## Thinking Mathematically (6th Edition)

(a) Consider the provided statement “Automobile accidents reduce life expectancy by 500 days or drowning does not reduce life expectancy by 30 days.” By using the following representation in statement, $p:$ Automobile accidents reduce life expectancy by 500 days. $q:$ Drowning reduce life expectancy by 30 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 false and q is false. Hence, put the truth values of p and q in symbolical statement as $\text{F}\vee \sim \text{F}\equiv \text{F}\vee \text{T}$ It implies that the provided statement is true. (b) Consider the provided statement “Automobile accidents reduce life expectancy by 500 days or drowning does not reduce life expectancy by 30 days.” By using the following representation in statement, $p:$ Automobile accidents reduce life expectancy by 500 days. $q:$ Drowning reduce life expectancy by 30 days. Using the following representation, the symbolic form of the provided statement is $p\vee \sim q$. The negation of $p\vee \sim q$ is \begin{align} & \sim p\wedge \sim \left( \sim q \right) \\ & \sim p\wedge q \\ \end{align} By using p and q, the statement form of $\sim p\wedge q$is “Automobile accidents does not reduce life expectancy by 500 days and drowning reduce life expectancy by 30 days.” (c) Consider the provided statement “Automobile accidents reduce life expectancy by 500 days or drowning does not reduce life expectancy by 30 days.” By using the following representation in statement, $p:$ Automobile accidents reduce life expectancy by 500 days. $q:$ Drowning reduce life expectancy by 30 days. Using the following representation, the symbolic form of the provided statement is $\sim p\wedge q$. From the graph, it can be seen that p is false and q is false. Hence, put the truth values of p and q in symbolical statement as $\sim \text{F}\wedge \text{F}\equiv \text{T}\wedge \text{F}\equiv \text{F}$ It implies that the provided statement is false.