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. complete parts (a) through (c) below for the following statement.

heart disease reduces life expectancy by 1555 days and being in a fire reduces life expectancy by 18 days.

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

true
false

b. write the compound statements negation. choose the correct answer below.

a. heart disease does not reduce life expectancy by 1555 days or being in a fire reduces life expectancy by 18 days.
b. heart disease reduces life expectancy by 1555 days and being in a fire reduces life expectancy by 18 days.
c. heart disease does not reduce life expectancy by 1555 days and being in a fire does not reduce life expectancy by 18 days.
d. heart disease does not reduce life expectancy by 1555 days or being in a fire does not reduce life expectancy by 18 days.

c. use the information in the graph to determine the truth value of the negation in part (b). choose the correct answer below.

false
true

Explanation:

Identify the compound statement

The given compound statement is: "Heart disease reduces life expectancy by 1555 days and being in a fire reduces life expectancy by 18 days."
Let \(p\) be "Heart disease reduces life expectancy by 1555 days."
Let \(q\) be "Being in a fire reduces life expectancy by 18 days."
The compound statement is a conjunction: \(p \land q\).

Determine truth value of part (a)

From the bar graph:

• Heart disease reduces life expectancy by 1555 days (this is true, \(p = \text{T}\)).
• The text states "being in a fire reduces life expectancy by 18 days." Since the graph shows five hazards (Heart Disease: 1555, Cancer: 1116, Homicide: 92, Home Accident: 60, Drowning: 23) and the prompt states "being in a fire reduces life expectancy by 18 days" as a given fact for the statement, both component statements are true.

Thus, \(p \land q\) is True.

Formulate the negation for part (b)

Using De Morgan's Laws, the negation of a conjunction \(p \land q\) is:

$$ eg(p \land q) \equiv eg p \lor eg q$$

In words: "Heart disease does not reduce life expectancy by 1555 days or being in a fire does not reduce life expectancy by 18 days."
This matches option D.

Determine truth value of part (c)

Since the original compound statement \(p \land q\) is True, its negation \(
eg(p \land q)\) must have the opposite truth value.
Therefore, the truth value of the negation is False.

Answer:

Question a

• True (Correct answer)
• False

Question b

• A. Heart disease does not reduce life expectancy by 1555 days or being in a fire reduces life expectancy by 18 days.
• B. Heart disease reduces life expectancy by 1555 days and being in a fire reduces life expectancy by 18 days.
• C. Heart disease does not reduce life expectancy by 1555 days and being in a fire does not reduce life expectancy by 18 days.
• D. Heart disease does not reduce life expectancy by 1555 days or being in a fire does not reduce life expectancy by 18 days. (Correct answer)

Question c

• False (Correct answer)
• True