1. Karnaugh Maps

2. Boolean algebra can be represented in a variety of ways. These include:
Boolean expressions
Truth tables
Circuit diagrams
Another method is the Karnaugh Map (also known as the K-map)
K-maps are particularly useful for simplfying boolean expressions
What are karnaugh maps?

3. 2 variable K-Map Starting with the Expression: A ∧ B
As a Truth Table this would be:
As a Circuit Diagram this would be:
A K-Map, will be a small grid with 4 boxes, one for each combination of A and B. Each grid square has a value for A and a value for B.
To complete the K-Map for the expression, you find the box in the row where A is true and the column where B is true, put a 1 in the box that is in both rows. A B
A ∧ B 1
2 variable K-Map 1

4. One way to view a K-Map is to figure out what the ‘address’ is for each box:
what its A and B values are for each position
These have been written in the format (A, B)
(0,0) (0,1)
(1,0) (1,1)
2 variable K-Map

5. 2 variable K-Map To create a K-map for the expression: A
Place 1s in all the boxes in the row where A is true
2 variable K-Map

6. 2 variable K-Map To create a K-Map for the expression B
Place 1s in all the boxes in the column where B is true
2 variable K-Map

7. 2 variable K-Map Expression: ~A ∧ B
Find the row where A is false and the column where B is true
Place a 1 in the overlapping position 1
2 variable K-Map

8. 2 variable K-Map Try working backwards!
Starting with the K-Map, interpret the results and write an expression
Highlight the row that contains the 1
Is it A or ~A?
Highlight the column that contains the 1
Is it B or ~B?
Write down your two variables and join them with an AND (∧)
2 variable K-Map 1

9. 2 variable K-Map Try working backwards!
Starting with the K-Map, interpret the results and write an expression
Highlight the row that contains the 1
Is it A or ~A?
Highlight the column that contains the 1
Is it B or ~B?
Write down your two variables and join them with an AND (∧)
2 variable K-Map
1 1

10. In a 3 variable K-Map, we need to accommodate 8 possible combinations of A, B and C
We’ll start by figuring out the ‘address’ for each position (A, B, C)
(0,0,0) (0,0,1)
(0,1,0) (0,1,1)
(1,1,0) (1,1,1)
(1,0,0) (1,0,1)
3 variable K-Map

11. 3 variable K-Map If we had a K-Map where the expression was B
We would place 1s in every box where B is true
We ignore the values of A and C, because they are not written in our expression
3 variable K-Map 1
1 1

12. 3 variable K-Map ~A ∧ C Create a K-map for the expression:
Highlight rows where A is false
Highlight columns where C is true
Place 1s in all the overlapping boxes
Note: We can ignore the values of B, because they are not mentioned in the expression 1
3 variable K-Map

13. 3 variable K-Map To create the K-Map for the expression A ∧B∧~C
Highlight rows where A is true
Highlight rows where B is false
Highlight columns where C is false
Place 1s in all the overlapping boxes
3 variable K-Map

14. This time we’ll start with a K-map and find the expression.
Identify any groupings (in this case we have a pair)
What is the value of A for both items in the pair?
What is the value of B for both items in the pair?
What is the value of C for both items in the pair? 1
3 variable K-Map

15. This time we’ll start with a K-map and find the expression.
Identify any groupings (in this case we have a pair)
What is the value of A for both items in the pair?
What is the value of B for both items in the pair?
What is the value of C for both items in the pair?
Because B is both true and false, it does not affect the answer and can be ignored 1
~A ∧ C
3 variable K-Map

16. 3 variable K-Map Determine the expression represented by this K-map
Identify any groupings
What is the value of A for both items in the pair?
What is the value of B for both items in the pair?
What is the value of C for both items in the pair?
3 variable K-Map 1

17. 3 variable K-Map Determine the expression represented by this K-map
Identify any groupings (I see two pairs)
What is the value of A for both items in the first pair?
What is the value of B for both items in the first pair?
What is the value of C for both items in the first pair?
First Pair:
What is the value of A for both items in the second pair?
What is the value of B for both items in the second pair?
What is the value of C for both items in the second pair?
Second Pair:
Put brackets around each pair and combine them with an OR
1 1
3 variable K-Map 1

