2.2 Subsets 57 31. = A a b c d e {, , , , } = B a c d e f {, , , , } None 32. = | A x x { is a sport that uses a ball} = B {basketball, soccer, tennis} B A B A , # , 33. Set A is the set of natural numbers between 2 and 7. Set B is the set of natural numbers greater than 2 and less than 7. A B A B B A , , = # # 34. Set A is the set of all cars manufactured by General Motors. Set B is the set of sports cars manufactured by General Motors. B A B A , # , In Exercises 35–38, list all the subsets of the sets given. 35. = ∅ D {} 36. = A { }○ { }, { }○ 37. = B {cow, horse} { }, {cow}, {horse}, {cow, horse} 38. π = C e i { , , } e i e i e i e i { },{ }, {}, { }, { , }, { , }, { , }, { , , } π π π π Problem Solving 39. For set = A a b c d {, , , }, a) list all the subsets of set A. * b) state which of the subsets in part (a) are not proper subsets of set A. a b c d {, , , } 40. A set contains nine elements. a) How many subsets does it have? 2 , 9 or 512, subsets b) How many proper subsets does it have? 511 proper subsets In Exercises 41–52, if the statement is true for all sets A and B, write “true.” If it is not true for all sets A and B, write “false.” Assume that ≠ ∅ ≠ ∅ A U, , and A U. , 41. If A B, # then A B. , False 42. If A B, , then A B. # True 43. A A# True 44. A A , False 45. ∅ A , True 46. ∅ A# True 47. A U# True 48. ∅ ∅, False 49. ∅ U, True 50. ∅ U # False 51. ∅ ∅# True 52. ∅ U , False 53. Building a House The Li family is planning to build a house in a new development. They can either build the base model offered by the builder or add any of the following options: security system, finished basement, deck, hardwood flooring. How many different variations of the house are possible? 2 , 4 or 16 54. Installing an In-ground Pool The Fitzgeralds are installing an in-ground pool in their backyard. They can either install the base model offered by the pool company or add any of the following options: automatic pool cleaner, solar cover, waterfall, hot tub, fountain, slide, diving board. How many different variations of the pool are possible? 2 , or 128 7 55. Salad Toppings Donald is making a salad for dinner. His salad can consist of just lettuce, or he can add any of the following items: cucumbers, onions, tomatoes, carrots, green peppers, olives, mushrooms. How many different variations of a salad are possible? 2 , or 128 7 56. Telephone Features A customer with Verizon can order telephone service with some, all, or none of the following features: call waiting, call forwarding, caller identification, three-way calling, voice mail, fax line. How many different variations of the set of features are possible? 2 , or 64 6 57. Pizza Toppings A cheese pizza at ' ' Za Za has the following vegetarian topping options: mushrooms, green peppers, black olives, pineapple, tomatoes, basil, and red onion. If Sharleen wishes to order a cheese pizza with at least one of these toppings, how many different choices does she have? 2 1, or 127 7 − 58. Garden Options Xeriscape landscaping involves growing plants that need little watering. Brenda would like to choose one or more plants from the following list to create her xeriscape garden: lantanas, hibiscus, coreopsis, begonias, pentas, and zinnias. How many different options does Brenda have for her garden? 2 1, or 63 6 − Challenge Problems 59. If E F# and F E, # what other relationship exists between E and F? E F = 60. Hospital Expansion A hospital has four members on the board of directors: Arnold, Benitez, Cathy, and Dominique. a) When the members vote on whether to add a wing to the hospital, how many different ways can they vote (abstentions are not allowed)? For example, Arnold— yes, Benitez—no, Cathy—no, and Dominique—yes is one of the many possibilities. 16 b) Make a listing of all the possible outcomes of the vote. For example, the vote described in part (a) could be represented as (YNNY). * c) How many of the outcomes given in part (b) would result in a majority supporting the addition of a wing to the hospital? That is, how many of the outcomes have three or more Y’s? 5 out of 16 61. How many elements must a set have if the number of proper subsets of the set is 1 2 of the total number of subsets of the set? 1 62. If A B , and B C, , must A C? , Yes 63. If A B , and B C, # must A C? , Yes 64. If A B# and B C, # must A C? , No *See Instructor Answer Appendix
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