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CHAPTER 2
2.1
The Basic Concepts of Set Theory
EXERCISES
Match each set in Column I with the appropriate description in Column II.
I
II
1. 2, 4, 6, 8
A. the set of all even integers
2. x x is an even integer greater than 4 and less than 6 B. the set of the five least positive integer powers of 2
3. . . . , 4, 3, 2, 1
C. the set of even positive integers less than 10
4. . . . , 6, 4, 2, 0, 2, 4, 6, . . .
D. the set of all odd integers
5. 2, 4, 8, 16, 32
E. the set of all negative integers
6. . . . , 5, 3, 1, 1, 3, 5, . . .
F. the set of odd positive integers less than 10
7. 2, 4, 6, 8, 10
G. 0
8. 1, 3, 5, 7, 9
H. the set of the five least positive integer multiples of 2
List all the elements of each set. Use set notation to describe the set.
9. the set of all counting numbers less than or equal
to 6
10. the set of all whole numbers greater than 8 and less
than 18
11. the set of all whole numbers not greater than 4
12. the set of all counting numbers between 4 and 14
13. 6, 7, 8, . . . , 14
14. 3, 6, 9, 12, . . . , 30
15. 15, 13, 11, . . . , 1
16. 4, 3, 2, . . . , 4
17. 2, 4, 8, . . . , 256
18. 90, 87, 84, . . . , 69
19. x x is an even whole number less than 11
20. x x is an odd integer between 8 and 7
Denote each set by the listing method. There may be more than one correct answer.
21. the set of all counting numbers greater than 20
22. the set of all integers between 200 and 500
23. the set of Great Lakes
24. the set of United States presidents who served after Lyndon Johnson and before William Clinton (See the
photograph.)
25. x x is a positive multiple of 5
26. x x is a negative multiple of 6
27. x x is the reciprocal of a natural number
28. x x is a positive integer power of 4
Denote each set by set-builder notation, using x as the variable. There may be more than one correct answer.
29. the set of all rational numbers
30. the set of all even natural numbers
31. 1, 3, 5, . . . , 75
32. 35, 40, 45, . . . , 95
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2.1
Symbols and Terminology
55
Identify each set as finite or infinite.
33. 2, 4, 6, . . . , 32
34. 6, 12, 18
35. 12, 23, 34, . . .
36. 10, 8, 6, . . .
37. x x is a natural number greater than 50
38. x x is a natural number less than 50
39. x x is a rational number
40. x x is a rational number between 0 and 1
Find nA for each set.
41. A 0, 1, 2, 3, 4, 5, 6, 7
42. A 3, 1, 1, 3, 5, 7, 9
43. A 2, 4, 6, . . . , 1000
44. A 0, 1, 2, 3, . . . , 3000
45. A a, b, c, . . . , z
46. A x x is a vowel in the English alphabet}
47. A the set of integers between 20 and 20
48. A the set of current U.S. senators
49. A 13, 24, 35, 46, . . . , 2729, 2830
50. A 12, 12, 13, 13, . . . , 110, 110
51. Explain why it is acceptable to write the statement
“x is a vowel in the English alphabet” in the set for
Exercise 46, despite the fact that x is a consonant.
52. Explain how Exercise 49 can be answered without
actually listing and then counting all the elements.
Identify each set as well-defined or not well-defined.
53. x x is a real number
54. x x is a negative number
55. x x is a good athlete
56. x x is a skillful typist
57. x x is a difficult course
58. x x is a counting number less than 2
Fill each blank with either or to make the following statements true.
59. 5
62. 12
65. 3
2, 4, 5, 7
3, 8, 12, 18
60. 8
63. 0
2, 3, 4, 6
61. 4
64. 0
3, 2, 5, 7, 8
2, 0, 5, 9
66. 6}
4, 7, 8, 12
3, 4, 6, 8, 10
3, 4, 5, 6, 7
Write true or false for each of the following statements.
67. 3 2, 5, 6, 8
68. 6 2, 5, 8, 9
69. b h, c, d, a, b
70. m 1, m, n, o, p
71. 9 6, 3, 4, 8
72. 2 7, 6, 5, 4
73. k, c, r, a k, c, a, r
74. e, h, a, n a, h, e, n
75. 5, 8, 9 5, 8, 9, 0
76. 3, 7, 12, 14 3, 7, 12, 14, 0
77. x x is a natural number less than 3 1, 2
78. x x is a natural number greater than 10 11, 12, 13, . . .
Write true or false for each of the following statements.
Let A 2, 4, 6, 8, 10, 12,
79. 4 A
B 2, 4, 8, 10,
80. 8 B
and
C 4, 10, 12.
81. 4 C
82. 8 B
83. Every element of C is also an element of A.
84. Every element of C is also an element of B.
85. This section opened with the statement, “The human
mind likes to create collections.” Why do you suppose this is so? In explaining your thoughts, utilize
one or more particular “collections,” mathematical
or otherwise.
86. Explain the difference between a well-defined set
and a not well-defined set. Give examples and utilize
terms introduced in this section.
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56
CHAPTER 2
The Basic Concepts of Set Theory
Recall that two sets are called equal if they contain
identical elements. On the other hand, two sets are
called equivalent if they contain the same number of elements (but not necessarily the same elements). For
each of the following conditions, give an example or explain why it is impossible.
87. two sets that are neither equal nor equivalent
88. two sets that are equal but not equivalent
89. two sets that are equivalent but not equal
(a) List the set of issues that had a share volume of
at least 188.7 million.
(b) List the set of issues that had a share volume of
at most 188.7 million.
92. Burning Calories Jaime Bailey is health conscious,
but she does like a certain chocolate bar, each of
which contains 220 calories. In order to burn off
unwanted calories, Jaime participates in her favorite
activities, shown below, in increments of one hour
and never repeats a given activity on a given day.
90. two sets that are both equal and equivalent
91. Volumes of Stocks The table lists the most active
stocks on the American Stock Exchange in a recent
year, with share volumes (in millions).
Issue
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Share Volume (millions)
Viacom (Class B)
Trans World Airlines
Harken Energy
Echo Bay Mines
JTS
Nabors Industries
Hasbro
Royal Oak Mines
Grey Wolf Industries
IVAX
223.1
222.8
214.5
188.7
177.0
172.9
159.1
140.6
132.9
132.5
Activity
Volleyball
Golf
Canoeing
Swimming
Running
Symbol
Calories Burned
per Hour
v
g
c
s
r
160
260
340
410
680
(a) On Monday, Jaime has time for no more than
two hours of activities. List all possible sets of
activities that would burn off at least the number
of calories obtained from three chocolate bars.
(b) Assume that Jaime can afford up to three hours
of time for activities on Saturday. List all sets of
activities that would burn off at least the number
of calories in five chocolate bars.
(Source: American Stock Exchange.)
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