Problem Solutions – Chapter 3
1
CHAPTER 3
© 2016 Pearson Education, Inc.
3-1.
Place a 1 in each K-map cell where 2 or more inputs are equal to 1.
X
Y
Z
1 1
F = XZ + XY + YZ
1
This is the same function as the
carry f or the full adder.
1
Place a 1 in each K-map cell where 2 or more inputs are equal to 1.
F = XZ + XY + YZ
This is the same function as the carry
for the full adder.
3-2.*
A
B
C
D
11
1 1 1
1
F = AB + AC
3-3.
Assuming inputs G3, G2, G1, G0 and outputs B3, B2, B1, B0, with G3 and B3 being the most significant bits, and treating the invalid
input combinations as don’t cares:
33
BG
2 3 2
B G G
1 2 1 3 2 1
B G G G G G
0 3 0 2 1 0 2 1 0
2 1 0 3 2 1 0
B G G G G G G G G
G GG G G G G
3-4. a) For the 3 x 3 pattern, there are exactly three row, three column and two diagonal combinations that represent a win for the X
player: W = X1 X2 X3 + X4 X5 X6 + X7 X8 X9 + X1 X4 X7 + X2 X5 X8 + X3 X6 X9 + X1 X5 X9 + X3 X5 X7 Gate Input cost = 32
b) W = X5 (X1 X9 + X2 X8 + X3 X7 + X4 X6) + X1 X2 X3 + X1 X4 X7 + X7 X8 X9 + X3 X6 X9 Gate Input Cost = 30
3-5. a) For the 4 x 4 pattern, there are exactly four row, four column and two diagonal combinations that represent a win for the X
player: W = X1 X2 X3 X4 + X5 X6 X7 X8 + X9 X10 X11 X12 + X13 X14 X15 X16 + X1 X5 X9 X13 X2 X6 X10 X14 + X3 X7 X11
X15 + X4 X8 X12 X16 + X1 X6 X11 X16 + X4 X7 X10 X13 Gate Input cost = 50
b) W = X1(X2 X3 X4 + X5 X9 X13 + X6 X11 X15) + X7(X5 X6 X8 + X3 X11 X15 + X4 X10 X13) + X9 X10 X 11 X12
+ X13 X14 X15 X16 + X2 X6 X10 X14 + X4 X8 X12 X16 Gate Input Cost = 48
1
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