18. 3 variable K-Map Working backwards to build an expression First Pair:
Identify any groupings (I see two pairs)
What are the value of A, B, C values for the first pair?
What are the value of A, B, C values for the second pair?
First Pair:
Second Pair:
Put brackets around each pair and combine them with an OR 1
3 variable K-Map 1

19. 3 variable K-Map Working backwards to build an expression
Identify any groupings (I see two pairs)
When I look more closely, I can see they are both ~B, so perhaps I can view it as a group of 4
What are the value of A, B, C values for the group?
1 1
3 variable K-Map
1 1

20. K-Map Notes Some notes to add at this point:
Groupings can only be exact binary values: 1, 2, 4, 8 You cannot have a groupings of 3
Groups can overlap
The bigger the groupings, the simpler the resulting expression/circuit
When converting from a truth table to a K-Map, simply use the values in the TT to find the position values within the K-Map
K-Map Notes

21. In a 4 variable K-Map, we need to accommodate 16 possible combinations of A, B, C and D
We’ll start by figuring out what the ‘address’ for each position (A, B, C, D)
(0,0,0,0) (0,0,0,1) (0,0,1,1) (0,0,1,0)
(0,1,0,0) (0,1,0,1) (0,1,1,1) (0,1,1,0)
(1,1,0,0) (1,1,0,1) (1,1,1,1) (1,1,1,0)
(1,0,0,0) (1,0,0,1) (1,0,1,1) (1,0,1,0)
4 variable K-Map

22. 4 variable K-Map If we had a K-Map where the expression was D
We would place 1s in every box where D is true
We ignore the values of A, B and C, because they are not written in our expression
4 variable K-Map

23. If we had a K-Map where the expression was: A ∧ ~B ∧ ~C ∧ ~D
We would highlight the
Rows where A is true
Rows where B is false
Columns where C is false
Columns where D is false
place 1s in all the overlapping boxes
(When you have all 4 variables in your expression, it will only result in a single value in the K-Map) 1
4 variable K-Map

24. 4 variable K-Map If we had a K-Map where the expression was: (A ∧D)
We highlight the:
Rows where A is true
Columns where D is true
Put 1s in all overlapping boxes
4 variable K-Map

25. If we had a K-Map where the expression was: (A ∧B) ∨(B∧C ∧~D)
We would work with one pair at a time.
For each part of the expression:
Identify the overlapping boxes
Place 1s in them
We have resulted in a:
Group of 4
Pair 1
4 variable K-Map

26. 4 variable K-Map Working backwards
Identify any groupings (I see a group of 4)
What is the value of A for the items in the group?
What is the value of B for the items in the group?
What is the value of C for the items in the group?
What is the value of D for the items in the group?
4 variable K-Map

27. 4 variable K-Map Working backwards
Identify any groupings (I see a group of 4 and a pair)
What are the values of A, B, C and D for the group?
What are the values of A, B, C and D for the pair?
Solution
4 variable K-Map

28. Often we are asked to write an expression based on a truth table, this can result in pretty complicated expressions that require simplification using logic laws
Using a K-Map can aid the simplification process
K-Maps & Truth Tables

29. K-Maps & Truth Tables Let’s start with a Truth Table
To build an expression we would normally write an expression (using AND’s) for each row resulting in true and combine those with ORs
This produces a really big expression that will be difficult to simplify using the logic laws A B C
Solution 1
(~A∧~B∧C)∨(~A∧B∧C)∨(A∧~B∧C)∨(A∧B∧~C)∨(A∧B∧C)
K-Maps & Truth Tables

30. Rather than converting straight to an expression, let’s try placing the values in a Karnaugh Map.
Place 1s in the appropriate positions, according to the Truth Table
This highlights a group of 4 and a pair From this we can write a simpler expression A B C
Solution 1 1
K-Maps & Truth Tables

31. Practise, Practise, Practise
The best way to learn how to work with K-Maps is to apply your new knowledge to some Karnaugh Maps worksheets
Practise, Practise, Practise