3.4 Equivalent Statements 133 Table 3.26 p q r p( ∧ q) → r T T T T T T T T T T F T T T F F T F T T F F T T T F F T F F T F F T T F F T T T F T F F F T T F F F T F F F T T F F F F F F T F 1 3 2 5 4 Let: p q r : You win your fantasy football league. : You are here Friday. : We will celebrate. In symbolic form, the statements are a) p q r ( ) . ∧ → b) p q r ( ) . ∼ ∨ ∼ →∼ The truth tables for these statements are given in Tables 3.26 and 3.27, respectively. The answers in the columns labeled 5 are not identical, so the statements are not equivalent. Now try Exercise 13 Table 3.27 p q r p (∼ ∨ q) ∼ → r∼ T T T F F F T F T T F F F F T T T F T F T T F F T F F F T T T T F T T T T F F F F T F T T F T T F F T T T T F F F F F T T T T T 1 3 2 5 4 7 RECREATIONAL MATH Kakuro Like Sudoku (see the Recreational Math box on Sudoku on page 100), Kakuro is a puzzle that requires logic. The aim of the game is to fill all the blank squares in the grid with only the digits 1-9 so that the numbers you enter add up to the corresponding clues. For example, in the puzzle shown below, the 17 near the upper left corner of the puzzle is a clue that indicates that the two numbers that you place in the two squares to the right of the 17 must add up to equal 17. The 9 near the upper left corner of the puzzle is a clue that indicates that the two numbers that you place in the two squares below the 9 must add up to equal 9. In addition, you may not repeat any digits in any given “word” or string of numbers. 12 15 4 45 11 9 17 22 17 15 16 22 14 15 14 6 11 7 33 14 5 8 6 13 17 19 16 12 6 26 3 28 45 16 17 6 17 15 9 3 30 4 For more information and more puzzles, see www.Kakuros.com. Furthermore, there are many apps that can be used to play Kakuro on your tablet or smartphone. The solution to the puzzle can be found in the answer section in the back of this book. For an additional puzzle, see Exercise 76. Table 3.28 p q ∼ p( ∧ q) T T F T T T T F T T F F F T T F F T F F T F F F 4 1 3 2 Example 3 Which Statements Are Logically Equivalent? Determine which, if any, of the following statements are equivalent to “It is not true that the story is on social media and the story is real.” a) If the story is not real, then the story is not on social media. b) The story is not on social media or the story is not real. c) The story is not real and the story is not on social media. d) If the story is not on social media, then the story is not real. Solution To determine whether any of the choices are equivalent to the given statement, first write the given statement and the choices in symbolic form. Then construct truth tables and compare the answer columns of the truth tables. Let p q : The story is on social media. : The story is real. The given statement is expressed in symbolic form as p q ( ). ∼ ∧ Using p and q as indicated, choices (a) through (d) may be expressed symbolically as a) q p. ∼ →∼ b) p q. ∼ ∨ ∼ c) q p. ∼ ∧ ∼ d) p q. ∼ →∼ Table 3.28 shows the truth table for the given statement and Table 3.29 shows the truth table for each statement (a) through (d). By examining the truth tables, we see that the given statement, p q ( ), ∼ ∧ is logically equivalent to choice (b), p q. ∼ ∨ ∼ Therefore, the correct answer is “The story is not on social media or the story is not real.” This statement is logically equivalent to the original statement “It is not true that the story is on social media and the story is real.”
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