I used this set of slides for the lecture on Computational Thinking I gave at the University of Zurich for the 1st year students following the course of Formale Grundlagen der Informatik.
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
71.
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