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CMPEN 471 Homework 1 Solution Fall 2015
5.3 (d) f(V,W,X,Y,Z) = Σm(0,2,3,4,5,11,18,19,20,23,24,28,29,31)
f(V,W,X,Y,Z) = W'X'Y + V'X'YZ + V'W'XY' + VWY'Z' + VXYZ + VWXZ + V'W'X'Z' + W'XY'Z' = W'X'Y + V'X'YZ + V'W'XY' + VWY'Z' + VXYZ + VWXZ + V'W'X'Z' + VXY'Z' = W'X'Y + V'X'YZ + V'W'XY' + VWY'Z' + VXYZ + VWXZ + VW'Y'Z' + W'XY'Z' = W'X'Y + V'X'YZ + V'W'XY' + VWY'Z' + VXYZ + VWXZ + VW'Y'Z' + VXY'Z' Total 4 solutions exist, any one is correct.
0 0 1 1
0 1 0 0
1 0 0 1
1 0 0
X
0
Z
Y
W f
1 1 1 0
0 0 1 0
1 1 1 0
1 0 0
X
0
Z
Y
W
V = 0 V = 1
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8 16
17
19
18
20
21
23
22
24
25
27
26
28
29
31
30
5.4
Wrong: z(y1,y0,x1,x0) = y1'y0' + y1'x0 + y1'x1 + y0'x1 + x1x0
Correct: z(y1,y0,x1,x0) = (y1' + x1)( y0' + x1 + x0)( y1' + y0' + x0)
1 0000
z
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
1
1
1
y1y0x1x0
0
1
1
1
0
0
1
1
0
0
0
1
0 0 0 1
1 1 0 0
1 1 1 1
1 1 0
y0
1
x0
x1
y1 z
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
5.10 (c) f(A,B,C,D,E) = Σm(1,2,3,4,5,11,18,19,20,21,23,28,31) + d(0,12,15,27,30)
Two solutions: f(A,B,C,D,E) = CD'E' + B'CD' + B'C'D + BDE + ADE + A'B'C' f(A,B,C,D,E) = CD'E' + B'CD' + B'C'D + BDE + ADE + A'B'D'
0 x 1 x
1 1 0 0
1 0 x 1
1 0 0
C
0
E
D
B f
0 1 1 0
0 1 0 0
1 1 1 x
1 0 x
C
0
E
D
B
A = 0 A = 1
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8 16
17
19
18
20
21
23
22
24
25
27
26
28
29
31
30
6.1 (a)
Prime implicants: x’y’ = xx00x = 0,1,8,9,16,17,24,25 w’x’z = 0x0x1 = 1,3,9,11 v’x’z = x00x1 = 1,3,17,19 w’vyz = 01x11 = 11,15 wvy’z = 11x01 = 25,29 vxyz = x1111 = 15,31 wvxz = 111x1 = 29,31 wvxy = 1111x = 30,31
0 00000
z
00001
00011
01000
01001
01011
01111
10000
10001
10011
11000
11001
11101
11110
11111
1
2
1
binary
2
3
4
1
2
3
2
3
4
4
5
m
9
11
0
1
3
8
15
16
17
19
24
25
29
30
31
c 00000
ck
01000
10000
00011
01001
10001
11000
01011
10011
11001
01111
11101
11110
11111
c
c
c
0-cubes
cubes
c
c
c
c
c
c
c
c
c
c
m
9
17
0
8
16
3
24
11
19
25
15
29
30
31
c 00001 1
c 0000x
ck
x0000
000x1
0x001
x0001
0100x
x1000
1000x
1x000
0x011
x0011
010x1
x1001
100x1
c
c
c
1-cubes
c
c
c
c
c
c
c
c
c
c
m
1,17
8,9
0,1
0,16
1,3
1,9
8,24
16,17
16,24
3,11
3,19
9,11
9,25
17,19
c 0x000 0,8
1x001
1100x
c
c
17,25
24,25
01x11
11x01
x1111
11,15
25,29
15,31
111x1
1111x
29,31
30,31
0,1,16,17
c 0x00x
ck
xx000
0x0x1
x00x1
xx001
x100x
1x00x
c
2-cubes
c
c
c
m
1,9,17,25
8,9,24,25
0,1,8,9
0,8,16,24
1,3,9,11
1,3,17,19
16,17,24,25
c x000x
16,17,24,25 xx00x
ck 3-cubes m
0,1,8,9,
6.1 (b)
6.9
6.9 continue
6.13 (c)
0 1 1 0
0 1 0 0
0 0 0 1
0 0 0
B
1
D
C
A f1
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
1 0 1 1
1 0 0 0
1 0 0 1
0 0 0
B
0
D
C
A f2
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
0 1 1 1
0 0 0 0
0 0 0 0
0 0 1
B
1
D
C
A f3
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
0 0 1 0
0 0 0 0
0 0 0 0
0 0 0
B
0
D
C
A f123
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
0 0 0 0
0 0 0 0
0 0 0 1
0 0 0
B
0
D
C
A f12
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
0 1 1 0
0 0 0 0
0 0 0 0
0 0 0
B
1
D
C
A f13
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
Prime implicants: A'BC'D' AB'CD BC'D' AB'CD' A'C'D' A'BC' AB'C B'C'D' B'CD A'B'C' A'B'D ABD' ACD'
0 0 1 1
0 0 0 0
0 0 0 0
0 0 0
B
0
D
C
A f23
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
1 0 1 1
1 0 0 0
1 0 0 1
0 0 0
B
0
D
C
A f2
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
0 1 1 1
0 0 0 0
0 0 0 0
0 0 1
B
1
D
C
A f3
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
0 1 1 0
0 1 0 0
0 0 0 1
0 0 0
B
1
D
C
A f1
0
1
3
2 6
7
5
4
14
15
13
12
10
11
9
8
6.18 (a)
6.18 (a) continue
6.18 (a) continue
