3.4 Equivalent Statements 131 Challenge Problems/Group Activities In Exercises 71 and 72, construct truth tables for the symbolic statement. 71. p q r p q [ ( )] ( ) ∨ →∼ ↔ ∧ ∼ * 72. r q p q r [( ) ] ( ) →∼ →∼ ∨ ↔∼ * 73. Construct a truth table for a) p q r s ( ) ( ). ∨ → ∧ T F F F T F F F T F F F T T T T b) q p r s ( ) ( ). →∼ ∨ ↔ T F F T T T T T T T T T T T T T Recreational Mathematics 74. Satisfiability Problem-Birthday Puzzle Refer to the Recreational Math box on page 126 and then solve the following satisfiability problem. Allen, Booker, Chris, and Dennis all were born in the same year—one in January, one in February, one in March, and one in April. Chris was born before Dennis. Dennis was born two months after Booker. Booker was born after Allen, but before Chris. Determine who was born in each month. * 75. Satisfiability Problem-Cat Puzzle Solve the following puzzle. Pablo has four cats. The parents are Tiger and Boots, and the kittens are Sam and Sue. Each cat insists on eating out of its own bowl. To complicate matters, each cat will eat only its own brand of cat food. The colors of the bowls are yellow, pink, blue, and green. The different types of cat food are Whiskas, Friskies, Nine Lives, and Meow Mix. Tiger will eat Meow Mix if and only if it is in a yellow bowl. If Boots is to eat her food, then it must be in a yellow bowl. Pablo knows that the label on the can containing Sam’s food is the same color as his bowl. Boots eats Whiskas. Meow Mix and Nine Lives are packaged in a brown paper bag. The color of Sue’s bowl is green if and only if she eats Meow Mix. The label on the Friskies can is pink. Match each cat with its food and the bowl of the correct color. * m See Exercise 75 76. The Youngest Triplet The Barr triplets have an annoying habit: Whenever a question is asked of the three of them, two tell the truth and the third lies. When I asked them which of them was born last, they replied as follows. Mary: Katie was born last. Katie: I am the youngest. Annie: Mary is the youngest. Which of the Barr triplets was born last? Katie was born last. Katie and Mary are saying the same thing. Research Activity 77. Advertisement Select an advertisement from the Internet, a newspaper, or a magazine that makes or implies a conditional statement. Analyze the advertisement to determine whether the consequent necessarily follows from the antecedent. Explain your answer. (See Example 3.) *See Instructor Answer Appendix SECTION 3.4 Equivalent Statements Consider the following statements: If you got straight A’s, then you are on the president’s list. If you are on the president’s list, then you got straight A’s. If you did not get straight A’s, then you are not on the president’s list. If you are not on the president’s list, then you did not get straight A’s. Are these statements all saying the same thing, or does each one say something completely different from the others? In this section, we will use logic symbols and truth tables to compare each of these statements. We will also learn several forms of equivalent statements. LEARNING GOALS Upon completion of this section, you will be able to: 7 Determine if two statements are equivalent. 7 Use De Morgan’s laws for logic. 7 Write conditional statements as disjunctions. 7 Write the negation of conditional statements. 7 Write variations of conditional statements, including inverses, converses, and contrapositives. Why This Is Important In our everyday lives we often encounter statements that are initially difficult to understand. The tools presented in this section can help us better understand such statements. Understanding when two statements are equivalent is important to understanding advertisers’ claims, political statements, and legal documents. Linn Currie/Shutterstock Stockfour/Shutterstock
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