122 CHAPTER 3 Logic The conditional statement p q → is true in every case except when p is a true statement and q is a false statement. Example 1 A Truth Table with a Conditional Construct a truth table for the statement ∼ →∼ p q. Solution Because this statement is a conditional, the answer will lie under the .→ Fill out the truth table by placing the appropriate truth values under ∼p, column 1, and under ∼q, column 2 (see Table 3.17). Then, using the information given in the truth table for the conditional (Table 3.16 above) and the truth values in columns 1 and 2, determine the solution, column 3. In row 1, the antecedent, ∼p, is false and the consequent, ∼q, is also false. Row 1 is F F, → which according to row 4 of Table 3.16, is T. Likewise, row 2 of Table 3.17 is F T, → which is T. Row 3 is T F, → which is F. Row 4 is T T, → which is T. Now try Exercise 7 7 Table 3.17 p q p∼ → ∼q T T F T F T F F T T F T T F F F F T T T 1 3 2 Learning Catalytics Keyword: Angel-SOM-3.3 (See Preface for additional details.) Translated into symbolic form, the statement becomes p q. → Let’s examine the four cases shown in Table 3.16. Table 3.16 Conditional p q p q → T T T T F F F T T F F T CASE 1: (T, T) You get an A, and I buy a car for you. I have met my commitment, and the statement is true. CASE 2: (T, F) You get an A, and I do not buy a car for you. I have broken my promise, and the statement is false. What happens if you don’t get an A? If you don’t get an A, I no longer have a commitment to you, and therefore I cannot break my promise. CASE 3: (F, T) You do not get an A, and I buy you a car. I have not broken my promise, and therefore the statement is true. CASE 4: (F, F) You do not get an A, and I don’t buy you a car. I have not broken my promise, and therefore the statement is true. The conditional statement is false when the antecedent is true and the consequent is false. In every other case the conditional statement is true. Table 3.18 p q r p → q (∼ ∧ r) T T T T F F F T T T F T F F F F T F T T T T T T T F F T F T F F F T T F T F F T F T F F T F F F F F T F T T T T F F F F T T F F 4 5 1 3 2 Example 2 A Conditional Truth Table with Three Simple Statements Construct a truth table for the statement → ∼ ∧ p q r ( ). Solution Because this statement is a conditional, the answer will lie under the →. Work within the parentheses first. Place the truth values under ∼q, column 1, and r, column 2 (Table 3.18). Then take the conjunction of columns 1 and 2 to obtain column 3. Next, place the truth values under p in column 4. To determine the answer, column 5, use columns 3 and 4 and the information of the conditional
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