ANSWERS A-9 69. True. If →p q is false, it must be of the form T F. → Therefore, the converse must be of the form F T, → which is true. 70. True. If →p q is false, it must be of the form T F. → Therefore, the inverse must be of the form F T, → which is true. 76. 17 20 14 23 15 12 18 14 11 9 16 6 11 16 12 30 7 9 2 8 1 7 2 5 6 1 8 9 7 2 3 1 8 7 5 7 6 7 9 7 8 9 6 8 1 5 9 7 8 4 7 9 3 8 6 1 7 3 4 8 3 9 9 9 3 9 6 4 12 6 17 13 21 34 22 16 3 15 11 4 22 12 16 9 16 19 11 15 16 19 17 22 11 49. a) p q q p → ∼ ∴∼ b) Valid 50. a) p q q p ∨ ∼ ∴ b) Valid SECTION 3.5, PAGE 151 33. a) p q p q → ∼ ∴∼ b) Invalid 34. a) p q q p → ∴ b) Invalid 35. a) p q p q → ∴ b) Valid 36. a) p q q r p r → → ∴ → b) Valid 37. a) p q q p → ∼ ∴∼ b) Valid 38. a) p q q p ∨ ∼ ∴ b) Valid 39. a) p q q p → ∴ b) Invalid 40. a) p q p q → ∴ b) Valid 41. a) p q p q ∨ ∼ ∴ b) Valid 42. a) p q q p → ∼ ∴∼ b) Valid 43. a) p q q r p r → → ∴ → b) Valid 44. a) p q p q → ∼ ∴∼ b) Invalid 45. a) p q q r r p ∧ → ∴ → b) Valid 46. a) p q q r p r ∨ → ∴ ∧ b) Invalid 47. a) p q p r q r ∨ → ∴ ↔ b) Invalid 48. a) p q q p q p ∨ ∼ → ∴ ∨ b) Valid 51. a) p q p q → ∼ ∴∼ b) Invalid 52. a) p q q p → ∴ b) Invalid 53. a) p q p q ∼ → ∼ ∼ ∴∼ b) Valid 54. a) p q q p → ∼ ∴∼ b) Valid 55. a) p q q p q ∧ ∼ → ∴ b) Invalid 56. a) p q p q p → ∼ ∧ ∴∼ b) Invalid 57. a) p q q r p r → → ∼ ∴ → b) Invalid 58. a) p q q r p r ∧ → ∴ ∧ ∼ b) Invalid 65. Yes, if the conclusion necessarily follows from the premises, the argument is valid, even if the conclusion is false. 66. Yes, if the conclusion does not necessarily follow from the premises, the argument is invalid, even if the conclusion is true. 67. Yes, if the conclusion does not necessarily follow from the premises, the argument is invalid, even if the premises are true. 68. Yes, if the conclusion necessarily follows from the premises, the argument is valid, even if the premises are false. 70. No. The conditional statement will always be true, and therefore the statement will be a tautology, and the argument will be valid. SECTION 3.7, PAGE 165 5. a) ∨p q b) The lightbulb will be on in all cases except when p is open and q is open. 6. a) ∧p q b) The lightbulb will be on when both p and q are closed. 7. a) p q q ( ) ∨ ∧ ∼ b) The lightbulb will be on only when p is closed and q is open. 8. a) p q p q ( ) ( ) ∧ ∨ ∼ ∧ b) The lightbulb will be on when p is closed and q is closed, or when p is open and q is closed. 9. a) p q p q r ( ) ( ) ∧ ∧ ⎡⎣ ∧ ∼ ∨ ⎤⎦ b) The lightbulb will be on only when p, q, and r are all closed. 10. a) p r q p q ( ) ( ) ⎡⎣ ∧ ∨ ∼ ⎤⎦ ∧ ∨ b) The lightbulb will be on when p, q, and r are all closed; when p and r are closed and q is open; or when p is closed and q and r are open. 11. a) p q r p ( ) ∨ ∨ ∧ ∼ b) The lightbulb will be on in all cases except when p, q, and r are all open.
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