124 CHAPTER 3 Logic In Section 3.2, we showed that determining the truth value of a compound statement for a specific case does not require constructing an entire truth table. Examples 5 and 6 illustrate this technique for the conditional and the biconditional. The biconditional statement p q ↔ is true only when p and q have the same truth value, that is, when both are true or both are false. Example 4 A Truth Table Using a Biconditional Construct a truth table for the statement ∼ ↔ ∼ → p q r ( ). Solution Since there are three letters, there must be eight cases. The parentheses indicate that the answer must be under the biconditional, as shown in Table 3.21. Use columns 3 and 4 to obtain the answer in column 5. When columns 3 and 4 have the same truth values, place a T in column 5. When columns 3 and 4 have different truth values, place an F in column 5. Now try Exercise 19 Table 3.21 p q r p~ ↔ q (~ → r) T T T F F F T T T T F F F F T F T F T F F T T T T F F F T T F F F T T T T F T T F T F T T F T F F F T T T T T T F F F T F T F F 4 5 1 3 2 7 Example 5 Determine the Truth Value of a Compound Statement Determine the truth value of the conditional statement ∼ ↔ → ∼ ↔ p q q r ( ) ( ) when p is true, q is true, and r is false. Solution Substitute the truth values for each simple statement and simplify. p q q r T T T F F T F F F T T ( ) ( ) ( ) ( ) ( ) ( ) ∼ ↔ → ∼ ↔ ∼ ↔ → ∼ ↔ ↔ → ↔ → For this specific case, the statement is true. 7 Now try Exercise 45 Exploring Truth Values for Conditional Statements Example 6 Determine the Truth Value of a Compound Statement Determine the truth value for each simple statement. Then use the truth values to determine the truth value of the compound statement. a) If 15 is an even number, then 29 is an even number. b) Stanford University is in California and Wake Forest University is in Alaska, if and only if Syracuse University is in Alabama.
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