In E
called
- the resulting statement is true.
ia, Colorado Connecticut} is expressed
method. The set {xx is a U.S. state whose
ne letter C} is expressed using
5. The
number of the
number is represented by
6. Two sets that contain the same number of elements are
sets.
7. Two sets that contain the same elements are called
sets.
70 elements is called the null set or the
et is represented by { } or
to indicate that an object
67. A
{17
68. A=
69.
B = {
70.
B =
71. C=
72. C-
lette
73. D =
75. A =
76. A =
77. B
78. B =
79. C
80. C =
In Exercise
Ai
a.
b. A
81. A is th
majori
82. A is ti
people
States.
83. A =
B
29. {x|x€N and 10 < x < 80}
30. {x|XEN and 15 < x < 60}
hich collections are not well defined
32. {x|x + 3 = 9}
31. {x|x + 5 = 7}
idents
In Exercises 33-46, determine which sets are the empty set.
34. {0,Ø}
and full-time students currently 33. {0,0}
35. {x|x is a woman who served as U.S. president before
orst U.S. presidents
2016}
-time students currently enrolled
36. {x/x is a living U.S. president born before 1200}
37. {x|x is the number of women who served as U.S. president
numbers greater than one before 2016}
38. {x|x is the number of living U.S. presidents born before
natural numbers greater 1200)
39. {x|x is a U.S. state whose name begins with the letter X}
cription of each set. (More than
40. {x|x is a month of the year whose name begins with the
ible)
letter X}
rs, Jupiter, Saturn, Uranus,
41. xx < 2 and x > 5}
42. {x|x < 3 and x > 7}
{January, June, July
43. X|XEN and 2 50}
110. {x|x is a country in which 7.6% = percentage having used
93. {x|XEN and x = 1,000,000}
marijuana < 9%}
94. {x|x€N and x = 2,000,000}
111. {x|x is a country in which the percentage having used
marijuana > 22.2%}
95. The set of natural numbers less than 1
112. {x|x is a country in which the percentage having used
96. The set of natural numbers less than 0
marijuana > 22.2%}
7.4
New
Zealand
Australia
United
States
United
Kingdom
Switzerland
Ireland
Spain
Canada
к
H
aps Lock
Shift
V BINM
z Xc
Ctrl
Alt
Alt
68
CHAPTER 2
Set Theory
76. {Ø}e{Ø,{0}}
Exercise Set 2.2
77. If A C Band de A, then d
78. If ASB and BCC, then
79. If set A is equivalent
n(A) = No.
А
Application Exercises
34. A = {x|x€N and 3 < x < 10}
B = {x|XEN and 2 sxs 8}
B
35. Ø {7, 8, 9, ... , 100}
36. Ø {101, 102, 103,...,
200}
37. {7, 8, 9, ...}
38. {101, 102, 103,... }
39. Ø
40. { }
We opened this section citin
shows the percentage of tatto
In Exercises 41-54, determine whether each statement is true or
false. If the statement is false, explain why.
Source: Harris Interactiv
Practice Exercises
In Exercises 1-18, write C or in each blank so that the resulting
statement is true
1. {1.2.5 {1, 2, 3, 4, 5, 6, 7}
2. {2,3,7) {1, 2, 3, 4, 5, 6, 7}
3. {-3, 0,3} {-3,-1,1.3}
4. {-4, 0,4} {-4, -3, -1,1,3,4}
5. Monday, Friday}
{Saturday, Sunday, Monday, Tuesday, Wednesday}
6. {Mercury, Venus, Earth}
{Venus, Earth, Mars, Jupiter}
7. {x/x is a cat} {x/x is a black cat}
8. {x/x is a dog} {xxlx is a pure-bred dog}
9. {c, o, n, v, e, r, s, a, t, i, o, n}
{v, o, i, c, e, s,r,a, n, t, o, n}
10. {r, e, v, o, 1, u, t, i, o, n} {t, 0, 1, 0, v, e, r, u, i, n}
11. {3} {2,} 12. {1,5} {2,3,5}
13. Ø {2, 4, 6}
14. Ø {1,3,5}
15. {2, 4, 6}
16. {1,3,5}
17. { }
18. Ø
In Exercises 19-40, determine whether C, C, both, or neither can
be placed in each blank to form a true statement.
19. {V, CR} {V, C, R, S}
20. {F, I, N} {F, I, N, K}
21. {0, 2, 4, 6, 8} {8,0, 6, 2,4}
22. {9, 1, 7,3,4} {1,3,4,7,9}
23. {x/x is a man} {x|x is a woman
24. {xx/x is a woman} {x|x is a man}
25. {x|x is a man} {x|x is a person
26. {x|x is a woman} {x|x is a person}
27. {x|x is a man or a woman} {x|x is a person)
28. {.x/x is a woman or a man} {x|x is a person
29. A = = {x|XEN and 5 < x < 12}
B = the set of natural numbers between 5 and 12
Sets and subsets allo
party affiliation. Th.
based on the numb
T = th
41. Ralph = {Ralph, Alice, Trixie, Norton}
42. Canada e {Mexico, United States, Canada)
43. Ralph C {Ralph, Alice, Trixie, Norton)
44. Canada C {Mexico, United States, Canada}
45. {Ralph} S {Ralph, Alice, Trixie, Norton}
46. {Canada} C {Mexico, United States, Canada)
47. Øe{Archie, Edith, Mike, Gloria}
48. Ø ${Charlie Chaplin, Groucho Marx, Woody Allen}
49. {5} c{{5}, {9}}
50. {1} c{{1}, {3}}
51. {1,4} & {4,1}
52. {1,4} & {4,1}
53. 0€
54. {0} €
In Exercises 55-60, list all the subsets of the given set.
