Karnaugh Maps are used for many small
design problems. It's true that many larger designs are done using computer
implementations of different algorithms. However designs with a small number of
variables occur frequently in interface problems and that makes learning
Karnaugh Maps worthwhile. In addition, if you study Karnaugh Maps you will gain
a great deal of insight into digital logic circuits.

In this section we'll examine some
Karnaugh Maps for three and four variables. As we use them be particularly
tuned in to how they are really being used to simplify Boolean functions.

The goals for this lesson include the
following.

Given a Boolean function described by a truth table or logic
function,

Draw the Karnaugh Mapfor
the function.
Use the information from a Karnaugh Map to determine the smallest
sum-of-products
function.

What Does a Karnaugh Map Look Like?

A Karnaugh Map is a grid-like
representation of a truth table. It is really just another way of presenting a
truth table, but the mode of presentation gives more insight. A Karnaugh map
has zero and one entries at different positions. Each position in a grid
corresponds to a truth table entry. Here's an example taken from the voting
circuit presented in the lesson on Minterms. The truth table is shown first.
The Karnaugh Map for this truth table is shown after the truth table.

A

B

C

V

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

1

1

0

0

0

1

0

1

1

1

1

0

1

1

1

1

1

How Can a Karnaugh Map Help?

At first, it might seem that the Karnaugh
Map is just another way of presenting the information in a truth table. In one
way that's true. However, any time you have the opportunity to use another way
of looking at a problem advantages can accrue to you. In the case of the
Karnaugh Map the advantage is that the Karnaugh Map is designed to present the
information in a way that allows easy grouping of terms that can be combined.

Let's start by looking at the Karnaugh Map
we've already encountered. Look at two entries side by side. We'll start by
focussing on the ones shown below in gray.

Let's examine the map again.

The term on the left in the gray area of the
map corresponds to:

The term on the right in the gray area of the
map corresponds to:

These two terms can be combined to give

The beauty of the Karnaugh Map is that it has
been cleverly designed so that any two adjacent cells in the map differ by a
change in one variable. It's always a change of one variable any time you cross
a horizontal or vertical cell boundaries. (It's not fair to go through the
corners!)

Notice that the order of terms isn't
random. Look across the top boundary of the Karnaugh Map. Terms go 00, 01, 11,
10. If you think binary well, you might have ordered terms in order 00, 01, 10,
11. That's the sequence of binary numbers for 0,1,2,3. However, in a Karnaugh
Map terms are not arranged in numerical sequence! That's done deliberately to
ensure that crossing each horizontal or vertical cell boundary will reflect a
change of only one variable. In the numerical sequence, the middle two terms,
01, and 10 differ by two variables! Anyhow, when only one variable changes that
means that you can eliminate that variable, as in the example above for the
terms in the gray area.

Let's check the claim made on above.
Click on the buttons to shade groups of terms and to find out what the reduced
term is.

The Karnaugh Map is a visual technique
that allows you to generate groupings of terms that can be combined with a
simple visual inspection. The technique you use is simply to examine the
Karnaugh Map for any groups of ones that occur. Grouping ones into the largest
groups possible and ensuring that all ones in the table have been included are
the first step in using a Karnaugh Map.

In the next section we will examine how
you can generate groups using Karnaugh Maps. First, however, we will look at
some of the kinds of groups that occur in Truth Tables, and how they appear in
Karnaugh Maps.

Click on these buttons to show some
groupings. There's one surprise, but it really is correct. In each case, be
sure that you understand the term that the group represents.

There is a small surprise in one grouping
above. The lower left and the lower right 1s
actually form a group. They differ only in having B and its' inverse.
Consequently they can be combined. You will have to imagine that the right end
and the left end are connected.

So far we have focussed on K-maps for
three variables. Karnaugh Maps are useful for more than three variables, and
we'll look at how to extend ideas to four variables here. Shown below is a
K-map for four variables.

Note the following about the four variable
Karnaugh Map.

There are
16 cells in the map.
Anytime you have N variables, you will have 2^{N} possible
combinations, and 2^{N} places in a truth table or Karnaugh Map.

Imagine moving around in the
Karnaugh Map. Every time you cross a horizontal or vertical boundary one -
and only one - variable changes value.

The two pairs of variables -
WX and YZ - both change in the same pattern.

Otherwise, if you can understand a Karnaugh Map for a
three-variable function, you should be able to understand one for a
four-variable function. Remember these basic rules that apply to Karnaugh maps
of any size.

In a Karnaugh Map of any
size, crossing a vertical or horizontal cell boundary is a change of only
one variable - no matter how many variables there are.

Each single cell that
contains a 1 represents a minterm in the function, and each minterm can be
thought of as a "product" term with N variables.

To combine variables, use
groups of 2, 4, 8, etc. A group of 2 in an N-variable Karnaugh map will
give you a "product" term with N-1 variables. A group of 4 will have N-2
variables, etc.

You will never have a group
of 3, a group of 5, etc. Don't even think about it. See the points above.

Let's look at some examples of groups in a 4-variable
Karnaugh Map.
Example 1
- A Group of 2

Here is a group of 2 in a 4-variable map.

Note that Y and Z are 00 and 01 at the top of the
two columns in which you find the two 1s. The
variable, Z, changes from a 0 to a 1 as you move from the left cell to the right
cell. Consequently, these two 1s are not dependent
upon the value of Z, and Z will not appear in the product term that results when
we combine the 1s in this group of 2. Conversely,
W, X and Y will be in the product term. Notice that in the row in which the
1s appear, W = 0 and X = 1. Also, in the two
columns in which the 1s appear we have Y = 0. That
means that the term represented by these two cells is: