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178 CHAPTER 4/LOGIC FUNCTION REPRESENTATION AND MINIMIZATION

or into its dual POS form

/(*„-!, ...,X2,X\,Xo)=[Xi + /(*„-!, ...,*,-+! , l,Xi-i, ...,X2,Xi,XQJ\

•[Xf + /(*„_!, ...,* / + i,0 , */-!, ...,X 2,X\,X 0)]
= \Xi + f-. \ • \Xj + f x I, (4.68)

p op p s pos
where fj° and f£ are the cofactors for x; and ;c, in Eq. (4.67), and f ° and f are
the cofactors for jc, and xi in Eq. (4.68).
Proof of Eqs. (4.67) and (4.68) is easily obtained by setting ;c, = 1 and then jt/ = 0 and
observing that in each case the surviving cofactors are identical to the left side of the
respective equation. For example, setting jc,- = 1 in Eq. (4.67) leads to

since f , = 0 when A;/ = 1.
Multiple applications of Eqs. (4.67) and (4.68) are possible. For example, if decompo-
sition is carried out with respect to two variables, x\ and XQ, Eq. (4.67) becomes

or generally for decomposition with respect to (xyt-i » • • • , -^2, -fi » ^o),

2*-l
x
/(*„_!, . . . , X 2, X] , X 0) = J2 m i( n-\, ...,X 2 ,X) , XQ) • f(x n-\ , ...,X k , m/_). (4.69)
(=0

Here, m/ are the canonical ANDed forms of variables Xj taken in ascending minterm code
k
order from / = 0 to (2 — 1), and m, represents their corresponding minterm code. As an
example, decomposition with respect to variables (x 2,X], XQ) gives

n-i, .. . , X 3,0, 0, 0)
(x n -i, . ..,jf 3 ,0 , 0, 1)H ----

for k = 3.
In similar fashion, the dual of Eq. (4.69) is the generalization of Eq. (4.68) given by

2*-l
Mi(x n-i, ...,X2,Xi,Xo ) + f(x n-l, ...,X k,Mi, (4.70)

where now M / represents the canonical ORed forms of variables Xj and Mj_ represents their
corresponding maxterm code.