Worksheet

A8.7 Half adders and full adders

Boolean algebra and logic circuits · A level · OCR H446 1.4.3, AQA 7517 4.6.4.1 · about 20 min

What this lesson is about

The full adder's expressions, building it from half adders, and chaining full adders into a ripple carry adder.

• The half adder, again
• Why "half"?
• The full adder
• Two half adders and an OR gate
• Adding whole numbers: the ripple carry adder

Questions 5 marks in all

1. [1 mark]Which expressions give a half adder's outputs?

   1. AS = A ⊻ B, C = A ∧ B
   2. BS = A ∨ B, C = A ∧ B
   3. CS = A ∧ B, C = A ⊻ B
   4. DS = A ⊻ B, C = A ∨ B

2. [1 mark]Why is a half adder not enough to add multi-bit numbers?

   1. AIt has no input for the carry from the previous column
   2. BIt cannot add 1 and 1
   3. CIt has no sum output
   4. DIt only works on the leftmost bit

3. [1 mark]A full adder has A = 1, B = 0 and Cin = 1. Give Cout and S as two digits, Cout first.

4. [1 mark]How is a full adder built from half adders?

   1. ATwo half adders, with their two carry outputs joined by an OR gate
   2. BTwo half adders, with their carry outputs joined by an AND gate
   3. CThree half adders in a row
   4. DOne half adder and a NOT gate

5. [1 mark]What does this program print?

   def full_adder(a, b, cin):
       s1, c1 = a ^ b, a & b
       return s1 ^ cin, c1 | (s1 & cin)
   
   carry = 0
   bits = ""
   for a, b in [(1, 1), (1, 0), (0, 1), (1, 0)]:
       s, carry = full_adder(a, b, carry)
       bits = str(s) + bits
   print(bits, carry)

The task: a 4-bit ripple carry adder

Write three functions. Every bit is the integer 0 or 1. - half_adder(a, b) returns a tuple (sum, carry). - full_adder(a, b, carry_in) returns a tuple (sum, carry_out), and must be built from two calls to half_adder and an OR. - add4(x, y) takes two strings of four characters, each "0" or "1", with the most significant bit first. It adds them with a ripple of full_adder calls, starting from the rightmost bit with a carry in of 0, and returns a tuple (total, carry): total a 4-character string of the sum bits, most significant first, and carry the final carry out (0 or 1). Your program may not use int(), bin(), format() or sum(). For each pair ("0110", "0111"), ("1011", "0110"), ("1111", "0001") and ("0101", "1010"), in that order, print one line in exactly this form: 0110 + 0111 = 1101 carry 0

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

def half_adder(a, b):
    return 0, 0

def full_adder(a, b, carry_in):
    return 0, 0

def add4(x, y):
    return "0000", 0

Plan your program here, then type it in and press Run.

Do it on the robot
www.bugbotlab.com/learn/a8-7-half-and-full-adders/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

1. Change add4 to add numbers of any length, and add two 8-bit numbers.
2. Count the gates in your 4-bit adder, counting XOR, AND and OR as one gate each.
3. Two's complement subtraction A minus B is A plus (NOT B) plus 1. Use your adder with a carry in of 1 to subtract 0011 from 0110.