ECE 349
Homework Assignment #2
Solutions
1. Find an expression for f = a0 b0 c0 + abc0 + a0 bc + ab0 c that uses only the exclusive-OR
and complement operations.
f = a(bc0 + b0 c) + a0 (b0 c0 + bc) = a(b ⊕ c) + a0 (b ⊕ c)0
f = a0 ⊕ b ⊕ c
2. The following table shows the input and output voltages for a 2-input logic gate circuit.
Translate the table to a truth table using positive logic and then negative logic. What
are the logic functions realized in each case?
e1 e2 e0
0
0
0
0 +5 +5
+5 0 +5
+5 +5 +5
e1
0
0
1
1
e2
0
1
0
1
e0
0
1
1
1
e1
1
1
0
0
OR
e2
1
0
1
0
e0
1
0
0
0
AND
3. Express each of the following as minterm expansions.
(a) f (a, b, c) = a + b0 c = a0 b0 c + ab0 c0 + ab0 c + abc0 + abc
Q
(b) f (w, x, y) = M (0, 4, 5, 7) = x0 y 0 z + x0 yz 0 + x0 yz + xyz 0
4. Express each of the following as maxterm expansions.
(a) f (a, b, c) = (a + b0 )(a0 + b + c)(b + c0 )
= (a + b0 + c0 )(a + b0 + c)(a0 + b + c)(a0 + b + c0 )(a + b + c0 )
P
(b) f (x, y, z) = m(1, 3, 5, 6) = (x + y + z)(x + y 0 + z)(x0 + y + z)(x0 + y 0 + z 0 )
5. For f (a, b, c) = (ab + c)(a0 + b0 ), find the minterm and maxterm expansions (use m and
M notation).
P
Q
f (a, b, c) = m(1, 3, 5) = M (0, 2, 4, 6, 7)
6. For the table below, express f as a minterm expansion (use m notation) and as a
maxterm expansion (use M notation).
a
0
0
0
0
1
1
1
1
f=
b
0
0
1
1
0
0
1
1
P
c f
0 0
1 1
0 X
1 1
0 0
1 X
0 1
1 0
m(1, 3, 6) +
P
d(2, 5) =
Q
M (0, 4, 7) ·
Q
D(2, 5)
7. For the truth table of the previous problem, find the simplest expression for f .
f = bc0 + a0 c
8. A bank vault has three locks with a different key for each lock. Each key is owned by a
different person. In order to open the door, at least two people must insert their keys
into the assigned locks. The signals A, B, and C are 1 if there is a key inserted into
lock 1, 2, or 3, respectively. Write an equation for the variable Z, which is 1 iff the
door should open.
Z = AB + AC + BC
9. A combinational switching network has four inputs (A, B, C, D) and one output Z.
The output is 1 iff three or more of the inputs are 1. Design the network using two
AND gates and three OR gates. Assume that each gate has a maximum of three inputs
so that it will be necessary to partially factor your logic equation before you realize it.
Z = AB(C + D) + CD(A + B) = BD(A + C) + AC(B + D) = BC(A + D) + AD(B + C)
10. (Optional 2 Points) A combinational network has four inputs (a, b, c, d) and three outputs (x, y, z). The outputs represent a binary number whose value equals the number
of 1’s at the input. For example, if abcd = 1011, then xyz = 011. Find the maxterm
expansion for x, y, and z.
Q
X = M (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14)
Q
Y = M (0, 1, 2, 4, 8, 15)
Q
Z = M (0, 3, 5, 6, 9, 10, 12, 15)
11. Provide the truth table for the following circuit.
A
L
L
L
L
H
H
H
H
B
L
L
H
H
L
L
H
H
C
L
H
L
H
L
H
L
H
F
L
H
L
H
L
L
H
H
A
B
F
A
C