ic
Section 8.2 Logic Networks
661
rightmost 0 bit of p stays a 0 bit in q, and the first 1 bit stays a 1 bit. The remaining bits of q, however, are
the reverse of the bits in p.
p = 1 0 0
q = 0 1 1
()*
q i = p i
′
1 0
1 0
()*
q i = p i
For each binary number p, find the 2’s complement of p, namely q, and then calculate p + q.
a. 1100
b. 1001
c. 001
28. For any bit x i in a binary number p, let r i be the corresponding bit in q, the 2’s complement of p (see
Exercise 27). The value of r i depends on the value of x i and also on the position of x i relative to the first
1 bit in p. For the ith bit, let c i−1 denote a 0 if the bits p j , 1 ≤ j ≤ i − 1, are 0 and a 1 otherwise. A value
c i must be computed to move on to the next bit.
a. Give a truth function for r i with inputs x i and c i−1 . Give a truth function for c i with inputs x i and c i−1 .
b. Write Boolean expressions for the truth functions of part (a). Simplify as much as possible.
c. Design a circuit module to output r i and c i from inputs x i and c i−1 .
d. Using the modules of part (c), design a circuit to find the 2’s complement of a three-bit binary number
zyx. Trace the operation of the circuit in computing the 2’s complement of 110.
29. a. Construct a network for the following expression using only NAND elements. Replace the AND and
OR gates and inverters with the appropriate NAND networks.
x 3 ′x 1 + x 2 ′x 1 + x 3 ′
b. Use the properties of a Boolean algebra to reduce the expression of part (a) to one whose network
would require only three NAND gates. Draw the network.
30. Replace the following network with an equivalent network using one AND gate, one OR gate, and one inverter.
x 2
x 1
x 3
31. Using only NOR elements, construct networks that can replace (a) an inverter, (b) an OR gate, and (c) an
AND gate.
32. Find an equivalent network for the half-adder module that uses exactly five NAND gates. Draw the network.
33. A thermostat controls a heating system that should raise the temperature above 67 °F during working
hours (between 7:00 am and 6:00 pm). The input values are
x 1 = 1 when the temperature is less than 67 °F
x 1 = 0 otherwise
x 2 = 1 when the time is less than 7:00 am or greater than 6:00 pm
x 2 = 0 otherwise
Find a truth function, a Boolean expression, and a logic network for when the heating system should be
on (value 1) or off (value 0).
First 1
Section 8.2 Logic Networks
661
rightmost 0 bit of p stays a 0 bit in q, and the first 1 bit stays a 1 bit. The remaining bits of q, however, are
the reverse of the bits in p.
p = 1 0 0
q = 0 1 1
()*
q i = p i
′
1 0
1 0
()*
q i = p i
For each binary number p, find the 2’s complement of p, namely q, and then calculate p + q.
a. 1100
b. 1001
c. 001
28. For any bit x i in a binary number p, let r i be the corresponding bit in q, the 2’s complement of p (see
Exercise 27). The value of r i depends on the value of x i and also on the position of x i relative to the first
1 bit in p. For the ith bit, let c i−1 denote a 0 if the bits p j , 1 ≤ j ≤ i − 1, are 0 and a 1 otherwise. A value
c i must be computed to move on to the next bit.
a. Give a truth function for r i with inputs x i and c i−1 . Give a truth function for c i with inputs x i and c i−1 .
b. Write Boolean expressions for the truth functions of part (a). Simplify as much as possible.
c. Design a circuit module to output r i and c i from inputs x i and c i−1 .
d. Using the modules of part (c), design a circuit to find the 2’s complement of a three-bit binary number
zyx. Trace the operation of the circuit in computing the 2’s complement of 110.
29. a. Construct a network for the following expression using only NAND elements. Replace the AND and
OR gates and inverters with the appropriate NAND networks.
x 3 ′x 1 + x 2 ′x 1 + x 3 ′
b. Use the properties of a Boolean algebra to reduce the expression of part (a) to one whose network
would require only three NAND gates. Draw the network.
30. Replace the following network with an equivalent network using one AND gate, one OR gate, and one inverter.
x 2
x 1
x 3
31. Using only NOR elements, construct networks that can replace (a) an inverter, (b) an OR gate, and (c) an
AND gate.
32. Find an equivalent network for the half-adder module that uses exactly five NAND gates. Draw the network.
33. A thermostat controls a heating system that should raise the temperature above 67 °F during working
hours (between 7:00 am and 6:00 pm). The input values are
x 1 = 1 when the temperature is less than 67 °F
x 1 = 0 otherwise
x 2 = 1 when the time is less than 7:00 am or greater than 6:00 pm
x 2 = 0 otherwise
Find a truth function, a Boolean expression, and a logic network for when the heating system should be
on (value 1) or off (value 0).
First 1