Page 119 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

100 rn Chapter Ten

Grouping Minterms
In the case of a 3-input Karnaugh Map, any two horizontally or vertically
adjacent minterms, each composed of three variables, can be combined to form
a new product term composed of only two variables. Similarly, in the case of a
4-input map, any two adjacent minterms, each composed of four variables, can
be combined to form a new product term composed of only three variables.
Additionally, the 1s associated with the minterms can be used to form multiple
groups. For example, consider the 3-input function shown in Figure 10-5, in
which the minterm corresponding to a = 1, b = 1, and c = 0 is common to three
groups.

abc Y

000
00 1
0 1 0
3-input Y 0 1 1
Function 100
1 0 1
1 1 0
1 1 1 1

Figure 10-5. Karnaugh Map minterms used to y=(b & Z) I (a&;) I (a& b)
form multiple groups

Groupings can also be formed from four adjacent minterms, in which case
two redundant variables can be discarded; consider some 4-input Karnaugh
Map examples (Figure 10-6).
In fact, any group of 2" adjacent minterms can be gathered together where
n is a positive integer. For example, 2l = two minterms, 22 = 2 x 2 = four
minterms, Z3 = 2 x 2 x 2 = eight minterms, etc.
As was noted earlier, Karnaugh Map input values are ordered so that the
values associated with adjacent rows and columns differ by only a single bit.
One result of this ordering is that the top and bottom rows are also separated
by only a single bit (it may help to visualize the map rolled into a horizontal
cylinder such that the top and bottom edges are touching). Similarly, the left
and right columns are separated by only a single bit (in this case it may help to