3.3 Truth Tables for the Conditional and Biconditional 129 In Exercises 17–24, construct a truth table for the statement. 17. ∼ → ∧ p q r ( ) * 18. ∼ → ∧ p q r ( ) * 19. ↔ ∼ → p q r ( ) * 20. ∼ ↔ ∨ ∼ p q r ( ) * 21. ∼ ∧ → ∼ p q r ( ) * 22. ∨ ∼ ↔ p q r ( ) * 23. → ↔ ∼ → ∼ p q q r ( ) ( ) * 24. ∼ ↔∼ → ∼ ↔ p q q r ( ) ( ) * In Exercises 25–30, write the statement in symbolic form. Then construct a truth table for the symbolic statement. 25. If I take vitamin C, then I will stay healthy and we can go to the concert. * 26. The house is made of brick if and only if Harrington Homes is the contractor, or Stone Hammer Homes is not the contractor. * 27. Blackberries are high in vitamin K if and only if mangos are high in B vitamins, or cherries are high in vitamin C. * 28. If the trail is open then we can go hiking, if and only if it is not raining. * 29. If it is not too cold then we can take a walk, or we can go to the gym. * 30. It is false that if you play spider solitaire, then you do not play free cell and you play minesweeper. * In Exercises 31–36, use a truth table to determine whether the statement is a tautology, self-contradiction, or neither. 31. ∼ ∨ ∨ p p q ( ) Tautology 32. ∼ ∧ ∧ p p q ( ) Self-contradiction 33. ∼ ∧ ∧ ∨ ∼ p q p q ( ) ( ) Self-contradiction 34. ∨ ∼ ∨ ∼ ∨ p q p q ( ) ( ) Tautology 35. ∼ ∨ ↔ p q q [( ) ] Neither 36. ∼ ∧ → p q p [( ) ] Self-contradiction In Exercises 37– 42, use a truth table to determine whether the statement is an implication. 37. ∼ → ∨ p p q ( ) Not an implication 38. ∧ → ∼ ∨ p q p q ( ) ( ) Implication 39. ∼ → ∼ ∧ p p q ( ) Implication 40. ∨ → ∨ ∼ p q p r ( ) ( ) Not an implication 41. p q q p p q [( ) ( )] ( ) → ∧ → → ↔ Implication 42. p q r p q [( ) ] ( ) ∨ ∧ → ∨ Implication In Exercises 43–50, if p is true, q is false, and r is false, determine the truth value of the statement. 43. p q r ( ) → → True 44. p q r ( ) ∧ ∼ →∼ True 45. p q q r ( ) ( ) ∧ ↔ ∨ ∼ False 46. r p q ( ) → ∼ ↔∼ True 47. p q r ( ) ∼ ∧ ∼ ∨ ∼ True 48. p q r [ ( )] ∼ → ∧ True 49. p r q r ( ) ( ) ∼ ↔ ∨ ∼ ↔ True 50. p q p r [( ) ( )] ∼ ∨ ↔ →∼ False Problem Solving In Exercises 51–58, determine the truth value for each simple statement. Then, using the truth values, determine the truth value of the compound statement. 51. If 4 8 12, + = then 15 7 8. − = True 52. If 7 49, 2 = then 49 7. = True 53. If Hong Kong is in Italy or the United States capital is Washington, D.C., then the capital of Canada is Rio de Janeiro. False m The U.S. Capitol Building 54. If there are seven days of the week and 52 weeks in a year, then there are 364 days in a non-leap year. False 55. Snickers is a brand of watches and Hershey’s is a brand of cell phones, if and only if Reese’s is a brand of trucks. True *See Instructor Answer Appendix Ddsignstock/123RF J Main/Shutterstock
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