How to simplify a Boolean function using Karnaugh map (k-map)?

How to simplify a Boolean function using Karnaugh map (k-map)?

How to simplify a Boolean function using Karnaugh map (k-map)?  Truth table representation of a function is always a unique expression. The complexity of a digital circuit can be reduced by just simplifying its Boolean expression. The more we reduce the redundant terms, the more simplified the digital circuit we have. One of the finest methods is the Karnaugh map. In this method, the minterms are represented in a block of box. The size of the box is determined from the number of literals of a functions. If there are n literals, then there will be 2n possible blocks in a box.  K-map method can be used from 2,3,4,5,….n variables. But because of the complexity of higher terms, we will just focus on upto four variables k-map. The k-map representation of a function is not unique. There can be different forms of a simplified function but each will contain equal number of literals.

2-variable k-map

The 2 variable k-map consists of four block where each block represents the minterm of a function. 2-variable k-map is shown below

2 variable k-map
2 variable k-map

It can be seen clearly that each block is representing a minterm. m0 is adjacent to m1 and m2. Similarly, m3 is adjacent to m2  and m1.

Two adjacent terms can be combined to get a simplified expression. Since adjacent terms change only one bit at a time. So this property helps to simplify the Boolean expression. Minterms can be combined horizontally, vertically, in squares, from edges. Only terms of the order of 2n can be combined. No diagonal terms can be considered as adjacent terms.

Examples of 2-variable k-map

2 variable k-map example
2 variable k-map example

In terms of summation of minterms this function is equal to F=x’y’+xy’

The simplified function expression is

F(x,y)=y’

since 0 is marked as primed variable. y=0 in selected column so mention it as y’. x changes its value from 0 to 1. so (x+x’)y’=y’ given that x+x’=1. Both results can be compared and it can be concluded that original function needs four literals, while simplified version needs only one literal.

2-variable m-map example
2-variable m-map example

In this example function is equal to

F(x,y)=y’+x

Since vertical adjacent terms only produce y’ and the horizontal adjacent terms are corresponding to row where x=1.

It should be noted that if all minterms of the function are equal to 1 then this represents an identity/constant function or F=1.

3-variable k map

The three variable k-map will consist of 8 blocks. The minterms are mentioned in such a way that each adjacent block changes only one bit at a time as shown below.

3-variable k-map
3-variable k-map

so if you notice m1 and m3 only change their 2nd bit (y 0—-> 1). This property causes the redundancy of Boolean expressions.

Examples of 3-variable k-map

example of 3 variables k-map
example of 3 variables k-map

This function is simplified as

F=z’+x’y

3-variable k-map example
3-variable k-map example

 

This function is simplified as

F=x’+z’

Solve some questions

practice question
practice question

4-variable k-map

The minterms of 4-variables k-map are represented as

4 variables k-map
4 variables k-map

Lets say we are given the following minterms of a function

Boolean function as summation of minterms
Boolean function as summation of minterms

Now lets just simplify it using k-map. Place 1 in all boxes corresponding to the given minterms.

example of 4 variables k-map
example of 4 variables k-map

Answer

F=w’x’y’+w’y’z+w’x’z+wy’z’+wxy+xyz’

Video lecture for Karnaugh map (k-map) simplification method

here is the link of video lecture for simplifying a Boolean function using k-map

Software to simplify the Boolean expression

Software to simplify the Boolean expression using k-map

Also read here

https://eevibes.com/digital-logic-design/how-to-represent-the-boolean-function-in-canonical-form/

what is the Canonical form representation of Boolean function?

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