1. Karnaugh Maps

2. What are karnaugh maps? 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

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. AB A ∧ B 000 010 100 111 1

4. 2 variable K-Map 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)

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 1

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 1

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

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 ( ∧ ) 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 ( ∧ ) 1

10. 3 variable K-Map 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)

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 1 1111 1111

12. 3 variable K-Map Create a K-map for the expression: ~A ∧ C 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 11 1

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 1

14. 3 variable K-Map 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? 1111

15. 3 variable K-Map 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 1111 ~A ∧ C

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? 1111

17. 3 variable K-Map Working backwards to build an expression Identify any groupings (I see 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 A is both true and false, it does not affect the answer and can be ignored This leaves: 1111 B ∧ ~C

18. 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 1111

19. 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: ~A ∧ ~B 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: A ∧ ~C Put brackets around each pair and combine them with an OR (~A ∧ ~B) ∨ (A ∧ ~C) 1 1111

20. 3 variable K-Map Working backwards to build an expression 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 1111 1111

21. 3 variable K-Map Working backwards to build an expression 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: ~A ∧ ~C Second Pair: C ∧ B Put brackets around each pair and combine them with an OR (~A ∧ ~C) ∨ (C ∧ B) 1111 1111

22. 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

23. 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? A is both true and false, so it can be ignored B is false C is both true and false, so it can be ignored This leaves: ~B 1

24. 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

25. 4 variable K-Map 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)

26. 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 1

27. 4 variable K-Map 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

28. 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 1

29. 4 variable K-Map 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 1 1

30. 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? 1 1

31. 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? False What is the value of B for the items in the group? False What is the value of C for the items in the group? True/False What is the value of D for the items in the group? True/False ~A ∧ ~B 1 1

32. 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 1 1 11111111

33. 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? (C ∧ ~D) What are the values of A, B, C and D for the pair? (A ∧ B ∧ ~C) Solution: (C ∧ ~D) ∨ (A ∧ B ∧ ~C) 1 1 11111111

34. K-Maps & Truth Tables 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

35. 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 ABCSolution 0000 0011 0100 0111 1000 1011 1101 1111 (~A ∧ ~B ∧ C) ∨ (~A ∧ B ∧ C) ∨ (A ∧ ~B ∧ C) ∨ (A ∧ B ∧ ~C) ∨ (A ∧ B ∧ C)

36. K-Maps & Truth Tables 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 ABCSolution 0000 0011 0100 0111 1000 1011 1101 1111 11111111 1

37. K-Maps & Truth Tables 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 ABCSolution 0000 0011 0100 0111 1000 1011 1101 1111 C ∨ (A ∧ B) 11111111 1

38. 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