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Transcript of mapas karnaught
![Page 1: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/1.jpg)
Maurice Karnaugh
Ingeniero de Telecomunicaciones
• AT&T Bell.
• 1953 Inventa el mapa-K o mapa de Karnaugh.
• Minimización de funciones por inspección visual.
Minimización de Funciones Booleanas
Mapas de Karnaugh
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Tabla o mapa de Karnaugh, Kmap
Procedimiento gráfico para la simplificación de funciones
algebraicas de un número de variables relativamente
pequeño
(en la práctica se puede utilizar para funciones de hasta seis variables).
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Tabla o mapa de Karnaugh
Un diagrama o
mapa de Karnaugh
es una tabla de
verdad dispuesta de
manera adecuada
para determinar por
inspección la
expresión mínima
de suma de
productos de una
función lógica.
![Page 5: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/5.jpg)
La factorización se efectúa cuando solo cambia una variable entre dos términos y esta variable se elimina
Con 2 variables A y B se pueden tener 4 Términos
Cada termino de dos variables tiene dos posibilidades de factorización
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Kmap para 2 variables
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Mapa de Karnaugh para dos variables
A’B’ AB’
A’B AB
m0 m2
m1 m3
0 2
1 3
0 1
0
1
AB
A
B
m A B S
0 0 0
1 0 1
2 1 0 AB’
3 1 1
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Kmap para 2 variables
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Kmap para 2 variables
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Como llenar el Kmap para 2 variables
1
0
1
1
F1 (A,B) = A’ B’ + A B’ + A B
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Como resolver Kmap para 2 variables
F1(A,B)= A
1
+ B’
0
F1 (A,B) = A’ B’ + A B’ + A B
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Kmap para 3 variables
Con 3 Variables se tienen 8 términos
y cada termino tiene 3 posibilidades
de factorización
![Page 13: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/13.jpg)
Kmap para 3 variables
Cada termino tiene 3 posibilidades de factorización
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Kmap para 3 variables
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Mapa de Karnaugh para 3 variables
A’B’C’ A’BC’ ABC’ AB’C’
A’B’C A’BC ABC AB’C
00 01 11 10
0
1
ABC
0 2 6 4
1 3 7 5
00 01 11 10
0
1
ABC
La idea con la codificación es poder usar el P9a. ab+ab’=a
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Mapa de Karnaugh para 3 variables
A’B’C’ A’BC’ ABC’ AB’C’
A’B’C A’BC ABC AB’C
00 01 11 10
0
1
ABC
0 2 6 4
1 3 7 5
00 01 11 10
0
1
ABC
La idea con la codificación es poder usar el P9a. ab+ab’=a
![Page 18: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/18.jpg)
Kmap para 3 variables
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Kmap para 3 variables
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
Kmap para 3 variables
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
A´ B C’
1
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
A B´ C´
1 1
Kmap para 3 variables
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
A B C´
1 11
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
A B C
1 11
1
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B
11
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
11
0
F (A, B, C) = B C’ +
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ +
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F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ +
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ +
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ +
1 1
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ + A
1 1
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ + A
0
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ + A C’
0
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ + A C´ +
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ + A C´ +
11
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Kmap para 3 variables
F (A, B, C) = A´ B C’ + A B´ C´ + A B C’ + A B C
1 11
1
F (A, B, C) = B C’ + A C´ + A B
11
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Kmap para 4 variables
Con 4 Variables se tienen 16 términos
y cada termino tiene 4 posibilidades
de factorización
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Kmap para 4 variables
Cada termino tiene 4 posibilidades de factorización
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Cada termino tiene 4 posibilidades de factorización
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K map para 4 variables
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AB00 01 11 1010
K map para 4 variables
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Kmap para 4 variables
AB
CD
00 01 11
00
01
11
1010
10
10
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Kmap para 4 variables
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Mapa de Karnaugh para 4 variables
A’B’C’D’ A’BC’D’ ABC’D’ AB’C’D’
A’B’C’D A’BC’D ABC’D AB’C’D
A’B’CD A’BCD ABCD AB’CD
A’B’CD’ A’BCD’ ABCD’ AB’CD’
00 01 11 10
00
01
11
10
ABCD
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
