138 CHAPTER 3 Logic To write the converse of the conditional statement, switch the order of the antecedent and the consequent. To write the inverse, negate both the antecedent and the consequent. To write the contrapositive, switch the order of the antecedent and the consequent and then negate both of them. Are any of the variations of the conditional statement equivalent? To determine the answer, we can construct a truth table for each variation, as shown in Table 3.30. It reveals that the conditional statement is equivalent to the contrapositive statement and that the converse statement is equivalent to the inverse statement . p q Conditional p q → Contrapositive q p ∼ → ∼ Converse q p → Inverse p q ∼ → ∼ T T T T T T T F F F T T F T T T F F F F T T T T Timely Tip From Table 3.30 we can see that a conditional statement and the contrapositive statement are always equivalent to each other. We can also see that the converse statement and the inverse statement are always equivalent to each other, but they are not equivalent to the original conditional statement. Example 10 The Converse, Inverse, and Contrapositive For the conditional statement “If the painting is by Salvador Dali, then the painting is valuable,” write the a) converse. b) inverse. c) contrapositive. Solution a) Let p q : The painting is by Salvador Dali. : The painting is valuable. The conditional statement is of the form p q, → so the converse must be of the form q p. → Therefore, the converse is “If the painting is valuable, then the painting is by Salvador Dali.” b) The inverse is of the form p q. ∼ →∼ Therefore, the inverse is “If the painting is not by Salvador Dali, then the painting is not valuable.” c) The contrapositive is of the form q p. ∼ →∼ Therefore, the contrapositive is “If the painting is not valuable, then the painting is not by Salvador Dali.” 7 Now try Exercise 43 Example 11 Determine the Truth Values Let p q : The number is divisible by 10. : The number is divisible by 5. Write the following statements and determine which are true. a) The conditional statement, p q → b) The converse of p q → c) The inverse of p q → d) The contrapositive of p q → Solution a) The conditional statement in symbols is p q. → Therefore, in words the conditional statement is If the number is divisible by 10, then the number is divisible by 5. This statement is true. A number divisible by 10 must also be divisible by 5, since 5 is a divisor of 10. Table 3.30 Peter Barritt/Alamy Stock Photo m Painting by Salvador Dali
RkJQdWJsaXNoZXIy NjM5ODQ=