128 CHAPTER 3 Logic In any implication the antecedent of the conditional statement implies the consequent. In other words, if the antecedent is true, then the consequent must also be true. That is, the consequent will be true whenever the antecedent is true. We will use implications when we study symbolic arguments in Section 3.5. Definition: Implication An implication is a conditional statement that is a tautology. Example 10 An Implication? Determine whether the conditional statement ∨ ∧ ∼ → p q q p [( ) ] is an implication. Solution If the conditional statement is a tautology, the conditional statement is an implication. Because the conditional statement is a tautology (see Table 3.24), the conditional statement is an implication. The antecedent ∨ ∧ ∼ p q q [( ) ] implies the consequent p. Note that the antecedent is true only in case 2 and the consequent is also true in case 2. Table 3.24 p q p q [( ) ∨ ∧ q] ∼ → p T T T F F T T T F T T T T T F T T F F T F F F F F T T F 1 3 2 5 4 Now try Exercise 37 7 Instructor Resources for Section 3.3 in MyLab Math • Objective-Level Videos 3.3 • Interactive Concept Video: Applying Four Rules to Truth Tables • Animation: Exploring Truth Values for Conditional Statements • PowerPoint Lecture Slides 3.3 • MyLab Exercises and Assignments 3.3 Exercises Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The conditional statement p q → is _______ only when p is true and q is false. False 2. In the conditional statement p q, → a) The lower-case letter p represents the _______. Antecedent b) The lower-case letter q represents the _______. Consequent 3. The biconditional statement p q ↔ is _______ only when p and q have the same truth value. True 4. A compound statement that is always true is known as a(n) _______ . Tautology 5. A compound statement that is always false is known as a(n) _______ . Self-contradiction 6. A conditional statement that is a tautology is known as a(n) _______ . Implication Practice the Skills In Exercises 7–16, construct a truth table for the statement. 7. ∼ →p q * 8. → ∼ p q * 9. ∼ → ∼ p q ( ) * 10. ∼ ∼ ↔p q ( ) * 11. p q p ( ) → ↔ * 12. p q p ( ) ↔ → * 13. p q p ( ) ↔ ∨ * 14. p q p ( ) ↔ ∨ * 15. ∼ → ↔ → ∼ p q p q ( ) ( ) * 16. ∼ ↔ → ∼ ↔∼ p q p q ( ) ( ) * SECTION 3.3 *See Instructor Answer Appendix
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