## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 3 - Logic - 3.4 Truth Tables for the Conditional and the Biconditional - Exercise Set 3.4 - Page 161: 90

#### Answer

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%is capitalists. $r$: 34% are members of the upper-middle class. The given compound statement can be written in the symbolic form as $p\to \left( q\vee r \right)$ A disjunction is false only when the truth values of all simple statements are false. In all other cases of truth values, it is true. 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{T} \\ & \text{ }\equiv \text{T} \\ \end{align}

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