02 - Boolean algebra
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The document presents an overview of Boolean algebra, detailing foundational concepts, operators, and laws, including conjunction, disjunction, negation, and various logical identities. Key figures in the development of Boolean algebra such as George Boole and Claude Shannon are mentioned, along with practical examples of Boolean functions and truth tables. It also covers important principles like De Morgan's laws and demonstrates how to derive various Boolean expressions.
02 - Boolean algebra
- 4. George Boole 1815 – 1864
- 5. George Boole 1815 – 1864 Claude Shannon 1916 – 2001
- 7. +
- 8. + a = 0
- 9. + y = 0 a = 0
- 10. ++ y = 0 a = 0
- 11. + a = 1 + y = 0 a = 0
- 12. + y = 1 a = 1 + y = 0 a = 0
- 13. +
- 14. b = 0 a = 0 +
- 15. b = 0 a = 0 y = 0 ∧ ba = y 00 0 +
- 16. b = 1 a = 0 ∧ ba = y 00 0 +
- 17. b = 1 a = 0 y = 0 ∧ ba = y 00 0 10 0 +
- 18. b = 0 a = 1 ∧ ba = y 00 0 10 0 +
- 19. b = 0 a = 1 y = 0 ∧ ba = y 00 0 10 0 01 0 +
- 20. b = 1 a = 1 ∧ ba = y 00 0 10 0 01 0 +
- 21. b = 1 a = 1 y = 1 ∧ ba = y 00 0 10 0 01 0 11 1 +
- 22. b = x a = 0 ∧ ba = y 00 0 10 0 01 0 11 1 +
- 23. b = x a = 0 y = 0 ∧ ba = y 00 0 10 0 01 0 11 1 x0 0 +
- 24. b = x a = 1 ∧ ba = y 00 0 10 0 01 0 11 1 x0 0 +
- 25. b = x a = 1 y = x ∧ ba = y 00 0 10 0 01 0 11 1 x0 0 x1 x +
- 26. b = x a = x ∧ ba = y 00 0 10 0 01 0 11 1 x0 0 x1 x +
- 27. b = x a = x y = x ∧ ba = y 00 0 10 0 01 0 11 1 x0 0 x1 x xx x +
- 28. a = 0 b = 0 +
- 29. a = 0 b = 0 y = 0 ∨ ba = y 00 0 +
- 30. a = 0 b = 1 ∨ ba = y 00 0 +
- 31. a = 0 b = 1 y = 1 ∨ ba = y 00 0 10 1 +
- 32. a = 1 b = 0 ∨ ba = y 00 0 10 1 +
- 33. a = 1 b = 0 y = 1 ∨ ba = y 00 0 10 1 01 1 +
- 34. a = 1 b = 1 ∨ ba = y 00 0 10 1 01 1 +
- 35. a = 1 b = 1 y = 1 ∨ ba = y 00 0 10 1 01 1 11 1 +
- 36. a = 0 b = x ∨ ba = y 00 0 10 1 01 1 11 1 +
- 37. a = 0 b = x y = x ∨ ba = y 00 0 10 1 01 1 11 1 x0 x +
- 38. a = 1 b = x ∨ ba = y 00 0 10 1 01 1 11 1 x0 x +
- 39. a = 1 b = x y = 1 ∨ ba = y 00 0 10 1 01 1 11 1 x0 x x1 1 +
- 40. a = x b = x ∨ ba = y 00 0 10 1 01 1 11 1 x0 x x1 1 +
- 41. a = x b = x y = x ∨ ba = y 00 0 10 1 01 1 11 1 x0 x x1 1 xx x +
- 42. ∨ ba = y 00 0 10 1 01 1 11 1 ∧ ba = y 00 0 10 0 01 0 11 1 Conjunction (AND) Disjunction (OR)
- 43. ∨ ba = y 00 0 10 1 01 1 11 1 ∧ ba = y 00 0 10 0 01 0 11 1 ? Conjunction (AND) Disjunction (OR)
- 44. a = 0
- 45. a = 0 y = 1 a =¬ y 0 1
- 46. a = 1 a =¬ y 0 1
- 47. a = 1 y = 0 a =¬ y 0 1 1 0
- 48. ∨ ba = y 00 0 10 1 01 1 11 1 ∧ ba = y 00 0 10 0 01 0 11 1 a =¬ y 0 1 1 0 Conjunction (AND) Disjunction (OR) Negation (NOT)
- 49. ¬ a a ∧ b a ∨ b
- 50. - a a * b a + b ¬ a a ∧ b a ∨ b
- 51. - a a * b a + b ¬ a a ∧ b a ∨ b ! a a & b a | b
- 52. - a a * b a + b ¬ a a ∧ b a ∨ b ! a a & b a | b NOT OR AND
- 53. - a a * b a + b ¬ a a ∧ b a ∨ b ! a a & b a | b NOT OR AND
- 55. a ∧ 1 = a a ∨ 0 = a Neutral elements
- 56. a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements
- 57. a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements a ∧ a = a a ∨ a = a Idempotence
- 58. a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements a ∧ a = a a ∨ a = a Idempotence a ∧ ¬ a = 0 a ∨ ¬ a = 1 Negation
- 59. a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements a ∧ a = a a ∨ a = a Idempotence a ∧ ¬ a = 0 a ∨ ¬ a = 1 Negation a ∨ b = b ∨ a a ∧ b = b ∧ a Commutativity
- 60. a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements a ∧ a = a a ∨ a = a Idempotence a ∧ ¬ a = 0 a ∨ ¬ a = 1 Negation a ∨ b = b ∨ a a ∧ b = b ∧ a Commutativity a ∧ (b ∧ c) = (a ∧ b) ∧ c a ∨ (b ∨ c) = (a ∨ b) ∨ c Associativity
- 61. a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements a ∧ a = a a ∨ a = a Idempotence a ∧ ¬ a = 0 a ∨ ¬ a = 1 Negation a ∨ b = b ∨ a a ∧ b = b ∧ a Commutativity a ∧ (b ∧ c) = (a ∧ b) ∧ c a ∨ (b ∨ c) = (a ∨ b) ∨ c Associativity a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) Distributivity
- 62. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) DeMorgan’s laws NAND NOR
