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4.12 WORKED EV K-MAP EXAMPLES 183

\DO r
A\ oo 01 ' 11 10
o 1 $ D "ol
0 1 3 2
1 1 1 ( 0 0 )
f
4 4 5 ^ — 7 ~~V — /'s
I
c c
(a) SOP cover (b) POS cover
FIGURE 4.52
(a) Minimum SOP cover and (b) minimum POS cover for function Z of Example 4.2.

(b) Extract minimum SOP and minimum POS cover for the function Z.
In Fig. 4.52 are the third-order K-maps showing the minimum SOP and minimum POS
cover for the function Z. Notice that the subfunction in cell 6 is interpreted differently in
the SOP and POS K-maps.
From reading this cover, the results are
Z SOP=ACD + C

ZPOS = (B + C + D}(B + C + D)(A + B),

which are seen to be logically equivalent but not algebraically equal. Notice that the 1's in
the SOP K-map are looped out as the octad B by using 0 8 = 1 in cell 4 of Fig. 4.5 1 to give
0 8 + D = 1 . Also, note that the 0 in cell 6 of the POS K-map in Fig. 4.5 1 is looped out as
the quad A + B by using 0 ]3 = 0 to give 0nD = 0. Thus, 0n is used as a 1 for minimum
SOP extraction but as a 0 for minimum POS extraction, meaning that the SOP and POS
expressions cannot be algebraically equal.

EXAMPLE 4.3 A four- variable function F(A, B, C, D) containing don't cares is com-
pressed into the truth table given in Fig. 4.53.

(a) Represent the function F in a second-order K-map, and express F in canonical SOP
and POS form by using coded notation.
(b) By proper interpretations of the don't care subfunctions, loop out the minimum SOP
and POS cover from the second-order K-map and give the gate/input tallies for each.

A B
0 0 C-D
0 1 D)
1 0 + D)
1 1
FIGURE 4.53
Compressed truth table for a function F of four variables.