Question

the bar graph shows six leading causes of death in a certain country, along with the average number of days of life lost for each hazard.
answer parts (a), (b), and (c) below for the following statement.
heart disease reduces life expectancy by 1652 days or cancer does not reduce life expectancy by 1110 days.

a. use the information given by the graph to determine the truth value of the compound statement. choose the correct answer below.
false
true

b. write the compound statements negation. choose the correct answer below.
a. heart disease does not reduce life expectancy by 1652 days or cancer reduces life expectancy by 1110 days.
b. heart disease does not reduce life expectancy by 1652 days and cancer reduces life expectancy by 1110 days.
c. heart disease does not reduce life expectancy by 1652 days or cancer does not reduce life expectancy by 1110 days.
d. heart disease does not reduce life expectancy by 1652 days and cancer does not reduce life expectancy by 1110 days.

Explanation:

Determine truth value of the compound statement

The statement is: "Heart disease reduces life expectancy by 1652 days or cancer does not reduce life expectancy by 1110 days."
Let \(p\) be "Heart disease reduces life expectancy by 1652 days." From the graph, this is True.
Let \(q\) be "Cancer reduces life expectancy by 1110 days." From the graph, this is True. Thus, \(\sim q\) ("cancer does not reduce life expectancy by 1110 days") is False.
The compound statement is \(p \lor \sim q\). Since \(p\) is True, the disjunction \(p \lor \sim q\) is True.

Write the compound statement's negation

The negation of \(p \lor \sim q\) is \(\sim(p \lor \sim q)\).
By De Morgan's Laws, \(\sim(p \lor \sim q) \equiv \sim p \land q\).
Translating \(\sim p \land q\) back to English: "Heart disease does not reduce life expectancy by 1652 days and cancer reduces life expectancy by 1110 days."
This matches option B.

Answer:

Question a

• False
• True (Correct answer)

Question b

• A. Heart disease does not reduce life expectancy by 1652 days or cancer reduces life expectancy by 1110 days.
• B. Heart disease does not reduce life expectancy by 1652 days and cancer reduces life expectancy by 1110 days. (Correct answer)
• C. Heart disease does not reduce life expectancy by 1652 days or cancer does not reduce life expectancy by 1110 days.
• D. Heart disease does not reduce life expectancy by 1652 days and cancer does not reduce life expectancy by 1110 days.