Page 135 - Engineering Digital Design

106 CHAPTER 3/BACKGROUND FOR DIGITAL DESIGN

from the truth tables for NOT, AND, and OR. In this section these laws are developed
exclusively within the logic domain with only passing reference to activation levels.

3.10.1 NOT, AND, and OR Laws

NOT Laws The unary operator NOT is the logic equivalent of complementation and
connotes inversion in the sense of supplying the lack of something. Although NOT is
purely a logic concept and complementation arises more from a physical standpoint, the
two terms, NOT and complementation, will be used interchangeably following established
practice.
The truth table for NOT is the positive logic interpretation of the physical truth table
given in Fig. 3.6b. It is from this truth table that the NOT laws are derived.

NOT
Truth Table NOT Laws
X X 3 6
0 = 1 ( - >
0 1 > 1=0
1 0

The NOT operation, like complementation, is designated by the overscore (or "bar"). A
double bar (or double complementation) of a function, sometimes called involution, is the
function itself, as indicated in Eqs. (3.6).
As examples of the applications of the NOT laws, suppose that X = AB. Then the
function^ = A Bis read as A AND B bar the quantity complemented, and X = AB — AB.
Or, if Y = 0, then Y = 0 = 1, etc. Finally, notice that Eqs. (3.2) can be generated one from
the other by involution — even in mixed logic notation. Thus, ot(L) = a(H) = <x(H), and
soon.

AND Laws The AND laws are easily deduced by taking the rows two at a time from the
truth table representing the logic AND interpretation of the AND gate given in Fig. 3.16c.
Thus, by taking Y equal to logic values 0, 1, X, and X, the four AND laws result and are
given by Eqs. (3.7).
AND
Truth Table AND Laws
X Y X -Y
X 0 = 0
0 0 0 X 1 = x (3-7)
0 1 0 — > X X -X
1 0 0 X X = 0
1 1 1

To illustrate the application of the AND laws, let X be the function X = A + B so that
(A+ 5)-0 = 0, (A + B)- 1 = A + B,(A + B)-( A + B) = A + B, and (A + B)-( A + B)
= 0. These laws are valid regardless of the complexity of the function X, which can represent
any multivariable function.