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- 2. Propositions Our discussion begins with an introduction to the basic building blocks of logic – propositions.
- 3. Propositions Example 1 All the following declarative sentences are propositions. 1. Minsk is the capital of
- 4. Propositions
- 5. Propositions
- 6. Propositions The area of logic that deals with propositions is called the propositional calculus or propositional
- 7. Compound propositions We now turn our attention to methods for producing new propositions from those that
- 8. Compound propositions Many mathematical statements are constructed by combining one or more propositions. New propositions, called
- 9. The negation of a proposition
- 10. The negation of a proposition
- 11. The negation of a proposition Example 3 Find the negation of the proposition “Vandana’s smartphone has
- 12. The conjunction of two propositions Definition 3 Let p and q be propositions. The conjunction of
- 13. The conjunction of two propositions
- 14. The conjunction of two propositions Example 4 Find the conjunction of the propositions p and q
- 15. The conjunction of two propositions Solution The conjunction of these propositions, p∧q, is the proposition “Rebecca’s
- 16. The disjunction of two propositions Definition 4 Let p and q be propositions. The disjunction of
- 17. The disjunction of two propositions
- 18. The disjunction of two propositions Example 5 Find the disjunction of the propositions p and q
- 19. The disjunction of two propositions Solution The disjunction of p and q, p∨q, is the proposition
- 20. The exclusive or The use of the connective or in a disjunction corresponds to one of
- 21. The exclusive or On the other hand, we are using the exclusive or when we say
- 22. The exclusive or Definition 5 Let p and q be propositions. The exclusive or of p
- 23. The exclusive or
- 24. Conditional statements
- 25. Conditional statements
- 26. Conditional statements
- 27. Conditional statements
- 28. Conditional statements
- 29. Converse, contrapositive and inverse
- 30. Converse, contrapositive and inverse
- 31. Biconditionals
- 32. Biconditionals
- 33. Biconditionals
- 34. Biconditionals
- 35. Truth tables of compound propositions We have now introduced four important logical connectives – conjunctions, disjunctions,
- 36. Truth tables of compound propositions We can use truth tables to determine the truth values of
- 37. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 38. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 39. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 40. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 41. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 42. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 43. Truth tables of compound propositions Example 9 Construct the truth table of the compound proposition (pq)
- 44. Precedence of logical operators
- 45. Precedence of logical operators
- 46. Precedence of logical operators
- 47. Precedence of logical operators Another general rule of precedence is that the conjunction operator takes precedence
- 48. Precedence of logical operators
- 49. Tautologies and contradictions Definition 8 A compound proposition that is always true, no matter what the
- 50. Tautologies and contradictions Example 10 We can construct examples of tautologies and contradictions using just one
- 51. Tautologies and contradictions
- 52. Logical equivalences
- 53. Logical equivalences One way to determine whether two compound propositions are equivalent is to use a
- 54. Logical equivalences Example 11 Show that (pq) and pq are logically equivalent.
- 55. Logical equivalences Example 11 Show that (pq) and pq are logically equivalent.
- 56. Logical equivalences Example 11 Show that (pq) and pq are logically equivalent.
- 57. Logical equivalences Example 11 Show that (pq) and pq are logically equivalent.
- 58. Logical equivalences Example 11 Show that (pq) and pq are logically equivalent.
- 59. Logical equivalences Example 2 Show that (pq) and pq are logically equivalent.
- 60. Logical equivalences Example 11 Show that (pq) and pq are logically equivalent.
- 61. Logical equivalences
- 62. Logical equivalences (pq) pq This logical equivalence is one of the two De Morgan laws,
- 63. Logical equivalences
- 64. Logical equivalences
- 65. Logical equivalences
- 66. Logical equivalences
- 67. Logical equivalences
- 68. Logical equivalences
- 69. Logical equivalences
- 70. Logical equivalences We will now establish a logical equivalence of two compound propositions involving three different
- 71. Logical equivalences Example 13 Show that p∨(q∧r) and (p∨q)∧(p∨r) are logically equivalent. This is the distributive
- 72. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 73. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 74. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 75. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 76. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 77. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 78. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent.
- 79. A Demonstration That p(qr) and (pq)(pr) Are Logically Equivalent. Because the truth values of p∨(q∧r) and
- 80. Logical equivalences Next table contains some important equivalences. In these equivalences, T denotes the compound proposition
- 83. Logical equivalences
- 84. Logical equivalences We also display some useful equivalences for compound propositions involving conditional statements and biconditional
- 87. Using De Morgan’s Laws Example 13 Use De Morgan’s laws to express the negations of “Miguel
- 88. Using De Morgan’s Laws Example 13 Use De Morgan’s laws to express the negations of “Miguel
- 89. Constructing new logical equivalences The logical equivalences in Table 1, as well as any others that
- 90. Constructing new logical equivalences This technique is illustrated in Examples 14 – 16, where we also
- 91. Constructing new logical equivalences Example 14 Show that (p q) and p q are
- 92. Constructing new logical equivalences Example 14 Show that (p q) and p q are
- 93. Constructing new logical equivalences Example 15 Show that (p (p q)) and (p
- 94. Constructing new logical equivalences Example 15 Show that (p (p q)) and (p
- 95. Constructing new logical equivalences Example 16 Show that (p q) (p q) is
- 96. Propositional satisfiability Definition 10 A compound proposition is satisfiable if there is an assignment of truth
- 97. Propositional satisfiability Definition 11 When we find a particular assignment of truth values that makes a
- 98. Propositional satisfiability However, to show that a compound proposition is unsatisfiable, we need to show that
- 99. Propositional satisfiability Example 17 Determine whether each of the compound propositions (p q) (q
- 100. Propositional satisfiability
- 101. Satisfiability problem Many problems, in diverse areas such as robotics, software testing, computer-aided design, machine vision,
- 102. Sudoku 99 A Sudoku puzzle is represented by a 9×9 grid made up of nine 3×3
- 103. Sudoku 99 The puzzle is solved by assigning a number to each blank cell so that
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