Page 136 - Engineering Digital Design

3.10 LAWS OF BOOLEAN ALGEBRA 107

OR Laws The four OR laws are deduced from the logic OR interpretation of the OR gate
given in Fig. 3.17d by assigning to Y the values 0, 1, X, and X and are given by Eqs. (3.8).
The OR laws are illustrated by letting X represent the function X — BC. Then, according
to the OR laws, BC + 0 = BC, BC + 1 = 1, BC + BC = BC, and BC + BC = 1. Here
again, use has been made of a multivariable function X to demonstrate the applicability of
a fundamental Boolean law, the OR law.
OR
Truth Table OR Laws
XYX+Y X + Q=x
0 0 0 X + 1 = 1 (3-8)
0 1 1 * X + X = X
1 0 1 X+X=l
1 1 1
Notice that the AND and OR laws are easily verified by substituting 0 and 1 for the
multivariable function X in the examples just given, and then comparing the results with
the AND and OR truth tables.

3.10.2 The Concept of Duality
An inspection of the AND and OR laws reveals an interesting relationship that may not be
obvious at first glance. If the 1's and O's are interchanged while the AND and OR operators,
(•) and (+), are interchanged, the AND laws generate the OR laws and vice versa. For
reference purposes, the interchange of 1 's and O's simultaneously with the interchange of
operators is represented by the double arrows (<•») as follows:

0 *+ 1
(•) o (+)
o <+ e

This simultaneous interchange of logic values and operators is called logic duality. The
duality between the AND and OR laws is given by Eqs. (3.9).

AND Laws OR Laws
X . 0 = 0 By x + 0 =X
X • 1 = X + + X + 1 - 1 (3-9)
X • X = X Duality x + X = X
X- X = 0 X + X = l

Perhaps the best way to demonstrate duality is by the two dual sets

(A O B)[AB + AB] = 0 + > (A © B) + [(A + B) • (A + B)] = 1

and
X O (X + Y) = X • Y + > X 0 (X • Y) = X + Y,