Page 111 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

92 ChupterNine

DeMorgan Transformations
A contemporary of Boole’s, Augustus DeMorgan ( 1806-1 87 1 ), also made
significant contributions to the field of symbolic logic, most notably a set of
rules which facilitate the conversion of Boolean expressions into alternate and
often more convenient forms. A DeMorgun Transfomtion comprises four steps:

1. Exchange all of the &a operators for I operators and vice versa.
2. Invert all the variables; also exchange Os for 1s and vice versa.
3. Invert the entire function.
4. Reduce any multiple inversions.

Consider the DeMorgan Transformation of a 2-input AND function
(Figure 9-11). Note that the NOT gate on the output of the new function can
be combined with the OR to form a NOR.

5tart a&b
Step#I a I b
-
5tep#2 a I 6
~tep#3 I 6

5tep#4 N/A

DeMorgan Transformation a
u
Reverse Transformation
a bt~J b 1 [Z!j)i 1

00 1 1
0
1
0 1010 0 1 1 0
1 0 0 1
1 1 1 5tep#4 a&b 1 1 00 0

Figure 9-1 1. DeMorgan Transformation of an AND function