Boolean Algebra and Simplification Techniques
209
The reverse is true in the case of a product-of-sums expression. The groups are then arranged,
beginning with the group having the least number of 1s in its included terms. Terms within the
same group are arranged in ascending order of the decimal numbers represented by these terms.
As an illustration, consider the expression
AABBC + AABBC + AABBC + AABBC + AABBC
The grouping of different terms and the arrangement of different terms within the group are shown
below:
AABBC
000
_____
AABBC
100
_____ −→
AABBC
011
AABBC
101
_____
ABC
111
First group
_____________
Second group
_____________
Third group
_____________
Fourth group
As another illustration, consider a product-of-sums expression given by
A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DDD
A + B + C + DDDA + B + C + DD
The formation of groups and the arrangement of terms within different groups for the product-ofsums expression are as follows:
AABBCCD
0000
______
______
AABBCCD
0011
AABBCCD
0101
AABBCCD −→
1010
______
______
AABBCCD
0111
AABBCCD
1110
______
______
AABBCCD
1111
_______
______
It may be mentioned here that the Boolean expressions that we have considered above did not
contain any optional terms. If there are any, they are also considered while forming groups. This
completes the first table.
3. The terms of the first group are successively matched with those in the next adjacent higherorder group to look for any possible matching and consequent reduction. The terms are considered
matched when all literals except for one match. The pairs of matched terms are replaced with a