Karnaugh Maps

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

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 inthe row where A is true and thecolumn where B is true, put a 1in the box that is in both rows.

A B A ∧ B

0 0 0

0 1 0

1 0 0

1 1 1

1

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)

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 1

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 1

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

• 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 andjoin them with an AND (∧)

1

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 andjoin them with an AND (∧)

1 1

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)

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

11

1

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 1

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

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?

1

1

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 1

1~A ∧ C

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?

1

1

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 andcombine them with an OR

1 1

1

1

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 andcombine them with an OR

1

1 1

1

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

1 1

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

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)

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 11 11 11 1

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 variablesin your expression, it will onlyresult in a single value in theK-Map)

1

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 overlappingboxes

1 11 1

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 1

1

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 1 1

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

1 111

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

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 Solution0 0 0 0

0 0 1 10 1 0 0

0 1 1 11 0 0 0

1 0 1 11 1 0 11 1 1 1

(~A∧~B∧C)∨(~A∧B∧C)∨(A∧~B∧C)∨(A∧B∧~C)∨(A∧B∧C)

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 pairFrom this we can write a simpler expression

A B C Solution0 0 0 0

0 0 1 10 1 0 0

0 1 1 11 0 0 0

1 0 1 11 1 0 11 1 1 1

1111

1

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