Karnaugh Maps (Cambridge International A-Level Computer Science)

Karnaugh maps

What is a Karnaugh map?

• A Karnaugh map (KMap) is a tool that is used for simplifying Boolean algebra expressions
• It offers a visual method of grouping together expressions with common factors
• The format of the map makes it easy to identify and eliminate redundant terms
• They are used in digital logic design, such as simplifying the logic of digital circuits

Steps:

1. Create the Map: Each cell in the grid corresponds to a term in the Boolean expression. Fill cells with 1s and 0s corresponding to the output of that term
2. Grouping: Group the 1s in the grid. Each group must be a rectangle and the size of the group must be a power of 2. A cell can be part of multiple groups
3. Simplifying: Write down a simplified Boolean expression for each group. The simplified expression for a group consists of the variables that remain constant in all terms in the group
4. Final Expression: Combine the simplified expressions from each group using OR operations to get the final simplified Boolean expression

Creating Karnaugh Maps

• A Karnaugh Map (KMap) can be used to simplify a Boolean expression with 2 inputs
• Here is an example for the expression A + B (A OR B)

Step 1

• Add each variable starting with A at the top and B down the side
• Add each possible state for A and B

Step 2

• Take each expression in turn separated by the V(OR)
• First look at A on it's own
• Find all cells where A is 1
• Add 1 to the cell

Step 3

• Repeat for B

Step 4

• This is now a completed KMap for the expression A + B (A OR B)

Simplifying expressions using Karnaugh Maps

• Simplify $\overline{A}.\overline{B}.C+\overline{A}.B.\overline{C}+A.\overline{B}.C+A.B$ using a KMap
• In this example there will be 3 variables A,B and C so the empty KMap will look like this:

Step 1:

• Split this long term at each OR giving 4 smaller expressions (subterms) to add to the table:

  • $\overline{A}.\overline{B}.C$
  • $\overline{A}.B.\overline{C}$
  • $A.\overline{B}.C$
  • $A.B$

Step 2:

• Take the first subterm $\overline{A}.\overline{B}.C$
• Put a 1 in the map for every cell where this term would be TRUE (1)
• So if A and B were 0 and C was 1 this subterm would be 1
• So put a 1 in every cell in the KMap where A is 0, B is 0 and C is 1

Step 3:

• The next subterm is $\overline{A}.B.\overline{C}$
• Put a 1 in the KMap where A is 0, B is 1 and C is 0

Step 4:

• The next subterm is $A.\overline{B}.C$
• Put a 1 in the KMap where A is 1, B is 0 and C is 1

Step 5:

• The final subterm is $A.B$
• Put a 1 in the KMap where A is 1 and B is 1 (2 cells this time)

Making the groups

• Once you have written the 1s and 0s into your Karnaugh Map, you can then use this to simplify the expression

• In order to do this, you need to identify the groups

• This is the key to using the Karnaugh map to derive the simplified expression

• The aim of making the groups is to

  • Make rectangular groups
  • Make groups that are as large as possible
  • Make groups that contain either 8,4,2 or 1 ones
  • Groups can overlap (i.e. some ones can be in multiple groups)
  • The Karnaugh map ‘grid’ wraps round in all directions so the groups can wrap round

Example grouping

• The completed map has 1s in five cells: AB = 00 with C = 1, AB = 01 with C = 0, AB = 11 with C = 0, AB = 11 with C = 1 and AB = 10 with C = 1

• Group 1:

  • The two 1s in the C = 1 row under AB = 00 and AB = 10 (note the wrapping around)
  • A changes, B stays at 0 and C stays at 1, so this group represents $\overline{B}.C$

• Group 2:

  • The two 1s in the C = 0 row under AB = 01 and AB = 11
  • A changes, B stays at 1 and C stays at 0, so this group represents $B.\overline{C}$

• Group 3:

  • The two 1s in the AB = 11 column (C = 0 and C = 1)
  • C changes, A and B both stay at 1, so this group represents $A.B$

• Final simplified expression

  • OR the terms for the three groups together
  • The simplified expression is $\overline{B}.C+B.\overline{C}+A.B$

Exam Hint In questions where you have to use a Karnaugh Map always show the groups by drawing a box/circle round them.