132 CHAPTER 3 Logic Definition: Equivalent Two statements are equivalent, symbolized ,⇔ if both statements have exactly the same truth values in the answer columns of the truth tables. Sometimes the words logically equivalent are used in place of the word equivalent. To determine whether two statements are equivalent, construct a truth table for each statement and compare the answer columns of the truth tables. If the answer columns are identical, the statements are equivalent. If the answer columns are not identical, the statements are not equivalent. Table 3.25 p q r p ∧ q r ( ) ∨ p q ( ) ∧ ∨ p r ( ) ∧ T T T T T T T T T T T F T T T T T F T F T T T T F T T T F F T F F F F F F T T F F T F F F F T F F F T F F F F F T F F T F F F F F F F F F F F F 1 3 2 1 3 2 Equivalent Statements Equivalent statements are an important concept in the study of logic. Example 1 Equivalent Statements Determine whether the following two statements are equivalent. p q r p q p r ( ) ( ) ( ) ∧ ∨ ∧ ∨ ∧ Solution Construct a truth table for each statement (see Table 3.25). Now try Exercise 9 Because the truth tables have the same answer (column 3 for both tables), the statements are equivalent. Therefore, we can write p q r p q p r ( ) ( ) ( ) ∧ ∨ ⇔ ∧ ∨ ∧ 7 Example 2 Are the Following Equivalent Statements? Determine whether the following statements are equivalent. a) If you win your fantasy football league and you are here Friday, then we will celebrate. b) If you do not win your fantasy football league or you are not here Friday, then we will not celebrate. Solution First write each statement in symbolic form, then construct a truth table for each statement. If the answer columns of both truth tables are identical, then the statements are equivalent. If the answer columns are not identical, then the statements are not equivalent.
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