AB-Algebra-Boole-Simplificacao-Circuitos
123 pág.

AB-Algebra-Boole-Simplificacao-Circuitos


DisciplinaOrganização de Computadores4.208 materiais77.343 seguidores
Pré-visualização7 páginas
para 5 Variáveis
A
\u100
99
Hexas (1)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região A
A\u100
Região B
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \u100
A\u100
Região \ufffd
100
Hexas (2)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região C
A\u100
Região D
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \ufffd
A\u100
Região ð
101
Hexas (3)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região E
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \u112
102
Oitavas (1/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região A.B
A\u100
Região A.\ufffd
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \u100.B
A\u100
Região \u100.\ufffd
103
Oitavas (2/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região A.C
A\u100
Região A.\ufffd
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \u100.C
A\u100
Região \u100.\ufffd
104
Oitavas (3/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região A.D
A\u100
Região A.ð
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \u100.D
A\u100
Região \u100.ð
105
Oitavas (4/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região A.E
A\u100
Região A.\u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \u100.E
A\u100
Região \u100.\u112
106
Oitavas (5/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região B.C
A\u100
Região B.\ufffd
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \ufffd.C
A\u100
Região \ufffd.\ufffd
107
Oitavas (6/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região B.D
A\u100
Região B.ð
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \ufffd.D
A\u100
Região \ufffd.ð
108
Oitavas (7/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região B.E
A\u100
Região B.\u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \ufffd.E
A\u100
Região \ufffd.\u112
109
Oitavas (8/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região C.D
A\u100
Região C.ð
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \ufffd.D
A\u100
Região \ufffd.ð
110
Oitavas (9/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região C.E
A\u100
Região C.\u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região \ufffd.E
A\u100
Região \ufffd.\u112
111
Oitavas (10/10)
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região D.E
A\u100
Região D.\u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
ð D
\ufffd
\ufffd
C
B
\ufffd
\u112 E \u112
A\u100
Região ð.E
A\u100
Região ð.\u112
112
Exemplo: Simplifique o Circuito 
representado pelo diagrama
ð D
\ufffd
0 0 0 0 \ufffd
0 1 0 1
C
B
1 1 1 1
0 0 0 0 \ufffd
\u112 E \u112
A\u100
ð D
\ufffd
1 0 1 0 \ufffd
1 1 1 0
C
B
0 1 0 1
1 1 0 1 \ufffd
\u112 E \u112
113
Exemplo: 2 Quadras
ð D
\ufffd
\ufffd
1 1
C
B
1 1 1 1
\ufffd
\u112 E \u112
A\u100
ð D
\ufffd
1 1 \ufffd
1 1 1
C
B
1 1
1 1 1 \ufffd
\u112 E \u112
114
Exemplo: 2 Quadras, 5 Pares
ð D
\ufffd
\ufffd
1 1
C
B
1 1 1 1
\ufffd
\u112 E \u112
A\u100
ð D
\ufffd
1 1 \ufffd
1 1 1
C
B
1 1
1 1 1 \ufffd
\u112 E \u112
S= A.B.C + C.ð.E + \u100.\ufffd.ð.\u112 + \u100.\ufffd.D.E + \u100.B.\ufffd.ð + \u100.B.D.\u112 + A.C.D.\u112
115
Exercício
ð D
\ufffd
0 0 0 1 \ufffd
0 1 1 1
C
B
0 1 1 0
1 0 0 0 \ufffd
\u112 E \u112
A\u100
ð D
\ufffd
0 0 0 1 \ufffd
0 1 1 1
C
B
0 1 1 0
1 0 0 0 \ufffd
\u112 E \u112
116
Solução
ð D
\ufffd
0 0 0 1 \ufffd
0 1 1 1
C
B
0 1 1 0
1 0 0 0 \ufffd
\u112 E \u112
A\u100
ð D
\ufffd
0 0 0 1 \ufffd
0 1 1 1
C
B
0 1 1 0
1 0 0 0 \ufffd
\u112 E \u112
S= C.E + \ufffd.D.\u112 + B.\ufffd.ð.\u112
117
Casos Sem Simplificação
\ufffd Seja a expressão
\ufffd S = \u100.B + A.B
\ufffd Ao tentar simplificar a expressão 
pelo diagrama de Veitch-Karnaugh, 
nota-se que não é possível agrupar 
termos
\ufffd Nesse caso, a expressão dada já 
se encontra minimizada
\ufffd O mesmo ocorre com a 
expressão
\ufffd S = A.B + \u100.\ufffd
\ufffd Que também se encontra 
minimizada
\ufffd B
\u100 0 1
A 1 0
\ufffd B
\u100 1 0
A 0 1
118
Casos Sem Simplificação
\ufffd O mesmo ocorre nas duas situações seguintes, 
que também não admitem simplificação
\ufffd Estes casos também ocorrem para 4 ou mais 
variáveis de entrada
\ufffd B
\u100 1 0 1 0
A 0 1 0 1
\ufffd C \ufffd
\ufffd B
\u100 0 1 0 1
A 1 0 1 0
\ufffd C \ufffd
119
Outra Maneira de Utilização
\ufffdOutra maneira de utilizar um diagrama 
Veitch-Karnaugh consiste em utilizar o 
complemento da expressão
\ufffdAssim, somente são considerados os casos 
onde a expressão S=0
\ufffd Com isso, têm-se o complemento da função, 
que precisa, portanto, ser invertida
\ufffd Isso corresponde a utilizar De Morgan
120
Diagrama de Veitch-Karnaugh pelo 
Complemento
\ufffd Usando o diagrama pelo 
método convencional, obtém-se
\ufffd S = A + C
Situação A B C S
0 0 0 0 0
1 0 0 1 1
2 0 1 0 0
3 0 1 1 1
4 1 0 0 1
5 1 0 1 1
6 1 1 0 1
7 1 1 1 1
\ufffd B
\u100 0 1 1 0
A 1 1 1 1
\ufffd C \ufffd
121
Diagrama de Veitch-Karnaugh pelo 
Complemento
\ufffd Usando o diagrama pelo 
método convencional, obtém-se
\ufffd S = A + C
\ufffd Usando o complemento, tem-se
\ufffd S = \u100.\ufffd
\ufffd Portanto,
\ufffd S = (\u100.\ufffd)\u2019
\ufffd Aplicando-se De Morgan na 
expressão acima, tem-se
\ufffd S = (\u100.\ufffd)\u2019 = A + C 
Situação A B C S
0 0 0 0 0
1 0 0 1 1
2 0 1 0 0
3 0 1 1 1
4 1 0 0 1
5 1 0 1 1
6 1 1 0 1
7 1 1 1 1
\ufffd B
\u100 0 1 1 0
A 1 1 1 1
\ufffd C \ufffd
122
Resumo
\ufffd Neste apresentação foram vistos os postulados e 
propriedades da álgebra de Boole
\ufffd