170 CHAPTER 3 Logic 58. Construct a diagram of a circuit that corresponds to the symbolic statement p q p q ( ) ( ). ∨ ∨ ∧ * 59. Determine whether the circuits shown on the right are equivalent. Equivalent p q q p q p p q *See Instructor Answer Appendix Test CHAPTER 3 In Exercises 1–3, write the statement in symbolic form p: Phobos is a moon of Mars. q: Callisto is a moon of Jupiter. r: Rosalind is a moon of Uranus. 1. Phobos is not a moon of Mars or Callisto is a moon of Jupiter, and Rosalind is not a moon of Uranus. p q r ( ) ~ ∼ ∨ ∧ 2. If Rosalind is a moon of Uranus then Callisto is a moon of Jupiter, or Phobos is not a moon of Mars. r q p ( ) ~ → ∨ 3. It is false that Rosalind is a moon of Uranus if and only if Callisto is not a moon of Jupiter. r q ~( ~ ) ↔ In Exercises 4 and 5, use p q, , and r as above to write each symbolic statement in words. 4. p r q (~ ) ~ ∧ ↔ * 5. p q r ( ~ ) ∨ → * In Exercises 6 and 7, construct a truth table for the given statement. 6. p r q ~( ) [ ] → ∧ * 7. q r p ( ~ ) ↔ ∨ * In Exercises 8 and 9, determine the truth value of the statement. 8. + = 2 6 8 or − = 7 12 5. True 9. A leap year has 366 days and a week has eight days if and only if an hour has 24 minutes. True In Exercises 10 and 11, given that p is true, q is false, and r is true, determine the truth value of the statement. 10. p q q r (~ ) ( ~ ) ∧ ←→ ∨ True 11. r p q p ~( ~ ) ( ) [ ] → ∧ → True 12. Determine whether the pair of statements are equivalent. Equivalent p q p q ~ , ~( ~ ) ∨ ∧ In Exercises 13 and 14, determine which, if any, of the three statements are equivalent. 13. a) If we leave home by 6:00 PM, then we will arrive on time. b) We do not leave home by 6:00 PM or we will arrive on time. c) If we do not leave home by 6:00 PM, then we will not arrive on time. a) and b) are equivalent. 14. a) It is not true that the test is today or the concert is tonight. b) The test is not today and the concert is not tonight. c) If the test is not today, then the concert is not tonight. a) and b) are equivalent. 15. Translate the following argument into symbolic form. Determine whether the argument is valid or invalid by comparing the argument to a recognized form or by using a truth table. If the soccer team wins the game, then Sue played fullback. If Sue played fullback, then the team is in second place. Therefore, if the soccer team wins the game, then the team is in second place. * 16. Use an Euler diagram to determine whether the syllogism is valid or invalid. All dogs are canines. Strider is a canine. Strider is a dog. ∴ Invalid In Exercises 17 and 18, write the negation of the statement. 17. All coffee beans contain caffeine. Some coffee beans do not contain caffeine. 18. Nick played football and Max played baseball. Nick did not play football or Max did not play baseball. 19. Write the converse, inverse, and contrapositive of the conditional statement “If the garbage truck comes, then today is Saturday.” * 20. Construct a diagram of a circuit that corresponds to p q p q ( ) (~ ~ ) ∧ ∨ ∨ * *See Instructor Answer Appendix
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