140 CHAPTER 3 Logic Solution Let p: You leave by 9 a.m. q: You will get to your destination on time. In symbolic form, the four statements are a) p q. → b) ∼ ∨ p q. c) ∼ ∨ ∼ q p ( ). d) ∼ →∼ q p. Which of these statements are equivalent? Earlier in this section, you learned that p q → is equivalent to ∼ ∨ p q. Therefore, statements (a) and (b) are equivalent. Statement (d) is the contrapositive of statement (a). Therefore, statement (d) is also equivalent to statement (a) and statement (b). Statements (a), (b), and (d) all have the same truth table (Table 3.31). Now try Exercise 55 Table 3.31 (a) (b) (d) p q p q → p q ∼ ∨ q p ∼ → ∼ T T T T T T F F F F F T T T T F F T T T Now let’s look at statement (c). To determine whether ∼ ∨ ∼ q p ( ) is equivalent to the other statements, we will construct its truth table (Table 3.32) and compare the answer column with the answer columns in Table 3.31. Table 3.32 (c) p q ∼ q( ∨ p) ∼ T T F T T F T F T F F F F T F T T T F F F F T T 4 1 3 2 None of the three answer columns of the truth table in Table 3.31 is the same as the answer column of the truth table in Table 3.32. Therefore q p ( ) ∼ ∨ ∼ is not equivalent to any of the other statements. Therefore, only statements (a), (b), and (d) are equivalent to each other. 7 Instructor Resources for Section 3.4 in MyLab Math • Objective-Level Videos 3.4 • Animation: Variations of the Conditional Statement • PowerPoint Lecture Slides 3.4 • MyLab Exercises and Assignments 3.4 Exercises Warm Up Exercises In Exercises 1–8, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. Statements that have exactly the same truth values in the answer columns of their truth tables are called ________ statements. Equivalent 2. De Morgan’s laws state that a) ∼ ∧p q ( ) is equivalent to ________, and ∼ ∨ ∼ p q b) ∼ ∨p q ( ) is equivalent to ________. ∼ ∧ ∼ p q 3. The conditional statement p q → is equivalent to the following disjunction statement: ________. ∼ ∨ p q SECTION 3.4
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