Chapter 4, Problem 66E

Explanation of Solution

Full adder:

• The full adder is an electronic circuit used to perform the addition of three binary digits.
• It takes the inputs of two binary digits and another input is the carry-in value.
• For computation, the half adder does not take the carry value. But the full adder takes the carry-in input value into calculation.
• It produces the output as sum and carry-out.
• To make a full adder, use the two half adders.
• Logic diagram for the full adder takes three inputs A, B, and Carry-in and produces the outputs Sum and Carry-out.

• From the circuit diagram:
  • Boolean expression for Sum:
    • First, the inputs A and B are passed to XOR gate to produce the output as $A\oplus B$.
    • Next, the output of first XOR gate “$\text{A}\oplus \text{B}$” and “Carry-in” as the input for second XOR gate and produce the output as $Sum=\left(\left(A\oplus B\right)\oplus Carry-in\right)$.
  • Boolean expression for Carry-out:
    • Next, pass the same inputs of A and B from the first XOR gate and it is passed to AND gate to produce the output as $A\cdot B$.
    • Next, pass the same output of first XOR gate “$\text{A}\oplus \text{B}$” and “Carry-in” as the input of second AND gate to produce the output as $\left(\left(A\oplus B\right)\cdot Carry-in\right)$