Digital Electronics

# Karnaugh Map K Map in Digital Electronics with Examples

## Description: Karnaugh Map K Map in Digital Electronics– In this tutorial, you will learn how to use the Karnaugh Map K Map in Digital Electronics for simplifying the algebraic expressions with Boolean algebra. This is going to be a very detailed tutorial. I will try to cover the maximum things including

• Karnaugh Map K Map introduction
• Karnaugh Map rules
• How to construct Karnaugh Map K Map
• Truth table to Karnaugh Map
• 2 variable K Map examples
• 3 variable K Map examples
• 4 variable K Map examples

Without any further delay, let’s get started!!!

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## Karnaugh Map K Map in Digital Electronics:

Many engineers simplify algebraic expressions with Boolean algebra. Boolean algebra is used to solve complex expressions and to get the minimum number of logic gates. Besides the Boolean algebra there is another method for expressing the Boolean functions named as Karnaugh Map K map. The Karnaugh Map is used for minimizing the complex Boolean expressions.

The K Map method provides a simple straight forward procedure for minimizing the Boolean expressions. This method is actually the pictorial form of the truth table.

The Karnaugh Map consists of squares. In these squares, we can make pairs, quad, and octal can be arranged. A pair eliminates one variable, A quad eliminates two variables, and octal eliminates three variables. The number of squares depends on the number of variables present in the given Boolean function.

If a Boolean function has two numbers of variables, then four squares will be there in the K map. Each square represents 1 Minterm or Maxterm.

## How to Construct a Karnaugh Map K Map:

Now, I will explain the method of constructing a Karnaugh Map K Map step by step with the help of an example. First, we will start with two variables so that you guys can easily understand the method.

### 2 variable K Map examples

#### Example 1:

Construct Karnaugh Map K Map for 2 variables Sol:

As you know there are only two variables a and b so,

2n = 22 = 4

Where n is the number of variables

So, we will have 4 squares in the Karnaugh Map. Now, we have to properly label the boxes for this let’s start with a truth table. 0 means that the corresponding variable will have the bar on it i.e. the compliment.

The Karnaugh Map K Map simply consists of rows and columns. So the table in figure 1 will become. Now, the next is to write the binary values, As per the truth table in figure 2,

00 = 0

01 = 1

10 = 2

11 = 3

So,

Now, we can write the corresponding decimal number in each box. So, this is how to construct a Karnaugh Map K Map for 2 variables. Now, this can be used for simplifying the algebraic functions. Now let’s solve an example.

#### Example 2:

Construct Karnaugh Map K Map for 2 variables and write down the simplified Boolean equation.

Sol:

As per the equations we will write 1’s in boxes 1, 2, and 3. So the Karnaugh Map K Map in figure 5 will become. Now, we can write down the common variables. In boxes 1 and 3 the variable b is common, while between the boxes 2 and 3 the variable a is common. So the simplified equation will be,

F = a + b

#### Example 3:

Construct Karnaugh Map K Map for 2 variables and write down the simplified Boolean equation.

Sol:

As per the equations we will write 1’s in the boxes 1, 2, and 3. So the Karnaugh Map K Map in figure 5 will become. As you can see all the boxes has 1’s so this makes the Quad. The Karnaugh Map K.Map in figure 7 will become. So the simplified equation will be #### Example 4:

Construct Karnaugh Map K Map for the algebraic equation given below. Sol:

As per the above algebraic equation, we have two variables a and b, which means we can use the same Karnaugh Map K Map given below. Now, we will simply write 1’s in the desired boxes, so figure 5 will become. So, now I am sure you have got the idea how to construct a Karnaugh Map K Map for 2 variables. Now we will construct the K Map for three variables a, b, and c.

### 3 Variables Karnaugh Map K Map:

For the 3 variables we will follow the same exact steps as we did for the 2 variables. So, first let’s construct the Karnaugh Map K Map for 3 variables and then we will solve some equations. 3 variables Karnaugh Map consists of 4 columns and 2 rows. Now we will simply multiple the rows and columns, and then will write their equivalent binary values. So, first, let’s start with the three variables truth table. The Karnaugh Map shown in figure 10 will become. Now we will write the binary equivalent values as per the truth table. As you can see in the truth table shown in figure 11.

0 = 000

1 = 001

2 = 010

3 = 011

4 = 100

5 = 101

6 = 110

7 = 111 This is how to construct a K Map for three variables. Now we can use this 3 variables Karnaugh Map for solving algebraic equations consisting of 3 variables. Let me give you an example.

### 3 variable K Map examples

#### Example 5:

Construct Karnaugh Map for 3 variables and write down the simplified Boolean equation.

Sol:

As per the equations, we will write 1’s in boxes 2, 3, 6, and 7. So the K.Map in figure 13 will become. This makes the quad. The variable b is common among all the boxes. So the output is

F = b

#### Example 6:

Minimize the following algebraic equation using the Karnaugh Map K Map. Sol:

Putting 1’s in the corresponding boxes of the Karnaugh Map K Map shown in figure 14. This makes a quad. As you can see clearly among all the 4 boxes the variable b bar is common. So the output is #### Example 7:

Let’s solve the same equation using Boolean algebra. Let’s see if we get the same output. Sol:

Rearranging the equation, As you can see we got the same output. Why I solved this using Boolean algebra? Just to explain how easily we can solve the algebraic equations using the Karnaugh Map K Map.

### Redundant Group in K Map Karnaugh Map:

A group who’s 1’s are all overlapping by other groups is called a redundant group. A redundant group can be neglected. Let me explain this with the help of an example.

#### Example 8:

Construct  K Map for 3 variables and write down the simplified Boolean equation.

Sol:

As per the equations, we will write 1’s in boxes 0, 1, 3, and 5. So the Karnaugh Map in figure 13 will become. Now we will make the pairs.  You can see clearly the 1 in box 5 is common with box 3 and also with box 1, which makes this group as the redundant group. As you can see the variable c is common among 1, 5, and 3 boxes. So we can remove the variable c. So, the final equation will become. #### Example 9:

Implement the expression using NAND Gates.

Sol:

This can be easily solved by using the 3 variables Karnaugh Map.  The K Map given in figure 13 will become. As per the 1’s in the boxes, now we can easily write the algebraic equation.   ### Karnaugh Map Don’t care:

An output without affecting the operation of the system. This output may be zero(0) or (1). Don’t care in Karnaugh Map is represented by X. I will explain this in detail in one of my future articles. For now, we will stick with the K Map basics. Now, let’s learn how to construct a 4 variables K.Map.

### 4 variables Karnaugh Map:

We will go through the same steps as we did for the 2 variables Karnaugh Map and 3 variables Karnaugh Map. As I have already explained the steps, and now you have the idea, how to find the number of boxes, how to write the binary values and their decimal equivalent values. As we are talking about 4 variables so

The total number of boxes in the 4 variables Karnaugh Map will be = 2n

Where n is the number of variables. So

Total number of boxes = 24 = 16. Now, we will write the binary values Now, we will write the decimal numbers in the boxes as per the binary numbers, which you can find in the truth table. This time you can make a truth table for 4 variables, as we did for the 3 variables and 2 variables previously.

This is how you can construct a Karnaugh Map for 4 variables. Now this 4 variables K Map can be used to simplify the algebraic expressions consisting of 4 variables.

#### Example 10:

Implement the expression Sol:

This can be easily solved by using the 4 variables K Map.  The K Map given in figure 21 will become. 