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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 2ⁿpossible 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

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 2ⁿcan be combined. No diagonal terms can be considered as adjacent terms.

Examples of 2-variable k-map

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.

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.