Homework 1 – Solutions
Chapter 1B
Multiple Negations. Explain the meaning of the given statement, then answer the question
that follows.
23. Sarah did not decline the offer to go to dinner. Did Sarah go to dinner?
Sarah accepted the offer to go to dinner, so Sarah did go to dinner.
And Statements. The following propositions have the form p and q. State p and q, and
give their truth values. Then determine whether the entire proposition is true or false,
and explain why.
35. Some people are happy and some people are short.
“Some people are happy” is a true statement, as is the statement “some people are
short,” so the conjunction is true.
Interpreting or. State whether or is used in the inclusive or exclusive sense in the following
propositions.
41. My next vacation will be in Mexico or Costa Rica.
This is an example of an exclusive or (assuming the planned vacation is not some
kind of tour).
If . . . then Statements. Identify the hypothesis and conclusion in the following propositions, and state their truth values. Then determine whether the entire proposition is
true or false.
67. If pigs can fly, then fish can brush their teeth.
The hypothesis is “pigs can fly” and the conclusion is “fish can brush their teeth”;
since the hypothesis is false, the entire proposition is vacuously true.
Chapter 1C
Classifying Numbers. Choose the first set in the list natural numbers, whole numbers,
integers, rational numbers, and real numbers that describes the following numbers.
15. 2/3
rational number
16. −5/2
rational number
19. π
24. 123/456
real number
rational number
Set Notation. Use set notation (braces) to write the members of the following sets, or state
that the set has no members. You may use “. . . ” to indicate patterns.
32. Every third number between 4 and 20 beginning with 4
{4, 7, 10, 13, 16, 19}
33. The perfect squares (such as 12 , 22 , 32 ) between 30 and 130
{62 , 72 , 82 , 92 , 102 , 112 } = {36, 49, 64, 81, 100, 121}
34. The kings of (the U.S. of) America
The set is empty.
35. Odd numbers between 2 and 30 that are multiples of 3
{3, 9, 15, 21, 27}
Venn Diagrams for Two Sets. Draw Venn diagrams with two circles showing the relationship between the following pairs of sets. Provide an explanation of the diagram you
drew.
39. words and verbs
words
40. reptiles and bacteria
reptiles
bacteria
verbs
Categorical Propositions. For the given categorical propositions, do the following.
a. If necessary, rephrase the statement in standard form.
b. State the subject and predicate sets.
c. Draw a Venn diagram for the propositions and label all regions of the diagram.
47. All U.S. presidents have been over 30 years old.
a. All U.S. presidents are people over 30 years old.
b. S = U.S. presidents; P = people over 30 years old
people
over 30
U.S.
presidents
c.
48. Every child can sing.
a. All children are people who can sing.
b. S = children; P = people who can sing
people
who can sing
children
c.
Venn Diagrams for Three Sets. Draw Venn diagrams with three overlapping circles (eight
regions) for the following groups of three sets. Describe the members of each region or
state that a region has no members.
56. oceans, bodies of salt water, bodies of fresh water
fresh & salt water ocean = empty
salt water oceans
oceans
salt water
bodies
fresh water
bodies
fresh water ocean = empty
fresh & salt body of water
Venn Diagram with Numbers. Use the Venn diagram to answer the following questions.
people at a party
men
under age 30
22
15
16
28
59. a. How many women at the party are under 30?
16
b. How many men at the party are not under 30?
22
c. How many women are at the party?
44
d. How many people are at the party?
81
Venn Diagram with Numbers. Use the Venn diagram to answer the following questions.
people at a conference
women
college degree
20
4
8
currently employeed
11
9
16
6
3
62. a. How many people at the conference are employed men without a college degree?
6
b. How many people at the conference are unemployed women?
24
c. How many people at the conference are unemployed men without a college degree?
3
d. How many people are at the conference?
77
70. Readership Survey. A (hypothetical) survey revealed the following results about the
news sources that a sample of 130 people use:
TV/radio only
Internet only
Newspapers only
None
20
29
15
6
TV/radio and Internet only
TV/radio and newspapers only
Internet and newspapers only
All three sources
12
18
22
8
a. Draw a three circle Venn diagram that summarizes the results of the survey.
TV/radio
Internet
12
20
18
Newspapers
8
29
22
15
6
b. How many people use (at least) TV/radio and newspapers?
c. How many people use TV/radio or Internet?
26
109
d. How many people use TV/radio or Internet, but not newspapers?
61
e. How many people use Internet, but not TV/radio?
51
f. How many people use TV/radio, but not newspapers?
32
Organizing Propositions. Draw a Venn diagram that represents the information in the
following statements. Use the diagram (and no other information) to answer the questions that follow. Explain your reasoning.
83. All meat has protein. All dairy products have protein. Some beans have protein. All
beans, but no meat or dairy products, are plants.
Questions: Could there be beans that are dairy products? Could there be meat that
is a dairy product? Could there be dairy products that are plants? Could there be
plants with protein?
protein
No beans can be dairy products (since
no dairy product are plants); Some
meat could be a dairy product; No dairy
products are plants (as given); and,
some plants have protein (e.g. some
beans).
plants
meat
beans
dairy
84. No Republicans are Democrats. No Republicans are Green Party members. All Republicans are conservative. Some liberals are Democrats. No liberals are conservatives.
Questions: Could there be conservative Democrats? Could there be liberal Green
Party members? Could there be liberal Republicans?
