Page 58 - Handout Digital Electronics

7.7 The three variable Karnaugh map

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A three variable K map has 2 cells or squares, and each variable occupies exactly half the total number
of cells. The cells are represented by numbers, for example (000) is 0, (001) is, (010) is 2, (011) is 3,
(100) is 4, (101) is 5, (110) is 6 and (111) is 7. This numbering quickens the process of inserting the
ones (1s) in cells during minimization of Boolean functions.

Example
Minimize the Boolean function below using the K map method:

F (A, B, C) = ∑ (0, 2, 3, 5, 6, 7). After filling in the values of F in their respective cell positions, three
groups are formed as shown below.

(I)
(II) (III)

Looking at group one (I), consider variables A and B. In this group B has toggled from 1 to 0, A is
constant, so variable B falls away or is discarded. Looking at C in the same group, it is seen that C is
constantly 0, so part answer is AC . Group two (II) is the vertical group with four 1s. In this group
looking at variables A and B, A has toggled from 0 to 1, so is discarded, C has also
toggled from 0 to 1 and so is discarded. On the other had B is constantly 1, so part answer is B. Looking
at the third group (III), B has toggled from 1 to 0 and A is constantly 1. C is also constantly 1, so the part
answer is AC. The full answer is:

F = AC + B + AC

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