3.4 Equivalent Statements 143 63. a) If the corn bag goes in the hole, then you are awarded three points. b) It is false that the corn bag goes in the hole and you are awarded three points. c) The corn bag does not go in the hole and you are not awarded three points. None are equivalent. 64. a) Fitz and the Tantrums will not go on tour if and only if James King does not play the saxophone. b) It is false that Fitz and the Tantrums will go on tour if and only if James King does not play the saxophone. c) If Fitz and the Tantrums go on tour, then James King plays saxophone. a) and b) are equivalent. 65. a) If the pay is good and today is Monday, then I will take the job. b) If I do not take the job, then it is false that the pay is good or today is Monday. c) The pay is good and today is Monday, or I will take the job. None are equivalent. 66. a) If you are 18 years old and a citizen of the United States, then you can vote in the presidential election. b) You can vote in the presidential election, if and only if you are a citizen of the United States and you are 18 years old. c) You cannot vote in the presidential election, or you are 18 years old and you are not a citizen of the United States. None are equivalent. 67. a) The spacecraft was made by SpaceX, or it was not made by Orbital and it was made by Boeing. b) The spacecraft was made by Boeing if and only if, it was made by SpaceX or it was not made by Orbital. c) If the spacecraft was not made by SpaceX, then it was not made by Orbital and it was made by Boeing. a) and c) are equivalent 68. a) The mortgage rate went down, if and only if Tim purchased the house and the down payment was 10%. b) The down payment was 10%, and if Tim purchased the house then the mortgage rate went down. c) If Tim purchased the house, then the mortgage rate went down and the down payment was not 10%. None are equivalent. Concept/Writing Exercises 69. If p and q represent two simple statements, and if p q → is a false statement, what must be the truth value of the converse, q p? → Explain. * 70. If p and q represent two simple statements, and if p q → is a false statement, what must be the truth value of the inverse, p q? ∼ →∼ Explain. * 71. If p and q represent two simple statements, and if p q → is a false statement, what must be the truth value of the contrapositive, q p? ∼ →∼ Explain. False. A conditional statement and its contrapositive always have the same truth values. 72. If p and q represent two simple statements, and if p q → is a true statement, what must be the truth value of the contrapositive, q p? ∼ →∼ Explain. True. A conditional statement and its contrapositive always have the same truth values. Challenge Problems/Group Activities 73. We learned that p q p q. → ⇔∼ ∨ Determine a conjunction that is equivalent to p q. → (Hint: There are many answers.) Answers will vary. 74. Determine whether p q p q ( ) . ∼⎡⎣∼ ∨∼ ⎤⎦⇔ ∨∼ Explain the method(s) you used to determine your answer. They are equivalent, since p p ( ) . ∼ ∼ ⇔ 75. Fuzzy Logic Read the Mathematics Today box on fuzzy logic on page 139. In symbolic logic, a statement is either true or false (consider true to have a value of 1 and false a value of 0). In fuzzy logic, nothing is true or false, but everything is a matter of degree. For example, consider the statement “The sun is shining.” In fuzzy logic, this statement may have a value between 0 and 1 and may be constantly changing. For example, if the sun is partially blocked by clouds, the value of this statement may be 0.25. In fuzzy logic, the values of connective statements are found as follows for statements p and q. Not p has a truth value of p 1 . − p q ∧ has a truth value equal to the minimum of p and q. p q ∨ has a truth value equal to the maximum of p and q. p q → has a truth value equal to the minimum of 1 and p q 1 . − + p q ↔ has a truth value equal to p q 1 , − − that is, 1 minus the absolute value of p minus q. Suppose the statement “p: The sun is shining” has a truth value of 0.25 and the statement “q: Mary is getting a tan” has a truth value of 0.20. Determine the truth value of a) p∼ 0.75 b) q∼ 0.80 c) p q ∧ 0.20 d) p q ∨ 0.25 e) p q → 0.95 f) p q ↔ 0.95 *See Instructor Answer Appendix Oranzy Photography/ Shutterstock
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