: 33

Answer

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

Work Step by Step

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 columns 3 and 4, use the negation table for $\sim p$ and $\sim q$ $\begin{array}{llllll} p & q & \sim p & \sim q & & \\ \hline T & T & F & F & & \\ T & F & F & T & & \\ F & T & T & F & & \\ F & F & T & T & & \end{array}$ Next column: $p\vee q$ (disjunction, inputs: columns 1 and 2) $\begin{array}{llllll} p & q & \sim p & \sim q & p\vee q & \\ \hline T & T & F & F & T & \\ T & F & F & T & T & \\ F & T & T & F & T & \\ F & F & T & T & F & \end{array}$ Next column: $\sim p\vee\sim q$ (disjunction, inputs: columns 3 and 4) $\begin{array}{llllll} p & q & \sim p & \sim q & p\vee q & \sim p\vee\sim q\\ \hline T & T & F & F & T & F\\ T & F & F & T & T & T\\ F & T & T & F & T & T\\ F & F & T & T & F & T \end{array}$ Final column: $(p\vee q)\wedge(\sim p\vee\sim q)$ (conjunction, inputs: columns $5$ and $6$) $\begin{array}{lllllll} p & q & \sim p & \sim q & p\vee q & \sim p\vee\sim q & (p\vee q)\wedge(\sim p\vee\sim q)\\ \hline T & T & F & F & T & F & F\\ T & F & F & T & T & T & T\\ F & T & T & F & T & T & T\\ F & F & T & T & F & T & F \end{array}$
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