Page 201 - Engineering Digital Design

172 CHAPTER 4 / LOGIC FUNCTION REPRESENTATION AND MINIMIZATION

2,3,6,7 W -- Y --
2,0,3,7 W-Y -
2,10 --XY Z
8, 9 WXY- -
^___ 8,10 WX-- Z
10 3,7(4) W"--Y Z /
3 •{ 0111 7
/ Indicates that an implicant is covered by a Prime Implicant in the columns to the right.
FIGURE 4.44
Tabular determination of Pis for the O's (treated as 1's) in function F of Eq. (4.58).

giving the final results

F POS = WY+WXZ
hos = FPOS = (W + Y)(W + X + Z). (4.59)

Notice that the PI (2, 3,6,1) is the EPI WY and that the remaining maxterms (treated as
minterms) are covered by the PI (8,10), the minimum set of Pis covering all minterms.
Had the Q-M algorithm been applied to Eq. (4.57), the minimum SOP result would be

F SOp = WY+WX+WZ, (4.60)

which is algebraically equal to the POS result of Eq. (4.59). The reason for this is that the
application of the Q-M algorithm uses the three 0's in the same way for the two cases,
a feature of the Q-M method. As a general rule, this is rarely the case for SOP and POS
minimized forms of incompletely specified functions obtained by other methods.

\ 3 6 8 10 Essential Pis
Pis
2,3,6,7 / / W -- Y -- = W Y
2,10 /
8,9 /
8,10 / / • WX-- Z = WX Z

FIGURE 4.45
Table of Pis (from Fig. 4.44) vs maxterms treated as minterms for function F of Eq. (4.58) showing
the essential Pis.