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Axiomas Teoremas Exemplos
Algebra Booleana
Prof. Ricardo Menotti
Universidade Federal de Sao Carlos (UFSCar)[email protected]
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0
2a 1.1 = 1
3a 0.1 = 1.0 = 0
4a Se x = 0, entao x = 1
1b 1 + 1 = 1
2b 0 + 0 = 0
3b 1 + 0 = 0 + 1 = 1
4b Se x = 1, entao x = 0
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0
2a 1.1 = 1
3a 0.1 = 1.0 = 0
4a Se x = 0, entao x = 1
1b 1 + 1 = 1
2b 0 + 0 = 0
3b 1 + 0 = 0 + 1 = 1
4b Se x = 1, entao x = 0
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0
2a 1.1 = 1
3a 0.1 = 1.0 = 0
4a Se x = 0, entao x = 1
1b 1 + 1 = 1
2b 0 + 0 = 0
3b 1 + 0 = 0 + 1 = 1
4b Se x = 1, entao x = 0
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Axiomas
1a 0.0 = 0
2a 1.1 = 1
3a 0.1 = 1.0 = 0
4a Se x = 0, entao x = 1
1b 1 + 1 = 1
2b 0 + 0 = 0
3b 1 + 0 = 0 + 1 = 1
4b Se x = 1, entao x = 0
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com uma variavel
5a x .0 = 0
6a x .1 = x
7a x .x = x
8a x .x = 0
5b x + 1 = 1
6b x + 0 = x
7b x + x = x
8b x + x = 1
9 x = x
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com uma variavel
5a x .0 = 0
6a x .1 = x
7a x .x = x
8a x .x = 0
5b x + 1 = 1
6b x + 0 = x
7b x + x = x
8b x + x = 1
9 x = x
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com uma variavel
5a x .0 = 0
6a x .1 = x
7a x .x = x
8a x .x = 0
5b x + 1 = 1
6b x + 0 = x
7b x + x = x
8b x + x = 1
9 x = x
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com uma variavel
5a x .0 = 0
6a x .1 = x
7a x .x = x
8a x .x = 0
5b x + 1 = 1
6b x + 0 = x
7b x + x = x
8b x + x = 1
9 x = x
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com uma variavel
5a x .0 = 0
6a x .1 = x
7a x .x = x
8a x .x = 0
5b x + 1 = 1
6b x + 0 = x
7b x + x = x
8b x + x = 1
9 x = x
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com duas ou mais variaveis
Comutativas
10a x .y = y .x 10b x + y = y + x
Associativas
11a x .(y .z) = (x .y).z 11b x + (y + z) = (x + y) + z
Distributivas
12a x .(y + z) = x .y + x .z 12b x + y .z = (x + y).(x + z)
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com duas ou mais variaveis
Comutativas
10a x .y = y .x 10b x + y = y + x
Associativas
11a x .(y .z) = (x .y).z 11b x + (y + z) = (x + y) + z
Distributivas
12a x .(y + z) = x .y + x .z 12b x + y .z = (x + y).(x + z)
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com duas ou mais variaveis
Comutativas
10a x .y = y .x 10b x + y = y + x
Associativas
11a x .(y .z) = (x .y).z 11b x + (y + z) = (x + y) + z
Distributivas
12a x .(y + z) = x .y + x .z 12b x + y .z = (x + y).(x + z)
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com duas ou mais variaveis
Absorcao
13a x + x .y = x 13b x .(x + y) = x
Combinacao
14a x .y + x .y = x 14b (x + y).(x + y) = x
DeMorgan
15a x .y = x + y
16a x + x .y = x + y
15b x + y = x .y
16b x .(x + y) = x .y
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com duas ou mais variaveis
Absorcao
13a x + x .y = x 13b x .(x + y) = x
Combinacao
14a x .y + x .y = x 14b (x + y).(x + y) = x
DeMorgan
15a x .y = x + y
16a x + x .y = x + y
15b x + y = x .y
16b x .(x + y) = x .y
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Teoremas com duas ou mais variaveis
Absorcao
13a x + x .y = x 13b x .(x + y) = x
Combinacao
14a x .y + x .y = x 14b (x + y).(x + y) = x
DeMorgan
15a x .y = x + y
16a x + x .y = x + y
15b x + y = x .y
16b x .(x + y) = x .y
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)
LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)
LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)
LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0
LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0
LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
(x1 + x2).(x2 + x3).(x1 + x3) = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
LHS = (x1 + x2).(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).(x1 + x3) + x2.(x2 + x3).(x1 + x3)LHS = x1.(x2 + x3).x1 + x1.(x2 + x3).x3 + x2.(x2 + x3).x1 + x2.(x2 + x3).x3
LHS = x1.x1.(x2 + x3) + x1.x3.(x2 + x3) + x2.x1.(x2 + x3) + x2.x3.(x2 + x3)LHS = x1.x1.x2 +x1.x1.x3 +x1.x3.x2 +x1.x3.x3 +x2.x1.x2 +x2.x1.x3 +x2.x3.x2 +x2.x3.x3
LHS = x1.x1.x2 +x1.x1.x3 +x1.x2.x3 +x1.x3.x3 +x1.x2.x2 +x1.x2.x3 +x2.x2.x3 +x2.x3.x3
LHS = 0.x2 + 0.x3 + x1.x2.x3 + x1.0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + x2.0LHS = 0 + 0 + x1.x2.x3 + 0 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3 + 0LHS = x1.x2.x3 + x1.x2.x2 + x1.x2.x3 + x2.x2.x3
LHS = x1.x2.x3 + x1.x2 + x1.x2.x3 + x2.x3
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.w
LHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.w
LHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.x
LHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w
LHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.w
RHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.w
RHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.z
RHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
Axiomas Teoremas Exemplos
Provar que:
x.z + x.z + y.z.w = x.z + x.z + x.y.w
LHS = x.z + x.z + y.z.wLHS = x.z.y + x.z.y + x.z.y + x.z.y + y.z.wLHS = x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + x.z.y.w + y.z.w.x + y.z.w.xLHS = x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.w + x.y.z.wLHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
RHS = x.z + x.z + x.y.wRHS = x.z.y + x.z.y + x.z.y+ x.z.y + x.y.wRHS = x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.z.y.w+ x.y.w.z+ x.y.w.zRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.wRHS = x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w+ x.y.z.w
Prof. Ricardo Menotti Algebra Booleana
![Aula 4: Álgebra booleana - Circuitos Digitaiscompscinet.org/hausen/courses/circuitos/aulas/aula4/aula4.pdf · Umaálgebradiferente Álgebra booleana [Boole,1854] Álgebraondeháapenasdoisvalores](https://img.document.onl/doc/110x75/5abb72107f8b9a24028ca8fa/aula-4-lgebra-booleana-circuitos-lgebra-booleana-boole1854-lgebraondehapenasdoisvalores.jpg)