55. {border collie, poodle} 56. {Romeo, Juliet}
57. {t, a, b}
58. {I, II, III)
59. {0}
60. Ø
In Exercises 61-68, calculate the number of distinct subsets and the
number of distinct proper subsets for each set.
61. {2,4,6,8}
62. {1,3,4,5)
63. {2, 4, 6, 8, 10, 12} 64. {a, b, c, d, e, f}
65. {x|x is a day of the week}
66. {x|x is a U.S. coin worth less than a dollar}
67. {x|XEN and 2 < x < 6}
68. {x|x€N and 2 < x < 6}
R = t
D=t
M =
W =
А A
B
30. A = {x|XEN and 3 < x < 10}
B = the set of natural numbers between 3 and 10
А B
31. A = {x/XEN and 5 < x < 12}
B = the set of natural numbers between 3 and 17
A B
32. A = {x|XEN and 3 < x < 10}
B = the set of natural numbers between 2 and 16
A B
33. A = {x|XEN and 5 < x < 12}
B = {x|x€N and 2 < x < 11}
А. B
Practice Plus
In Exercises 69–82, determine whether each statement is true or
false. If the statement is false, make the necessary change(s) to
produce a true statement.
69. The set {1, 2, 3, ..., 1000} has 21000 proper subsets.
70. The set {1, 2, 3, ..., 10,000} has 210,000 proper subsets.
71. {x|x€N and 30 < x < 50} ({x|x€N and 30 < x < 50}
72. {x|x€N and 20 < x < 60}&{x|x€N and 20 < x < 60}
73. Ø¢{Ø,{@}}
74. {@} & {0,{0}}
75. Øc{Ỡ,{8}}
In Exercises 8
false. If the se
produce a true
83. DET
85. MC T
87. If x ED
89. If xer
90. If xe
91. The se
92. The
set
93. Hou
can
opt
ala
the
94. A
fo
P
(
78. If ACB and BCC, then ACC.
If set A is equivalent to the set of natural numbers, then
NA)
79.
81. The set of subsets of {a, e, i, o, u} contains 64 elements.
82. The set of subsets of {a,b,c, d, e, f} contains 128 elements.
No
Application Exercises
We opened this section citing a 2012 Harris poll that estimated 45 million Americans have at least one tattoo. The bar graph on the left
shows the percentage of tattooed Americans, by party affiliation and gender.
Percentage of Tattooed Americans,
by Party Affiliation and Gender
30%
20%
Percent of Anton Aus
Number of Tattooed Americans per 10,000 Adults,
by Party Affiliation and Gender
315 Men
699 Democrats
→ 384 Womer
1890 Tattooed
238 Men
Americans
529 Republicans
→ 291 Wome
per 10,000 Adults
298 Men
662 Independents
364Won
15%
ton
Democrat
Republic
Independent
fence
All Adults
By Party
Affiliation
By Gender
Source: Harris Interactive
Sets and subsets allow us to order and structure the data. On the right, the set of tattooed Americans is divided into subsets cate
party affiliation. These subsets are further broken down into subsets categorized by gender. All numbers in the branching treed
based on the number of people per 10,000 American adults. Based on the tree diagram, let
T= the set of tattooed Americans
95. Based on more than 1500 ballots sent to film
R the set of tattooed Republicans
American Film Institute rated the top U.S.
D the set of tattooed Democrats
Institute selected Citizen Kane (1941), The God
M the set of tattooed Democratic men
Casablanca (1942), Raging Bull (1980), Singi
(1952), and Gone with the Wind (1939) as th
W the set of tattooed Democratic women.
Suppose that you have all six films on DVD
view some, all, or none of these films. How
In Exercises 83-92, determine whether each statement is true or options do you have?
false. If the statement is false, make the necessary change(s) to 96. A small town has four police cars. If a
produce a true statement.
receives a call, depending on the nature o
83. DET
84. RET
cars, one car, two cars, three cars, or all fou
85. MCT
86. WCT
How many options does the dispatcher h-
87. If xe D. then xeW. 88. If xe, then xe M.
police cars to the scene of the caller?
97. According to the U.S. Census Bureau,
89. If XeR, thenx & D.
diverse U.S. cities are New York City,
90. If xe D. then x € R.
Chicago, Washington, D.C., Houston, S:
91. The set of elements in M and W combined is equal to set D.
If you decide to visit some, all, or nor
92. The set of elements in M and W combined is equivalent to
many travel options do you have?
98. Some of the movies with all-time-
include
93. Houses in Euclid Estates are all identical. However, a person
can purchase a new house with some, all, or none of a set of
Avatar ($761 million), The Avengers
options. This set includes { pool, screened-in balcony, lake view,
Titanic ($601 million), The Dark K
alarm system, upgraded landscaping). How many options are
Star Wars ($461 million), Shrek 2
there for purchasing a house in this community?
E.T. the Extra Terrestrial ($435 millic
94. A cheese pizza can be ordered with some, all, or none of the
following set of toppings: {beef, ham, mushrooms, sausage, Suppose that you have all seven
over the course of a week, to vie
peppers, pepperoni, olives, prosciutto, onion). How many
films. How many viewing option
different variations are available for ordering a pizza?
set D.
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