A-10 ANSWERS 7. a) p q q ( ) ∨ ∧ ∼ b) The lightbulb will be on only when p is closed and q is open. 9. a) p q p q r ( ) ( ) ∧ ∧ ⎣⎡ ∧ ∼ ∨ ⎤⎦ b) The lightbulb will be on only when p, q, and r are all closed. 11. a) p q r p ( ) ∨ ∨ ∧ ∼ b) The lightbulb will be on in all cases except when p, q, and r are all open. 13. p q r 15. r p q 17. p q r s 19. p q r p q 21. p q p q ; ∨ ∼ ∼ ∧ not equivalent 23. p q r p q r p ( ) ; ( ) ; [ ] ∧ ∨ ∧ ∨ ∧ equivalent 25. p p q r p q r ( ) ; ; ∨ ∼ ∧ ∧ ∧ ∧ not equivalent 27. It is a series circuit; therefore, both switches must be closed for current to flow and the lightbulb to go on. When the p switch is closed, the p switch is open and no current will flow through the circuit. When the p switch is closed, the p switch is open and no current will flow through the circuit. 29. a) p q SECTION 3.5, PAGE 151 1. Valid 3. Fallacy 5. Valid 7. Syllogism 9. Inverse 11. Syllogism 13. Invalid 15. Valid 17. Valid 19. Invalid 21. Valid 23. Valid 25. Valid 27. Invalid 29. Invalid 31. Valid 33. a) p q p q → ∼ ∴∼ b) Invalid 35. a) p q p q → ∴ b) Valid 37. a) p q q p → ∼ ∴∼ b) Valid 39. a) p q q p → ∴ b) Invalid 41. a) p q p q ∨ ∼ ∴ b) Valid 43. a) p q q r p r → → ∴ → b) Valid 45. a) p q q r r p ∧ → ∴ → b) Valid 47. a) p q p r q r ∨ → ∴ ↔ b) Invalid 49. a) p q q p → ∼ ∴∼ b) Valid 51. a) p q p q → ∼ ∴∼ b) Invalid 53. a) p q p q ∼ → ∼ ∼ ∴∼ b) Valid 55. a) p q q p q ∧ ∼ → ∴ b) Invalid 57. a) p q q r p r → → ∼ ∴ → b) Invalid 59. Therefore, we go on vacation. 61. Therefore, have strep throat. 63. Therefore, you did not close the deal. 65. Yes, if the conclusion necessarily follows from the premises, the argument is valid, even if the conclusion is false. 67. Yes, if the conclusion does not necessarily follow from the premises, the argument is invalid, even if the premises are true. 69. Valid 71. a) p q p q → ∼ ∴∼ b) No c) This argument is the fallacy of the inverse. SECTION 3.6, PAGE 158 1. Euler 3. Invalid 5. No 7. Valid 9. Invalid 11. Valid 13. Invalid 15. Invalid 17. Invalid 19. Valid 21. Invalid 23. Invalid 25. Invalid 27. Valid 29. Yes, if the conclusion necessarily follows from the premises, the argument is valid. SECTION 3.7, PAGE 165 1. Series 3. Closed 5. a) p q ∨ b) The lightbulb will be on in all cases except when p is open and q is open. b) p q REVIEW EXERCISES, PAGE 168 1. Some diamonds are not made of carbon. 2. Some soft drinks have caffeine. 3. No women are presidents. 4. All pine trees are green. 5. The coffee is hot or the coffee is strong. 6. The coffee is not hot and the coffee is strong. 7. If the coffee is hot, then the coffee is strong and the coffee is not Maxwell House. 8. The coffee is Maxwell House if and only if the coffee is not strong. 9. The coffee is not Maxwell House, if and only if the coffee is strong and the coffee is not hot. 10. The coffee is Maxwell House or the coffee is not hot, and the coffee is not strong.
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