118 CHAPTER 3 Logic Solution Let p q r : : : There are more subscribers to Disney than Prime. There are more subscribers to Spotify than Apple Music. The most subscribed service is Netflix. + The given compound statement can be written in symbolic form as p q r ( ) . ∧ ∨ ∼ From Fig. 3.1, we see that statement p is false: There are actually more subscribers to Prime than Disney+. We also see that statement q is true: There are more subscribers to Spotify than Apple Music. We also see that statement r is true: The most subscribed service is Netflix. Since r is true, its negation, r, ∼ is false. Therefore, we substitute F for p, T for q, and F for r, ∼ which gives ∧ ∨ ∼ ∧ ∨ ∨ p q r ( ) (F T) F F F F Thus, the original compound statement is a false statement. 7 Now try Exercise 65 Instructor Resources for Section 3.2 in MyLab Math • Objective-Level Videos 3.2 • Interactive Concept Video: Constructing Truth Tables • Animation: Exploring Truth Values for Compound Statements • PowerPoint Lecture Slides 3.2 • MyLab Exercises and Assignments 3.2 Exercises Warm Up Exercises In Exercises 1– 4, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The negation ∼p will always have the ________ truth value of p. Opposite 2. The conjunction p ∧ q is true only when both p and q are ________. True 3. The disjunction p ∨ q is false only when both p and q are ________. False 4. A truth table for a compound statement with a) one distinct simple statement will have ________ cases. Two b) two distinct simple statements will have ________ cases. Four c) three distinct simple statements will have ________ cases. Eight Practice the Skills/Problem Solving In Exercises 5–18, use the method described in Examples 1–5 to construct a truth table for the statement. 5. ∧ ∼ p p F F 6. ∨ ∼ p p T T 7. ∼ ∨p q * 8. ∼ ∨ ∼ p q * 9. ∧ ∼ p q * 10. ∼ ∧ ∼ p q * 11. ∼ ∧ ∼ p q ( ) * 12. ∼ ∨ ∼ p q ( ) * 13. ∨ ∼ ∨ p q r ( ) * 14. ∼ ∧ ∨ p q r ( ) * 15. ∧ ∼ ∨ p q r ( ) * 16. ∨ ∼ ∧ p q r ( ) * 17. ∼ ∧ ∨ ∼ p q r ( ) * 18. ∼ ∨ ∼ ∧ p q r ( ) * In Exercises 19–32, use the alternate method as described in Examples 6–8 to construct a truth table for the statement. 19. ∧ ∼ q q * 20. ∨ ∼ q q * 21. ∨ ∼ p q * 22. ∼ ∧p q * 23. ∼ ∨p q * 24. ∼ ∨ ∼ p q * 25. ∼ ∼ ∨p q ( ) * 26. ∼ ∼ ∧p q ( ) * 27. ∧ ∼ ∧ p q r ( ) * 28. ∼ ∨ ∧ p q r ( ) * 29. ∼ ∧ ∨ p q r ( ) * 30. ∼ ∨ ∧ p q r ( ) * 31. ∼ ∧ ∼ ∨ p q r ( ) * 32. ∧ ∨ p q r ~ ( ~ ) * In Exercises 33– 40, write the statement in symbolic form and construct a truth table. 33. Sammy made tacos and Wendy made salad. * 34. I have not worked overtime, but I have been paid. * 35. Jaime volunteered at the shelter or she did not go on vacation. * 36. DeJuan will not kayak or he will paddle board. * 37. It is false that the car is not in the shop but the car is a Kia. * 38. It is false that Kelvin does not take Chemistry or Janice takes history. * SECTION 3.2 *See Instructor Answer Appendix
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