- 63. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 64. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 65. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 66. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 67. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 68. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 69. De Morgan (1806 – 1871) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) a b a ∨ b ¬ (a ∨ b) ¬ a ¬ b (¬ a) ∧ (¬ b) 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 DeMorgan’s laws NAND NOR
- 70. a ∧ 1 = a a ∨ b = b ∨ a a ∨ 0 = a a ∧ 0 = 0 a ∨ 1 = 1 a ∧ a = a a ∨ a = a a ∧ ¬ a = 0 a ∨ ¬ a = 1 Neutral elements Zero elements Idempotence Negation a ∧ (b ∧ c) = (a ∧ b) ∧ c a ∧ b = b ∧ a a ∨ (b ∨ c) = (a ∨ b) ∨ c a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) ¬ (a ∧ b) = (¬ a) ∨ (¬ b) ¬ (a ∨ b) = (¬ a) ∧ (¬ b) Commutativity Associativity Distributivity DeMorgan’s
- 72. ≠ ba = y 00 0 10 1 01 1 11 0 Exclusive OR (XOR)
- 73. ≠ ba = y 00 0 10 1 01 1 11 0 Exclusive OR (XOR) ba = y 00 1 10 1 01 0 11 1 Implication
- 74. ≠ ba = y 00 0 10 1 01 1 11 0 Exclusive OR (XOR) ba = y 00 1 10 1 01 0 11 1 ba = y 00 1 10 0 01 0 11 1 Implication Equivalence
- 75. ≠ ba = y 00 0 10 1 01 1 11 0 Exclusive OR (XOR) ba = y 00 1 10 1 01 0 11 1 ba = y 00 1 10 0 01 0 11 1 Implication Equivalence ∨ ba = y 00 0 10 1 01 1 11 1 ∧ ba = y 00 0 10 0 01 0 11 1 Conjunction (AND) Disjunction (OR) a =¬ y 0 1 1 0 Negation (NOT)
- 76. ≠ ba = y 00 0 10 1 01 1 11 0 Exclusive OR (XOR) ba = y 00 1 10 1 01 0 11 1 ba = y 00 1 10 0 01 0 11 1 Implication Equivalence ∨ ba = y 00 0 10 1 01 1 11 1 ∧ ba = y 00 0 10 0 01 0 11 1 Conjunction (AND) Disjunction (OR) a =¬ y 0 1 1 0 Negation (NOT) 4 3 3 3 2 1
- 77. Tautology f = 1 (a ∧ (a b)) b Contradiction f = 0 a ∧ (¬ a) Satisfiable f = 1 sometimes a b
- 78. How many basic boolean functions with 2 parameters are possible?
- 79. How many basic boolean functions with 2 parameters are possible?
- 80. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 81. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 82. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 83. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 84. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 85. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 86. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 87. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 88. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 89. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 90. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 91. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 92. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 93. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 94. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 95. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 96. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 97. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible?
- 98. 0011 0101 a b 0000 y0 = 0 0001 y1 = a ∧ b 0010 y2 = a ∧ ¬ b 0011 y3 = a 0100 y4 = ¬ a ∧ b 0101 y5 = b 0110 y6 = a ≠ b 0111 y7 = a ∨ b How many basic boolean functions with 2 parameters are possible? 0011 0101 a b 1111 y15 = 1 1110 y14 = ¬ a ∨ ¬ b 1101 y13 = ¬ a ∨ b 1100 y12 = ¬ a 1011 y11 = a ∨ ¬ b 1010 y10 = ¬ b 1001 y9 = a b 1000 y8 = ¬ a ∧ ¬ b
- 99. How to create a half adder?
- 100. How to create a half adder? + ba = co 00 0 10 0 01 0 11 1 q 0 1 1 0
- 101. How to create a half adder? a b q co + ba = co 00 0 10 0 01 0 11 1 q 0 1 1 0 AND XOR
- 102. How to create a full adder?
- 103. How to create a full adder? + ba = co 00 0 10 0 01 0 11 1 q 0 1 1 0 ci + 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1
- 104. How to create a full adder? + ba = co 00 0 10 0 01 0 11 1 q 0 1 1 0 ci + 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 Full adder a b coci q
- 105. How to create a full adder? Full adder a b coci q
- 106. How to create a full adder? Full adder a b coci q Full adder a b coci q Full adder a b coci q