00 01 11 10
00
01
11
10
ABCD
![Page 45: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/45.jpg)
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Kmap para 4 variables
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Kmap para 5 variables
Con 5 Variables se tienen 32 términos
y cada termino tiene 5 posibilidades
de factorización
![Page 48: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/48.jpg)
Kmap para 5variables
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Kmap para 5variables
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Kmap para 5 variables
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C DA B
E F
0 0
0 0
0 1
0 1
11
11
1 0
1 0
01 6 3 2
4 8
42 0 3 6
5 2
52 1 3 7
5 3
62 2 3 8
5 4
72 3 3 9
5 5
82 4 4 0
5 6
92 5 4 1
5 7
1 02 6 4 2
5 8
112 7 4 3
5 9
1 22 8 4 4
6 0
1 32 9 4 5
6 1
1 43 0 4 6
6 2
1 53 1 4 7
6 3
11 7 3 3
4 9
21 8 3 4
5 0
31 9 3 5
5 1
0 00 1 1 0
1 1
Kmap para 6 variables
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Reglas para el uso del Kmap
1.- Formar el menor numero de grupos
2.- Cada grupo lo mas grande posible
3.- Todos los unos deberán de ser agrupados
4.- Un solo uno puede formar un grupo
5.- Casillas de un grupo pueden formar parte de otro grupo
Grupo = Unos adyacentes enlazados (paralelogramos) en una cantidad igual a una potencia entera de dos, eje. (1, 2, 4, 8,
…).
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Ejemplos del Kmap
m X Y F0 0 0 11 0 1 12 1 0 03 1 1 1
F 0
1F (X, Y)= X’ + Y
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ejemplos del Kmap
F2(X, Y, Z) =m(1, 2, 5, 7)
1
1 1 1
00
0
0
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F2(X, Y, Z) =m(1, 2, 5, 7)
1
1 1 1
00
0
0
F2(X, Y, Z) = X Z
1 1
1
+
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+ Y’
F2(X, Y, Z) =m(1, 2, 5, 7)
1
1 1 1
00
0
0
F2(X, Y, Z) = X Z Z
00
1
+
![Page 57: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/57.jpg)
Y’
F2(X, Y, Z) =m(1, 2, 5, 7)
1
1 1 1
00
0
0
F2(X, Y, Z) = X Z Z + X’ Y Z’
01
0
+
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B
01
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B
0
0
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A
01
C’D
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A
01
C’D
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A C’D+
0
A’
0
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A C’D+ A’
11
CD
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1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A C’D+ A’CD+
11
AB
![Page 67: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/67.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
FX(A, B, C, D) =m(3, 4, 5, 7, 9, 13, 14, 15)
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A C’D+ A’CD+AB
1
C
1
![Page 68: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/68.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F X0
1
1 0
1
1
1
1
11
1
1
000
0
0
0 0 0
FX(A, B, C, D) = A’ B C’ + A C’D+ A’CD+ABC
1.- Formar el menor número de grupos
2.- Cada grupo lo más grande posible
![Page 69: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/69.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
0
0
0
0
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
0
0
![Page 70: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/70.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
1.- Formar el menor número de grupos
2.- Cada grupo lo más grande posible
![Page 71: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/71.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) =
![Page 72: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/72.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) =
00
B’
![Page 73: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/73.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) = B’
10
CD’
![Page 74: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/74.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) = B’ C D’ +
1 1
B
![Page 75: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/75.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) = B’ C D’ + B
1
1
D
![Page 76: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/76.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) = B’ C D’ + B D +
0 0
A’
![Page 77: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/77.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) = B’ C D’ + B D + A’
0
0
D’
![Page 78: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/78.jpg)
1 24 8
3 1 57 1 1
1 35 9
2 1 46
0 0
0 0
0 1
0 1
1 1
1 1
1 0
1 0
A B
C D
F Y0
1
1 0
0
1
1
1
01
0
1
001
0
0
1 1 1
FY(A, B, C, D) = m (1, 3, 8, 9, 11, 12, 14)
FY(A, B, C, D) = B’ C D’ + B D + A’ D’
![Page 79: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/79.jpg)
F3(A, B, C, D) =m(0,2,5,6,7,8,12,14)
![Page 80: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/80.jpg)
F3(A, B, C, D) =m(0,2,5,6,7,8,12,14)
F3= A'B'D' + A C'D' + A'B D + B C D‘F3= B'C'D' + A'C D' + A'B D + A B D'
![Page 81: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/81.jpg)
F4(A, B, C) =m(2, 7)
0
0
1 1 1
1 1 1
![Page 82: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/82.jpg)
Reglas para el uso del Kmap1.- Formar el menor numero de grupos.
