## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 3 - Logic - 3.6 Negations of Conditional Statements and De Morgan's Law - Exercise Set 3.6 - Page 179: 57

#### Answer

(a) Consider the provided statement “Homicide reduces life expectancy by 74 days or fire does not reduce life expectancy by 25 days.” By using the following representation in statement, $p:$Homicide reduces life expectancy by 74 days. $q:$Fire reduces life expectancy by 25 days. Using the following representation, the symbolic form of the provided statement is$p\vee \sim q$. From the graph, it can be seen that p is true and q is false. Hence, put the truth values of p and q in symbolical statement as $\text{T}\vee \sim \text{F}\equiv \text{T}\vee \text{T}\equiv \text{T}$ It implies that the provided statement is true. (b) Consider the provided statement “Homicide reduces life expectancy by 74 days or fire does not reduce life expectancy by 25 days.” By using the following representation in statement, $p:$Homicide reduces life expectancy by 74 days. $q:$Fire reduces life expectancy by 25 days. Using the following representation, the symbolic form of the provided statement is $p\vee \sim q$. Negation of $p\vee \sim q$ is \begin{align} & \sim p\vee \sim \left( \sim q \right) \\ & \sim p\wedge q \\ \end{align} By using p and q, statement form of $\sim p\wedge q$is “Homicide does not reduces life expectancy by 74 days and fire reduces life expectancy by 25 days.” (c) Consider the provided statement “Homicide reduces life expectancy by 74 days or fire does not reduce life expectancy by 25 days.” By using the following representation in statement, $p:$Homicide reduces life expectancy by 74 days. $q:$Fire reduces life expectancy by 25 days. Using the following representation, the symbolic form of the provided statement is$p\vee \sim q$. From the graph, it can be seen that p is true and q is false. Hence, put the truth values of p and q in symbolical statement as $\text{T}\wedge \text{F}\equiv \text{F}$ It implies that the provided statement is false.

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