Karnaugh Map (K-MAP)

KARNAUGH MAP (K-MAP)

The K-map is a graphical technique for simplifying boolean expression. The K map is three dimensional representation of a truth table. This K-map technique is ideally suited for four or less variables.

K-map can be a 2-variable K-map, 3-variable K-map; 4-variable K-map.

Two variable K-map

The first row is for Ā and second for A. In the same way, first column is for B and second column for B. For any square, both row and column are observed. For first square of first row, Ā and are the variables and so this square indicates the product term or minterm Similarly, second square indicates ĀB. In map, 0 represents Ā and 1 represents A.

Similarly, 0 and 1 logic are for and B respectively.

Problem 4.44

Simplify with help of K-map. 

Solution :

Problem 4.45

Simplify with help of K-map.

Solution:

Boolean simplification

In group 1:

In group 2:

By summing both the results

Y = A + B 

K-map for three variables

For three variables K-map, the ordering of the variables is done in gray code.

Problem 4.46

Simplify the function Y = + ABC using K-map method.

Solution:

Problem 4.47

Simplify the expression Y =

Solution :

Hence the resulting expression is

Problem 4.48

Simplify the expression

Solution :

Using K-map simplify the expression,

Placing is in the respective squares, we observe that all the squares containing 1s are adjacent and so they can be grouped together to give A which is common to all squares.

Therefore Y = A

K-map for four variables

The representation of logical variables and product terms for four variables K- map is depicted in figure 4.12.

Problem 4.49

Simplify the boolean expression.

Solution :

Problem 4.50

Simplify the boolean expression.

Solution: