The K-Map or Karnaugh Map is a systematic method of simplifying a complex Boolean function or expression. The K-Map is basically a graph or a chart which consists of a certain number of adjacent cells. Each cell represents a particular combination of variables in either sum or product form.

However, we can use the K-Map for simplifying a Boolean function in any number of variables, but it becomes tedious for functions involving five or more variables. In actual practice, we usually use K-Map for simplification of Boolean functions in upto six variables.

A Boolean function in n variables can have 2ⁿ possible combinations of product terms in sum of products (SOP) form, or 2ⁿ possible combinations of sum terms in product of sums (POS) form.

Therefore, for a Boolean function in 2 variables, the K-map will have 2² = 4 cells, for a function in 3 variables, it will have 2³ = 8 cells, and so on.

A Boolean function can be expressed in two canonical or standard form namely SSOP (Standard Sum of Products) Form and SPOS (Standard Product of Sums) Form.

A SSOP form is one in which a Boolean function is expressed as a sum of product terms, where each term of the expression contains all the variables of the function in either complemented or un-complemented form. Each product term of the logic expression in SSOP form is referred to as a minterm.

For example,

$$\mathrm{Y \: = \: AB \: + \: \overline{A}B}$$

Here, Y is a Boolean function in two variables A and B. The terms AB and AB' are the minterms of the function.

In the SPOS form, a Boolean function is expressed as a product of sum terms, where each sum term, called maxterm, contains all the variables of the function in either complemented or non-complemented form.

For example,

$$\mathrm{Y \: = \: \lgroup A \: + \: B\rgroup \: .\: \lgroup \overline{A} \: + \: B \rgroup}$$

Here, Y is a Boolean function in two variables A and B, and the terms (A+B) and (A'+B) are the two maxterms of the function.

This article is primarily meant for explaining how to represent a Boolean function in Max Term form on a K-map. So, let us discuss the maxterm representation or plotting zeros on K map.

Plotting Zeros (Maxterm Representation)

As we already discussed that each sum term in a standard POS form expression is called a maxterm. A maxterm is denoted by the uppercase letter M with a subscript which denotes the decimal designation of that maxterm.

For representing a standard POS expression on to the K-map, zeros are plotted in the cells corresponding to the maxterms which are represented in the expression, and no entries are made in the cells corresponding to the maxterms which are not present in the expression.

Now, for better understanding of the concept of plotting zeros or maxterm representation, let us discuss some solved examples.

Example 1

Plot the following 2 variable Boolean expression on the K-Map.

$$\mathrm{Y \: = \: \lgroup A \: + \: B \rgroup\lgroup A \: + \: \overline{B} \rgroup\lgroup \overline{A} \:+ \: B \rgroup}$$

Solution

The given Boolean expression in terms of maxterms can be represented as,

$$\mathrm{Y \: = \: M_{3} \cdot M_{2} \cdot M_{1} \: = \: \prod M \lgroup 0, \: 1, \: 2 \rgroup}$$

The maxterm representation of this function on the K-map is shown in Figure-1.

Example 2

Plot the following 3 variables Boolean function on the K-Map.

$$\mathrm{Y \: = \: \lgroup A \: + \: B \: + \: C \rgroup\lgroup \overline{A} \: + \: \overline{B} \: + \: C\rgroup\lgroup A \: + \: B \: + \: \overline{C}\rgroup\lgroup \overline{A} \: + \: B \: + \: \overline{C} \rgroup\lgroup \overline{A} \: + \: \overline{B} \: + \: \overline{C}\rgroup}$$

Solution

The given Boolean function in terms of maxterms can be represented as,

$$\mathrm{Y \: = \: M_{0} \cdot M_{1}\cdot M_{5} \cdot M_{6} \cdot M_{7} \: = \: \prod M \lgroup 0, \: 1, \: 5, \: 6, \: 7 \rgroup}$$

The maxterm representation of this function on the K-map is shown in Figure 2.

Hence, this is all about plotting zeros or maxterm representation of Boolean expression on the K-map.

Tutorial Problems

Try to solve the following tutorial problems to understand the concept more clearly.

Q1. Plot the following Boolean expression in maxterm representation on the K-map.

$$\mathrm{f( A, \: B) \: = \: (A \: + \: B)\cdot(\overline{A} \: + \: B).(\overline{A} \: + \: \overline{B})}$$

Q2. Plot the following 3 variable Boolean function in maxterm representation on the K map.

$$\mathrm{f(A, \: B, \: C) \: = \: (A \: + \: B \: + \: \overline{C})\cdot(A \: + \: \overline{B} \: + \: C)\cdot( \overline{A} \: + \: \overline{B} \: + \: C)\cdot(\overline{A} \: + \: B \: + \: \overline{C})}$$

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