Dem
conservatives
liberals
Rep
GP
Yes,
there can be conservative
Democrats; Yes, there can be liberal Green Party members; but, since
“all Republicans are conservative” and
“no liberals are conservative,” there
cannot be liberal Republicans.
Chapter 2A
Working with Fractions. The following exercises require the skills covered in the Brief
Review on pp. 84–85
15. Write each of the following as a common fraction.
a. 3.5
e. 2.15
a. 7/2
e. 43/20
b. 0.3
f. 0.35
c. 0.05
g. 0.98
d. 4.1
h. 4.01
b. 3/10
f. 7/20
c. 1/20
g. 49/50
d.
h.
41/10
401/100
17. Convert the following fractions to decimal form; round to the nearest thousandth if
necessary.
a.
e.
1
4
13
2
b.
f.
3
8
23
6
c.
g.
2
3
103
50
d.
h.
3
5
42
26
a. 0.25
b. 0.375
c. 0.667
d. 0.6
e. 6.5
f. 3.833
g. 2.06
h. 1.615
Identifying Units. Identify the units of the following quantities. State the units mathematically (for example, mi/hr) and in words (for example miles per hour).
21. The cost of a piece of carpet, found by dividing the price in dollars by its area in
square yards
dollars per square yard (USD/yd2 )
22. The flow rate of a river in which 5000 cubic feet of water flow past a particular location
every second
5000 cubic feet per second (ft3 /s)
Unit Conversions. Carry out the following unit conversions.
27. Convert 24 feet to inches.
12 in
24 ft = 24 ft ×
= 288 in
1 ft
28. Convert 24 feet to yards.
24 ft = 24 ft ×
1 yd
= 8 yd
3 ft
Conversions with Square and Cubic Units.
39. Find a conversion factor between square feet and square inches. Write it in three
forms.
1 ft2 = 12 in ×12 in = 144 in2 ;
144 in2 /1 ft2 = 1;
1 ft2 /144 in2 = 1
42. Find the area in square feet of a rectangular yard that measures 20 yards by 12 yards.
ft
ft
20 yd ×12 yd = 20 yd 13yd
× 12 yd 13yd
= 2160 ft2
Currency Conversions. Use the currency exchange rates in Table 2.1 for the following
question.
49. You return from Mexico with 3000 pesos. How much are they worth in U.S. dollars?
3000 pesos ×
0.06584 USD
= 197.52 USD
1 peso
Working with Units. Use unit conversions to answer the following questions.
55. An airliner travels 45 miles in 5 minutes. What is its speed in miles per hour?
45 mi 60 min
×
= 540 mi
hr
5 min
1 hr
56. What is the total cost of 1.2 cubic yards of soil if it sells for $24 per cubic yard?
1.2 yd3 ×
$24
= $28.80
yd3
Gas Mileage. Answer the following practical gas mileage questions.
75. Gas mileage actually varies slightly with the driving speed of a car (as well as with
highway vs. city driving). Suppose your car averages 38 miles per gallon on the
highway if your average speed is 55 miles per hour, and it averages 32 miles per gallon
on the highway if your average speed is 70 miles per hour.
a. What is the driving time for a 2000-mile trip if you drive at an average speed of 55
miles per hour? What is the driving time at 70 miles per hour?
55 mi
70 mi
2000 mi ÷
≈ 36.36 hr; 2000 mi ÷
≈ 28.57 hr
1 hr
1 hr
b. Assume a gasoline price of $2.90 per gallon. What is the gasoline cost for a 2000mile trip if you drive at an average speed of 55 miles per hour? What is the gasoline
cost at 70 miles per hour?
38 mi
$2.90
32 mi
$2.90
2000 mi ÷
×
≈ $152.63; 2000 mi ÷
×
= $181.25
1 gal
1 gal
1 gal
1 gal
76. Suppose your car averages 32 miles per gallon on the highway if your average speed is
60 miles per hour, and it averages 25 miles per gallon on the highway if your average
speed is 75 miles per hour?
a. What is the driving time for a 1500-mile trip if you drive at an average speed of 60
miles per hour? What is the driving time at 75 miles per hour?
60 mi
75 mi
1500 mi ÷
= 25 hr; 1500 mi ÷
= 20 hr
1 hr
1 hr
b. Assume a gasoline price of $2.90 per gallon. What is the gasoline cost for a 1500mile trip if you drive at an average speed of 60 miles per hour? What is the gasoline
cost at 75 miles per hour?
32 mi
$2.90
25 mi
$2.90
1500 mi ÷
×
≈ $135.94; 1500 mi ÷
×
= $174
1 gal
1 gal
1 gal
1 gal
83. Shower vs. Bath Assume that when you take a bath, you fill a tub to the halfway point
and the tub measures 6 feet by 3 feet by 2.5 feet. When you take a shower, you use a
shower head with a flow rate of 1.75 gallons per minute, and you typically spend 10
minutes in the shower. There are 7.5 gallons in one cubic foot.
a. Do you use more water taking a shower or taking a bath?
bath = 1/2 (6 ft ×3 ft ×2.5 ft) = 22.5 ft3
is greater than
shower =
1.75 gal
1 ft3
× 10 min ×
≈ 2.33 ft3
min
7.5 gal
b. How long would you need shower in order to use as much water as you use taking
a bath?
1.75 gal
1 ft3
22.5 ft3 ÷
≈ 96 min
×
min
7.5 gal
c. Assuming your shower is in a bath tub, propose a nonmathematical way to compare,
in one experiment, the amounts of water you use taking a shower and a bath.
You could simply plug the drain in the tub and take a shower as normal.