2.- Cada grupo lo mas grande posible.
3.- Todos los unos deberán de ser agrupados.
4.- Un solo uno puede formar un grupo.
5.- Casillas de un grupo pueden formar parte de otro grupo.
Grupo = Unos adyacentes enlazados (paralelogramos) en una cantidad igual a una potencia entera de dos ejemplo (1, 2, 4,
8,…).
![Page 83: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/83.jpg)
F5(X, Y, Z, W) =m(0,2,7,8,10,12,13,14)F6(A, B, C, D) =m(0,15)F7(A, B, C, D) =m(9, 11,15)F8(X, Y, Z, W) =m(0,2,3,5,6,7,8,10,11,14,15)F9 ( A,B,C,D )= m ( 2, 5, 7, 13, 15)F10 ( X,Y,Z,W )= m ( 5, 13, 15)F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’ F12 ( X,Y,Z,W )= m ( 4,7,9,10,12,13,14,15) F13 ( X,Y,Z,W )= m ( 1, 3, 6, 7, 9, 11, 12) F14 (A,B,C,D) = m ( 3,5,6,7, 9,10,11,12,13,14)F15 (A,B,C,D) =(B’+C+D)(B’+C’+D)(A’+B’+C’+D’)(A’+B +C+D’) F16 (A,B,C,D) = m ( 0, 2, 4, 5, 6, 7, 8, 9, 10, 13, 15)F17 (A,B,C,D) = m ( 0, 1, 2, 3, 5, 8, 9, 10, 13, 14, 15)
La mejor forma de Huir de un problema es resolverlo.
![Page 84: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/84.jpg)
F5(X, Y, Z, W) =m(0,2,7,8,10,12,13,14)
![Page 85: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/85.jpg)
F6(A, B, C, D) =m(5,15)
![Page 86: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/86.jpg)
F6(A, B, C, D) =m(5,15)
![Page 87: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/87.jpg)
F7(A, B, C, D) =m(9, 11,15)
![Page 88: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/88.jpg)
F7(A, B, C, D) =m(5, 7,15)Agrupando ceros POS
F7(A, B, C, D)=(B'+C'+D')(A+B'+D') (POS)
![Page 89: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/89.jpg)
+Y’ W’+Z
F8(X, Y, Z, W) =m(0,2,3,5,6,7,8,10,11,14,15)
1
1
1
1
0
1
1
0
1
1
0
0
1
1
0
1
F8(X, Y, Z, W)=X’YW
![Page 90: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/90.jpg)
F8(X, Y, Z, W) =m(0,2,3,5,6,7,8,10,11,14,15)
1
1
110
1101
1
00
110
1
![Page 91: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/91.jpg)
F8(X, Y, Z, W) =m(0,2,3,5,6,7,8,10,11,14,15)
1
1
110
1101
1
00
110
1
![Page 92: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/92.jpg)
F9 (A,B,C,D )= m ( 2, 5, 7, 13, 15)
![Page 93: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/93.jpg)
F9 (A,B,C,D )= m ( 2, 5, 7, 13, 15)
0
1
001
1111
1
01
111
0
F9 = B D' + B'D + A D' + C'D' F9 = B D' + B'D + A D' + B'C' F9 = B D' + B'D + A B' + C'D' F9 = B D' + B'D + A B' + B'C' ***********************************F9 = (B'+ D') (A + B + C'+ D )
![Page 94: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/94.jpg)
F9 (A,B,C,D )= m ( 2, 5, 7, 13, 15)
0
1
001
1111
1
01
111
0
F9 = B D' + B'D + A D' + C'D' F9 = B D' + B'D + A D' + B'C' F9 = B D' + B'D + A B' + C'D' F9 = B D' + B'D + A B' + B'C' ***********************************F9 = (B'+ D') (A + B + C'+ D )
