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54 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 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 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. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 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.) An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.