## PHYS345 Electricity and Electronics

Example of Karnaugh Mapping

Boxing Zeroes for Product of Sums:

Consider again the following truth table for three inputs:
A   B   C     O
0 0 0   0
0 0 1   0
0 1 0   1
0 1 1   1
1 0 0   1
1 0 1   0
1 1 0   1
1 1 1   0

The truth table expressed in a form suitable for Karnaugh mapping:

BC 00  01  11  10
A\
0     0   0   1   1
1     1   0   0   1

Boxing the zeroes:

BC 00  01  11  10
A\
0     0   0   1   1
1     1   0   0   1

The red cells form a 1x2 supercell represented by A'.B'.
The green cells form a 1x2 supercell represented by A.C.
The resulting Boolean expression for this truth table is the complement of the sum of these cells, (A'.B' + A.C)'.
This still looks like a sum of products, just NOTted!

Several applications of De Morgan's theorems will yield the desired product of sums:

 (A'.B' + A.C)' = (A'.B')' . (A.C)' by one application of De Morgan's theorem. = (A+B) . (A.C)' by application of the other De Morgan's theorem. = (A+B) . (A'+C') again by application of De Morgan's theorem. Thus results the product of sums!

This realization again requires five gates as shown below:

NOTE: The minimum realization requires only three gates.
This results from the next to last line in the application of De Morgan's theorems above.