![Page 95: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/95.jpg)
F10 ( X,Y,Z,W )= m ( 4,7,9,10,12,13,14,15)
![Page 96: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/96.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
X Y’1
1
1
1
![Page 97: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/97.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
1
1
1
1
X Y W’1
1
![Page 98: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/98.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
1
1
1
1
X’ Y’ W1
1
1
1
![Page 99: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/99.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
1
1
1
1
X’ Y’ Z’ W’
1
1
1
1
1
![Page 100: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/100.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
1
1
1
11
1
1
1
1
![Page 101: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/101.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
1
1
1
11
1
1
1
1 0
0
0
0
0
0
0
![Page 102: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/102.jpg)
F11 (X, Y, Z, W ) = X Y’ + X Y W’ + X’ Y’ W + X’ Y’ Z’ W’
F11X, Y
Z, W
1
1
1
1
1
1
1
1
1
![Page 103: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/103.jpg)
F12 ( X,Y,Z,W )= m ( 1, 3, 6, 7, 9, 11, 12)
F12X,Y
Z,W1
1
1
1
1
1
1
![Page 104: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/104.jpg)
F12 ( X,Y,Z,W )= m ( 1, 3, 6, 7, 9, 11, 12)
F12X,Y
Z,W1
1
1
1
1
1
10 0
0
0
0
0
0
0 0
![Page 105: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/105.jpg)
F13 (A,B,C,D) = m (3,5,6,7, 9,10,11,12,13,14)
F13
![Page 106: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/106.jpg)
M A B C D P0 0 0 0 0 11 0 0 0 1 12 0 0 1 0 13 0 0 1 1 04 0 1 0 0 15 0 1 0 1 06 0 1 1 0 07 0 1 1 1 08 1 0 0 0 19 1 0 0 1 010 1 0 1 0 011 1 0 1 1 012 1 1 0 0 013 1 1 0 1 014 1 1 1 0 015 1 1 1 1 0
![Page 107: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/107.jpg)
M A B C D S0 0 0 0 0 11 0 0 0 1 12 0 0 1 0 13 0 0 1 1 14 0 1 0 0 15 0 1 0 1 06 0 1 1 0 07 0 1 1 1 08 1 0 0 0 19 1 0 0 1 110 1 0 1 0 011 1 0 1 1 012 1 1 0 0 113 1 1 0 1 014 1 1 1 0 015 1 1 1 1 0
![Page 108: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/108.jpg)
F14 ( A, B , C ,D)= Σm(4, 8, 9, 10, 11, 12, 14, 15)
![Page 109: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/109.jpg)
F15 (A,B,C,D) =(B’+C+D)(B’+C’+D)(A’+B’+C’+D’)(A’+B +C+D’)
![Page 110: mapas karnaught](https://reader031.fdocument.pub/reader031/viewer/2022011721/587ec2d81a28abf37b8b5693/html5/thumbnails/110.jpg)
F15 (A,B,C,D) =(B’+C+D)(B’+C’+D)(A’+B’+C’+D’)(A’+B +C+D’)
0
0
0
0
0
0
1
1
1
1
1
1
1
1 1
1