ANSWERS A-11 47. a) and c) are equivalent. 48. None are equivalent. 49. Invalid 50. Valid 51. Valid 52. Invalid 53. Invalid 54. Valid 55. Invalid 56. Invalid 57. a) p q r p [( ) ] ∧ ∧ ∨ ∼ b) The lightbulb will be on only when p, q, and r are all closed. 58. q p p q 59. Equivalent CHAPTER TEST, PAGE 170 1. p q r ( ) ∼ ∨ ∧ ∼ 2. r q p ( ) → ∨ ∼ 3. r q ( ) ∼ ↔∼ 4. Phobos is not a moon of Mars and Rosalind is a moon of Uranus, if and only if Callisto is not a moon of Jupiter. 5. If Phobos is a moon of Mars or Callisto is not a moon of Jupiter, then Rosalind is a moon of Uranus. 6. F T F F F F F F 7. T T T T F T T F 8. True 9. True 10. True 11. True 12. Equivalent 13. a) and b) are equivalent. 14. a) and b) are equivalent. 15. s f f p s p Valid → → ∴ → 16. Invalid 17. Some coffee beans do not contain caffeine. 18. Nick did not play football or Max did not play baseball. 19. Converse: If today is Saturday, then the garbage truck comes. Inverse: If the garbage truck does not come, then today is not Saturday. Contrapositive: If today is not Saturday, then the garbage truck does not come. 20. q p q p 11. q r ∧ ∼ 12. r p → ∼ 13. r q p ( ) → ∨ ∼ 14. q p r ( ) ↔ ∧ ∼ 15. r q p ( ) ∧ ∨ ∼ 16. r q ( ) ∼ ∧ 17. F F T F 18. T F F F 19. T T T T T F F T 20. T F T T F F F F 21. F T F F T T T T 22. F T T T T T T T 23. True 24. False 25. False 26. True 27. True 28. True 29. False 30. False 31. Not equivalent 32. Equivalent 33. Equivalent 34. Not equivalent 35. It is false that if an apple is a fruit then a potato is a fruit. 36. If Lynn Swann did not play for the Steelers, then Jack Tatum played for the Raiders. 37. Altec Lansing does not produce only speakers and Harman Kardon does not produce only stereo receivers. 38. It is false that we did go to the beach or we did find sharks’ teeth. 39. The temperature is above 32° or we will go ice fishing at O’Leary’s Lake. 40. a) If Mikel is a linebacker, then he plays football. b) If Mikel does not play football, then he is not a linebacker. c) If Mikel is not a linebacker, then he does not play football. 41. a) If we will learn the table’s value, then we take the table to Antiques Roadshow. b) If we do not take the table to Antiques Roadshow, then we will not learn the table’s value. c) If we will not learn the table’s value, then we do not take the table to Antiques Roadshow. 42. a) If you do not sell more doughnuts, then you do not advertise. b) If you advertise, then you sell more doughnuts. c) If you sell more doughnuts, then you advertise. 43. a) If you fail the course, then you do not study for the math test. b) If you study for the math test, then you do not fail the course. c) If you do not fail the course, then you study for the math test. 44. a) If I let you attend the prom, then you will get straight A’s on your report card. b) If you do not get straight A’s on your report card, then I will not let you attend the prom. c) If I will not let you attend the prom, then you did not get straight A’s on your report card. 45. a), b), and c) are equivalent. 46. None are equivalent.
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