Chapter 3, Problem 23E

a.

Explanation of Solution

Truth table for the circuit with at least 7 input combinations:

• With “4” bits, one can produce “${\text{2}}^4$”, which means “16” unique input combinations.
• The different “16” input combinations are as follows,

A	B	C	D
0	0	0	0
0	0	0	1
0	0	1	0
0	0	1	1
0	1	0	0
0	1	0	1
0	1	1	0
0	1	1	1
1	0	0	0
1	0	0	1
1	0	1	0
1	0	1	1
1	1	0	0
1	1	0	1
1	1	1	0
1	1	1	1

• Let’s assume, AND gate is used to implement the required condition.
• AND operation produces the output “1” only if both the inputs are “1”. Otherwise, it produces the output “0”.
• Hence, for the above input combinations, AND operation produces “1” only when the inputs “A”, “b”, “C”, and “D” has the value “1”.
• Therefore, if AND gate is used, the required condition cannot be achieved.
• Let’s assume, Exclusive OR (OR) gate is used to implement the required condition.
• OR operation produces the output “1” if one or more inputs are “1” and it produces the output “0” when both the inputs are “0”.
• Hence, for all the above input combinations, OR operation produces the output “1” except the input combination “0000”

b.

Explanation of Solution

Gate-level logic circuit:

• The function of Exclusive OR gate (XOR) is obtained by joining the standard logic gates together to make more complex gate functions used for constructing arithmetic logic circuits.
• The truth table in terms of the inputs “A”, “B”, “C”, “D” and the output “Z” is as follows,

Inputs				Output
A	B	C	D	Z
0	0	0	0	0
0	0	0	1	1
0	0	1	0	1
0	0	1	1	0
0	1	0	0	1
0	1	0	1	0
0	1	1	0	0
0	1	1	1	1
1	0	0	0	1
1	0	0	1	0
1	0	1	0	0
1	0	1	1	1
1	1	0	0	0
1	1	0	1	1
1	1	1	0	1
1	1	1	1	0

• The gate-level logic circuit for the above truth table is implemented with the combinations of AND gates and OR gate.
• The number of OR gates and AND gates utilized for gate-level logic circuit is based on the truth table.
• The required number of AND gate is the possible number of input combinations.
• Few number of AND gates are reduced, when the input combination produces the output “0”.
• The required number of OR gate is the number of output columns given in the truth table.
• To draw the gate-level logic circuit for the given truth table, “$2^4$” which means “16” number of AND gates are used. But, the number of AND gates going to be used are reduced to “8” because, “8” number of input combinations in the truth table, produces the output “0”