Half-Adders and Full Adders

Overview

Recall questions on the half-adder's circuit and truth table, its limitation, building a full adder from two half-adders, tracing both for given inputs, and chaining full adders to add eight-bit numbers.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   Which single gate performs binary addition of two bits in the sum column, and why?

   Model answer

   XOR. Adding two single bits gives 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 and 1 + 1 = 10, whose sum bit is 0 with a carry of 1. The sum column, 0, 1, 1, 0, is exactly the XOR truth table.

2. 2

   Describe the circuit of a half-adder so that you could construct it.

   Model answer

   Two inputs A and B, each of which branches so that it feeds two gates. A and B go into an XOR gate whose output is the Sum. A and B also go into an AND gate whose output is the Carry. That is the whole circuit: two inputs, two gates side by side, two outputs, with a dot on each input line where it branches.

3. 3

   Give the truth table of a half-adder.

   Model answer

   Inputs A and B, outputs Sum and Carry. A = 0, B = 0 gives Sum 0, Carry 0. A = 0, B = 1 gives Sum 1, Carry 0. A = 1, B = 0 gives Sum 1, Carry 0. A = 1, B = 1 gives Sum 0, Carry 1. The Sum column is A XOR B and the Carry column is A AND B.

4. 4

   Trace the half-adder for A = 1, B = 1.

   Model answer

   The XOR gate receives 1 and 1, which are the same, so Sum = 0. The AND gate receives 1 and 1, so Carry = 1. Read together the outputs are Carry 1, Sum 0, which is binary 10, that is 2, the correct answer to 1 + 1.

5. 5

   What is the limitation of a half-adder, and what fixes it?

   Model answer

   A half-adder adds two bits only. It produces a carry out but has nowhere to accept a carry in from a less significant column, so it cannot be used for any column but the rightmost when adding multi-bit numbers. A full adder fixes this by taking a third input, the carry in.

6. 6

   Describe the circuit of a full adder built from two half-adders so that you could draw it.

   Model answer

   Three inputs A, B and Cin. A and B go into the first half-adder, that is into an XOR gate and an AND gate in parallel. Call the XOR output S1 and the AND output C1. S1 and Cin then go into the second half-adder, that is into a second XOR gate and a second AND gate in parallel. The output of the second XOR gate is the final Sum. Call the second AND output C2. Finally C1 and C2 go into an OR gate, whose output is the final Carry out. So the circuit is two XOR gates, two AND gates and one OR gate.

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