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136 CHAPTER 4 / LOGIC FUNCTION REPRESENTATION AND MINIMIZATION

AB C f

00 0 1 m o
00 1 0 -
0 1 0 1 m 2
0 1 1 0 - M 3
1 0 0 0 -
1 0 1 1
1 1 0 1
1 1 1 0 - M 7
FIGURE 4.4
Truth table for Eqs. (4.8).

form of (X + Y)(X + Y) = X. Notice that the AND law M 3 • M 3 = M 3 is applied since
this maxterm occurs twice in the canonical expression.
The results expressed by Eq. (4.7) are represented by the truth table in Fig. 4.4, where
use is made of both minterm and maxterm codes. Function / values equal to logic 1 are
read as minterms, while function values equal to logic 0 are read as maxterms. From this
there emerges the result
f(A, B,C) = Y^, m(0, 2, 5, 6) = ]~] M(l, 3, 4, 7), (4.8)

which shows that a given function can be represented in either canonical SOP or canonical
POS form. Moreover, this shows that if one form is known, the other is found simply by
using the missing code numbers from the former.
By applying DeMorgan's laws given by Eqs. (3.15a), it is easily shown that the comple-
ment of Eqs. (4.8) is

f(A, B,C) = Y[ M(Q, 2, 5, 6) = ]Tm(l, 3,4, 7). (4.9)

This follows from the result

, 2, 5, 6) = n

= ra 0 • m 2 • m 5 • m 6
= MO • A/2 • M 5 • Me
= ]~J Af(0, 2, 5, 6) = ^m(l, 3, 4, 7)

A similar set of equations exist for / = J~[ M(l, 3, 4, 7). Equations (4.8) and (4.9), viewed
as a set, illustrate the type of interrelationship that always exists between canonical forms.
There is more information that can be gathered from the interrelationship between canon-
ical forms. By applying the OR law, X + X = 1, and the OR form of the commutative laws