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 2N
possible combinations, and 2N 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.
