3.4 Equivalent Statements 141 4. The negation of the conditional statement p q ( ) ∼ → is equivalent to the following conjunction statement: ________. ∧ ∼ p q 5. Given the conditional statement p q, → the converse of the conditional statement in symbolic form is ________. q p → 6. Given the conditional statement p q, → the inverse of the conditional statement in symbolic form is ________. ∼ → ∼ p q 7. Given the conditional statement p q, → the contrapositive of the conditional statement in symbolic form is ________. ∼ → ∼ q p 8. Of the converse, inverse, and contrapositive, only the contrapositive of the conditional statement is ________ to the conditional statement. Equivalent Practice the Skills In Exercises 9–16, use a truth table to determine whether the two statements are equivalent. 9. ∼ → ∧ ∼ p q p q ( ), Equivalent 10. p q p q , → ∼ ∨ Equivalent 11. ∼ → ∼ → q p p q , Equivalent 12. q p p q , → ∼ → ∼ Equivalent 13. p q r p q r ( ) , ( ) ∨ ∧ ∨ ∧ Not equivalent 14. p q r p q r ( ), ( ) ∧ ∨ ∧ ∨ Not equivalent 15. p q q p p q ( ) ( ), ( ) → ∧ → ↔ Equivalent 16. p q r p q r ( ) , ( ) → → → → Not equivalent In Exercises 17–24, use De Morgan’s laws to determine whether the two statements are equivalent. 17. ∼ ∧p q ( ), ∼ ∧ ∼ p q Not equivalent 18. ∼ ∧p q ( ), ∼ ∨ ∼ p q Equivalent 19. ∼ ∨p q ( ), ∼ ∧ ∼ p q Equivalent 20. ∼ ∼ ∧p q ( ), ∧ ∼ p q Not equivalent 21. ∼ ∨p q ( ), ∼ ∨ ∼ p q Not equivalent 22. ∼ ∼ ∧ ∨ ∼ p q p q ( ), Equivalent 23. p q r p q r ( ) , ( ) ∼ ∨ ∼ → ∼ ∧ → Equivalent 24. → ∼ ∧ ∼ → ∼ ∨ q p r q p r ( ), Equivalent Problem Solving In Exercises 25–30, use De Morgan’s laws to write an equivalent statement for the given sentence. 25. It is false that Kenny Chesney sings country and Drake sings opera. Kenny Chesney does not sing country or Drake does not sing opera. 26. It is false that the printer is an HP or the laptop is a Dell. The printer is not an HP and the laptop is not a Dell. 27. The cat was not a Persian and the cat was not a Siamese. It is false that the cat was a Persian or the cat was a Siamese. 28. Saniaa is an artist, but she is not famous. It is false that Saniaa is not an artist or she is famous. 29. If I am late for class, then I will not get bonus points or I will fail the test. If I am late for class, then it is false that I will get bonus points and I will not fail the test. 30. If Phil buys us dinner, then we will not go to the top of the CN Tower but we will be able to walk to the Red Bistro Restaurant. * In Exercises 31–36, use the fact that p q → is equivalent to ∼ ∨p q to write an equivalent form of the given statement. 31. If I see a movie, then I buy popcorn. I do not see a movie or I buy popcorn. 32. The path is not steep or we went hiking. If the path is steep then we went hiking. 33. The sky is not clear or we see the moon. If the sky is clear then we see the moon. 34. If Joanne goes to the Lightning game, then she will not go to the Rays game. Joanne does not go to the Lightning game or she will not go to the Rays game. 35. Garrett goes apple picking or he does not go on a hayride. If Garrett does not go apple picking, then he does not go on a hayride. 36. If the interest rate does not go up, then we buy a new home. The interest rate goes up or we buy a new home. In Exercises 37– 42, use the fact that ∼ →p q ( ) is equivalent to ∧ ∼ p q to write the statement in an equivalent form. 37. It is false that if we go to Chicago, then we will go to Navy Pier. We go to Chicago and we do not go to Navy Pier. 38. It is not true that if Ghana is in the United Nations then Ghana is in the World Trade Organization. Ghana is in the United Nations and Ghana is not in the World Trade Organization. 39. The water is cold and I am not going swimming. It is false that if the water is cold then I go swimming. 40. The Bills beat the Patriots and the Orange beat the Blue Devils. It is false that if the Bills beat the Patriots then the Orange did not beat the Blue Devils. 41. It is not true that if Amazon has a sale then we will buy $100 worth of books. Amazon has a sale and we will not buy $100 worth of books. 42. Dexter works in Miami, but Debra does not work for him. It is false that if Dexter works in Miami then Debra works for him. *See Instructor Answer Appendix Jovan Mandic/123RF
RkJQdWJsaXNoZXIy NjM5ODQ=