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112 CHAPTER 3 / BACKGROUND FOR DIGITAL DESIGN

(X-F)0(X-Z)=X.(r0Z ) Factoring Law
Distributive Laws { } (3.19)
(X + Y)Q(X + Z) = X + (YQZ) Distributive Law

Absorptive Laws { _ [ (3.20)

Consensus Laws { } (3.21)
\(X + Y) 0 (X + Z) • (7 + Z) = (X + F) O (X + Z)f

DeMorgan's Laws { _ \. (3.22)
\x e y = x o F = x o n

Notice that each of the six sets of equations are presented in dual form. Thus, by interchang-
ing AND and OR operators while simultaneously interchanging EQV and XOR operators,
duality of the set is established. The first of the distributive laws given in Eqs. (3.19) can be
termed the factoring law of XOR algebra owing to its similarity with the factoring law of
Cartesian algebra and that of Eqs. (3.12).
Generalizations of DeMorgan's XOR laws follow from Eqs. (3.15a) and (3.22) and are
given by

and (3.22a)

Verification of the associative, commutative, and distributive laws is easily accomplished
by using truth tables. For example, the second of the distributive laws in Eqs. (3.19) is verified
by the truth table in Fig. 3.36. Here, Eq. (3.5) is used together with the OR laws [Eqs. (3.8)]
to show the identity of the terms (X + Y) O (X + Z) and X + (Y Q Z).
The distributive laws may also be proven by using Boolean algebra. For example, the
factoring law of Eqs. (3.19) is proven by applying the defining relation of the XOR function

X Y Z X + Y x + z YQZ (X + F) 0 (X + Z) X + (Y 0 Z)
00 0 0 0 1 1 1
0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1
1 0 1 1 1 0 1 1
1 1 0 1 1 0 1 1
1 1 1 1 1 1 1 1

FIGURE 3.36
Truth table for the XOR distributive law given in Eqs. (3.19).