78 CHAPTER 2 Sets In Exercises 65– 68, use a set statement to write a description of the shaded area. Use union, intersection, and complement as necessary. More than one answer may be possible. 65. <A B ( )′ U A B U A B U A B C U A B C 66. >A B ( )′ 67. < > A B C ( ) ′ 68. > < > A B B C ( ) ( ) 33. VI 34. VIII 35. III 36. IV 37. III 38. I 39. V 40. III 41. II 42. VIII 43. VII 44. VI Senate Bills In Exercises 45–50, use Fig. 2.24. During a session of the U.S. Senate, three bills were voted on. The votes of six senators are shown. Determine in which region of the figure each senator should be placed. The set labeled Bill 1 represents the set of senators who voted yes on Bill 1, and so on. SENATOR BILL 1 BILL 2 BILL 3 45. Dhaliwal yes no no I 46. Gottlieb no no yes VII 47. Nguyen no no no VIII 48. Runde yes yes yes V 49. Washington no yes yes VI 50. Yusuf no yes no III In Exercises 51–56, use Venn diagrams to determine whether the following statements are equal for all sets A and B. 51. < > ′ ′ ′ A B A B ( ) , Yes 52. > < ′ ′ A B A B ( ) , No 53. < > ′ ′ A B A B , No 54. < < ′ ′ ′ A B A B , ( ) No 55. > < ′ ′ ′ A B A B , No 56. > < A B A B ( ) , ′ ′ ′ Yes In Exercises 57– 64, use Venn diagrams to determine whether the following statements are equal for all sets A B, , and C. 57. > < A B C ( ), > < A B C ( ) No 58. < > A B C ( ), > < B C A ( ) Yes 59. > < A B C ( ), < > B C A ( ) Yes 60. < > A B C ( ) ,′ > < A B C ( ) ′ ′ No 61. > < A B C ( ), > < > A B A C ( ) ( ) Yes I U Bill 1 Bill 2 II III IV V VI VII Bill 3 VIII Figure 2.24 62. < > A B C ( ), < > < A B A C ( ) ( ) Yes 63. < > < A B B C ( ) ( ), < > B A C ( ) Yes 64. < > A B C ( ) , ′ < > < A C B C ( ) ( ) ′ ′ ′ No 69. Let U A B {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 6} {3, 6, 7} = = = a) Apply De Morgan’s Law 1 on page 75 to show that > < A B A B ( )′ = ′ ′ for these sets. Both equal {1, 2, 4, 5, 7, 8} b) Make up your own sets A and B. Verify that > < A B A B ( )′ = ′ ′ for your sets A and B. Answers will vary. 70. Let U abcdef ghi A a e g h i B a g i {,,, ,,,,,} {,,,,} { , , } = = = a) Apply De Morgan’s Law 2 on page 75 to show that < > A B A B ( )′ = ′ ′ for these sets. Both equal b c d f {, , , } b) Make up your own sets A and B. Verify that > < A B A B ( )′ = ′ ′ for your sets A and B. Answers will vary. 71. World Blood Types The following table contains data from WorldAtlas.com regarding the percentage of people, worldwide, who have each blood type. Type Positive Blood, % Negative Blood, % O 42% 3% A 31% 2.5% B 15% 1% AB 5% 0.5% Construct a Venn diagram similar to the one in Example 2 and place the correct percentage in each of the eight regions. * *See Instructor Answer Appendix
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