Karnaugh Map

Karnaugh Map or simply known as K-map is a two-dimensional way of presenting the truth table to simplify Boolean functions. It is another method for the simplification of Boolean problems. K-map is a table like a diagram consisting of squares. Each square represents a minterm and logic expression can be written in the SOP form.

K-Map for n-variables is made up of 2n squares. Either 1s or 0s are written as the assigned values of a function. Group of 1s gives the value of minimal SOP and on the other hand, a group of 0s gives the minimal POS. The 1s are put together by surrounding them with rectangular boxes.

The types of grouping in K-map can be:

• Pair

It is the grouping of two 1s or two 0s.

1 row x 2 columns, 2 rows x 1 column

• Quad

Quad is the grouping of 4 1s or 0s.

4 rows x 1 column, 1-row x 4 columns

• Octad

Octad is the grouping of 8 1s or 8 0s.

2 rows x 4 columns, 4 rows x 2 columns

We cannot use 0s in a grouping. Rules for grouping in SOP

• Diagonals are not allowed in a grouping.
• Each group must be as larger as possible.
• Every 1s must be in at least 1 group.
• We can do overlapping.

To simplify a Boolean function using K-map, the first step is to plot all ones in the functions of the truth table. The next step is to combine 1s into a single group using squares or ovals. The group of the minterm should be as large as possible. Because a single large group of minterm gives a simpler expression than the smaller grouping.

Now, a square having 1 may belong to more than more terms in the sum of product expression. Finally, all the grouped minterms are taken under an OR operation to find the simplified expression.
And, if needed we can express the simplified expression in the diagram form.