168 CHAPTER 3 Logic Section 3.7 Switching Circuits as Symbolic Statements Switches in series will always be represented with a conjunction, ∧. Switches in parallel will always be represented with a disjunction, ∨. Examples 1– 4, pages 162–164 3.1 In Exercises 1– 4, write the negation of the statement. 1. All diamonds are made of carbon. Some diamonds are not made of carbon. 2. No soft drinks have caffeine. Some soft drinks have caffeine. 3. Some women are presidents. No women are presidents. 4. Some pine trees are not green. All pine trees are green. In Exercises 5–10, write each compound statement in words. p q r : The coffee is Maxwell House. : The coffee is hot. : The coffee is strong. 5. ∨ q r * 6. q r ~ ∧ * Review Exercises CHAPTER 3 3.2, 3.3 In Exercises 23–26, determine the truth value of the statement. You may need to use the Internet as a reference. 23. ESPN is a sports network and the Hallmark Channel is a news network, or Netflix is a streaming service. True 24. If a minute has 60 seconds, then an hour has 60 minutes if and only if a day has 20 hours. False 25. If Oregon borders the Pacific Ocean or California borders the Atlantic Ocean, then Minnesota is south of Texas. False 26. President’s Day is in February, or Memorial Day is in May and Labor Day is in December. True 3.3 In Exercises 27–30, determine the truth value of the statement when p is T, q is F, and r is F. 27. p q p q (~ ) ~( ~ ) ∨ → ∧ True 28. p q p r ( ) (~ ) ↔ → ∨ True 29. [ ] ↔ ∨ ↔ r p q p ~ ( ) ~ False 30. q r p r ~ ( ) (~ ) [ ] ∧ → ∨ False 3.4 In Exercises 31–34, determine whether the pairs of statements are equivalent. You may use De Morgan’s laws, the fact that p q p q ( ) (~ ), → ⇔ ∨ the fact that p q p q ~( ) ( ~ ), → ⇔ ∧ truth tables, or equivalent forms of the conditional statement. 31. p q ~( ~ ), ∧ p q ~ ∧ Not equivalent 32. p q p q , ~ ∨ → Equivalent 33. p q r p q p r ~ ( ), (~ ) (~ ) ∨ ∧ ∨ ∧ ∨ Equivalent 34. q p p p q p (~ ) , ~(~ ) → ∧ ↔ ∨ Not equivalent In Exercises 35–39, use De Morgan’s laws, the fact that p q p q ( ) (~ ), → ⇔ ∨ or the fact that p q ~( ) → ⇔ p q ( ~ ), ∧ to write an equivalent statement for the given statement. 35. An apple is a fruit and a potato is not a fruit. It is false that if an apple is a fruit then a potato is a fruit. 7. q r p ( ~ ) → ∧ * 8. p r~ ↔ * 9. p r q ~ ( ~ ) ↔ ∧ * 10. ∨ ∧ p q r ( ~ ) ~ * 3.2 In Exercises 11–16, use the statements for p, q, and r as in Exercises 5–10 to write the statement in symbolic form. 11. The coffee is hot, but the coffee is not strong. q r~∧ 12. If the coffee is strong, then the coffee is not Maxwell House. r p~ → 13. If the coffee is strong then the coffee is hot, or the coffee is not Maxwell House. r q p ( ) ~ → ∨ 14. The coffee is hot if and only if the coffee is Maxwell House, and the coffee is not strong. q p r ( ) ~ ↔ ∧ 15. The coffee is strong and the coffee is hot, or the coffee is not Maxwell House. r q p ( ) ~ ∧ ∨ 16. It is false that the coffee is strong and the coffee is hot. r q ~( ) ∧ In Exercises 17–22, construct a truth table for the statement. 17. ∨ ∧ p q p ( ) ~ * 18. ↔ ∨ q p q ( ~ ) * 19. ∨ ↔ ∨ p q p r ( ) ( ) * 20. ∧ ∨ p q r (~ ) * 21. → ∧ p q r ( ~ ) * 22. ∧ →∼ p q r ( ) * *See Instructor Answer Appendix
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