Statements Truth Tables. Definition of a Statement

Transcription

1 Aendix E Introduction to Logic E.1 Introduction to Logic Statements Truth Tables Statements In everyday seech and in mathematics you make inferences that adhere to common laws of logic. These methods of reasoning allow you to build an algebra of statements by using logical oerations to form comound statements from simler ones. A rimary goal of logic is to determine the truth value (true or false) of a comound statement knowing the truth value of its simler comonents. For instance, the comound statement The temerature is below freezing and it is snowing is true only if both comonent statements are true. Definition of a Statement 1. A statement is a sentence to which only one truth value (either true or false) can be meaningfully assigned. 2. An oen statement is a sentence that contains one or more variables and becomes a statement when each variable is relaced by a secific item from a designated set. NOTE In this definition, the word statement can be relaced by the word roosition. EXAMPLE 1 Statements, Nonstatements, and Oen Statements Statement A suare is a rectangle. T 3 is less than 5. F Nonstatement Do your homework. Did you call the olice? No truth value can be meaningfully assigned. No truth value can be meaningfully assigned. Oen Statement x is an irrational number. We need a value of x. She is a comuter science We need a secific erson. major. Symbolically, statements are reresented by lowercase letters,, r, and so on. Statements can be changed or combined to form comound statements by means of the three logical oerations and, or, and not, which are reresented by (and), (or), and ~ (not). In logic the word or is used in the inclusive sense (meaning and/or in everyday language). That is, the statement or is true E1

2 E2 APPENDIX E Introduction to Logic if is true, is true, or both and are true. The following list summarizes the terms and symbols used with these three oerations of logic. Oerations of Logic Oeration Verbal Statement Symbolic Form Name of Oeration ~ not ~ Negation and Conjunction or Disjunction Comound statements can be formed using more than one logical oeration, as demonstrated in Examle 2. EXAMPLE 2 Forming Negations and Comound Statements The statements and are as follows. : The temerature is below freezing. : It is snowing. Write the verbal form for each of the following. (a) (b) ~ (c) ~ (d) ~ ~ (a) The temerature is below freezing and it is snowing. (b) The temerature is not below freezing. (c) It is not true that the temerature is below freezing or it is snowing. (d) The temerature is not below freezing and it is not snowing. EXAMPLE 3 Forming Comound Statements The statements and are as follows. : The temerature is below freezing. : It is snowing. (a) Write the symbolic form for: The temerature is not below freezing or it is not snowing. (b) Write the symbolic form for: It is not true that the temerature is below freezing and it is snowing. (a) The symbolic form is: ~ ~ (b) The symbolic form is: ~

3 APPENDIX E Introduction to Logic E3 Truth Tables To determine the truth value of a comound statement, you can create charts called truth tables. These tables reresent the three basic logical oerations. Negation Conjunction Disjunction ~ ~ T T F F T F F T F T T F F F T T T T T T F F F T F F F F T T T T F T F T T F F F For the sake of uniformity, all truth tables with two comonent statements will have T and F values for and assigned in the order shown in the first two columns of each of these three tables. Truth tables for several oerations can be combined into one chart by using the same two first columns. For each oeration, a new column is added. Such an arrangement is esecially useful with comound statements that involve more than one logical oeration and for showing that two statements are logically euivalent. Logical Euivalence Two comound statements are logically euivalent if they have identical truth tables. Symbolically, we denote the euivalence of the statements and by writing. EXAMPLE 4Logical Euivalence Use a truth table to show the logical euivalence of the statements ~ ~ and ~. ~ ~ ~ ~ ~ T T F F F T F T F F T F T F F T T F F T F F F T T T F T Identical Because the fifth and seventh columns in the table are identical, the two given statements are logically euivalent.

4 E4 APPENDIX E Introduction to Logic The euivalence established in Examle 4 is one of two well-known rules in logic called DeMorgan s Laws. Verification of the second of DeMorgan s Laws is left as an exercise. ~ is a tautology ~ ~ T F T F T T DeMorgan s Laws 1. ~ ~ ~ 2. ~ ~ ~ Comound statements that are true, no matter what the truth values of comonent statements, are called tautologies. One simle examle is the statement or not, as shown in the table at the left. E.1 Exercises In Exercises 1 12, classify the sentence as a statement, a nonstatement, or an oen statement. 1. All dogs are brown. 2. Can I hel you? 3. That figure is a circle. 4. Substitute 4 for x. 5. x is larger than is larger than x y Hockey is fun to watch. 10. One mile is greater than 1 kilometer. 11. It is more than 1 mile to the school. 12. Come to the arty. In Exercises 13 20, determine whether the oen statement is true for the given values of x. Oen Statement Values of x 13. x 2 5x 6 0 (a) x 2 (b) x x 2 x 6 0 (a) x 2 (b) x x 2 4 (a) x 2 (b) x x 3 4 (a) x 1 (b) x x 2 (a) x 0 (b) x x 2 x (a) x 3 (b) x x x 1 (a) x 4 (b) x x 2 (a) x 8 (b) x 8 In Exercises 21 24, write the verbal form for each of the following. (a) ~ (b) ~ (c) (d) 21. : The sun is shining. 22. : The car has a radio. : It is hot. : The car is red. 23. : Lions are mammals. : Lions are carnivorous. 24. : Twelve is less than 15. : Seven is a rime number. In Exercises 25 28, write the verbal form for each of the following. (a) ~ (b) ~ (c) ~ (d) ~ 25. : The sun is shining. : It is hot. 26. : The car has a radio. : The car is red. 27. : Lions are mammals. : Lions are carnivorous. 28. : Twelve is less than 15. : Seven is a rime number.

5 APPENDIX E Introduction to Logic E5 In Exercises 29 32, write the symbolic form of the given comound statement. In each case, let reresent the statement It is four o clock, and let reresent the statement It is time to go home. 29. It is four o clock and it is not time to go home. 30. It is not four o clock or it is not time to go home. 31. It is not four o clock or it is time to go home. 32. It is four o clock and it is time to go home. In Exercises 33 36, write the symbolic form of the given comound statement. In each case, let reresent the statement The dog has fleas, and let reresent the statement The dog is scratching. 33. The dog does not have fleas or the dog is not scratching. 34. The dog has fleas and the dog is scratching. 35. The dog does not have fleas and the dog is scratching. 36. The dog has fleas or the dog is not scratching. In Exercises 37 42, write the negation of the given statement. 37. The bus is not blue. 38. Frank is not 6 feet tall. 39. x is eual to x is not eual to Earth is not flat. 42. Earth is flat. In Exercises 43 48, construct a truth table for the given comound statement. 43. ~ 44. ~ 45. ~ ~ 46. ~ ~ 47. ~ 48. ~ In Exercises 49 54, use a truth table to determine whether the given statements are logically euivalent. 49. ~, ~ 50. ~ ~, ~ 51. ~ ~, ~ 52. ~, ~ ~ 53. ~, ~ ~ 54. ~, ~ ~ In Exercises 55 58, determine whether the statements are logically euivalent. 55. (a) The house is red and it is not made of wood. (b) The house is red or it is not made of wood. 56. (a) It is not true that the tree is not green. (b) The tree is green. 57. (a) The statement that the house is white or blue is not true. (b) The house is not white and it is not blue. 58. (a) I am not 25 years old and I am not alying for this job. (b) The statement that I am 25 years old and alying for this job is not true. In Exercises 59 62, use a truth table to determine whether the given statement is a tautology. 59. ~ 60. ~ 61. ~ ~ ~ 62. ~ ~ ~ 63. Use a truth table to verify the second of DeMorgan s Laws: ~ ~ ~