Thinking Mathematically (6th Edition)

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

Chapter 3 - Logic - 3.4 Truth Tables for the Conditional and the Biconditional - Exercise Set 3.4 - Page 161: 89


The given compound statement can be written in simple statements as \[p,\text{ }q,\text{ and }r\].Here, \[p,\text{ }q,\text{ and }r\] represent three simple statements. \[p\]: There are more people in the lower-middle class than in the capitalist and upper-middle classes combined. \[q\]: 1% percent is capitalists. \[r\]: 34% are members of the upper-middle class. The given compound statement can be written in the symbolic form as \[p\to \left( q\wedge r \right)\] A conjunction is true only when the truth values of all simple statements are true. In all other cases of truth values, it is false. In case of a conditional statement, it is false only when the antecedent is true and the consequent is false. From the provided bar graph, it can be observed that \[p\] is true,\[q\] is true, and \[r\] is false. Put the truth values of \[p,\text{ }q,\text{ and }r\] in symbolic form to get\[\text{T}\to \left( \text{T}\vee \text{F} \right)\]. It can be further simplified as \[\begin{align} & \text{T}\to \left( \text{T}\vee \text{F} \right)\equiv \text{T}\to \text{F} \\ & \text{ }\equiv \text{F} \\ \end{align}\]
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