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Logic Design 187

Y Z A Y Y Z B Y
0 0 1 0 1 0 0 1 0 1
0 1 0 0 1 1 0 1 1 0 1 0
Z Z
1 0 1 1 0 0 1 0 0 1 1 1
1 1 0 1 1 1
A = Z B = Y + Z
Figure 6-11 Two-variable examples.

one set of one cell, but this would not give the simplest sum. To find the
minimal sum with the fewest and simplest terms, we should use the
fewest number of sets and the largest sets possible. For the vertical set,
the input Y is 0 for both cells, meaning this set corresponds to the term
___
Y. For the horizontal set, the input Z is 1 for both cells giving the term
Z.
For two input functions, it might be possible to write the minimal sum
by inspection. Karnaugh maps are valuable when minimizing functions
of three variables or more.
In a three-variable Karnaugh map, two cells in the same row and in
the leftmost and rightmost columns are considered neighbors (Fig. 6-12).
For example, F is a neighbor of F and F is a neighbor of F . Functions
5
0
4
1
are minimized as before by including all the 1’s in the fewest number
of largest possible sets.
In the first example in Fig. 6-13, two sets of two cells are sufficient to
cover all the 1’s. Picking these two sets allows us to quickly find the same
minimal function that was found by boolean transformations in Fig. 6-8.
In the second example, one set of two cells is chosen including cells F 0
and F 4 . The final 1 in the map has no neighboring cells that are 1’s, so
it must be included in a set by itself.
In a four-variable Karnaugh maps, just like three-variable maps, two
cells in the same row and in the leftmost and rightmost columns are con-
sidered neighbors. In addition, two cells in the same column and in the

X Y Z F
0 0 0 F 0
0 0 1 F 1 XY
0 1 0 F 2 00 01 11 10
0 1 1 F 3 Z 0 F 0 F 2 F 6 F 4
1 0 0 F 4 1 F 1 F 3 F 7 F 5
1 0 1 F 5 Figure 6-12 Three-variable
1 1 0 F 6 Karnaugh map.
1 1 1 F 7