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142 CHAPTER 4/LOG 1C FUNCTION REPRESENTATION AND MINIMIZATION

All that is AHthatis _
NOT AB = (A+B)

A 00 01 ' 11 / 10

D
eL/
All that is^' ' ' ' ' ' ' Y
AC ~~ " \-AllthatisABC All that is A -
(a) (b)
FIGURE 4.11
(a) K-map for the reduced function Y of Eq. (4.16). (b) K-map showing minimum SOP and POS
cover for function Y.

as read in maxterm code. The minimum POS results are easily read from the K-maps of
Fig. 4.10b by combining adjacent O's as indicated. Thus, F\ is read as "all that is NOT Z"
or Z = Z. Similarly, F 2 can be read as "all that is NOT YZ + XY" or YZ + XY = (Y + Z)
(X+Y). Notice that the distributive law in Eqs. (3.12) is applied as (Y+Z)(X+ Y) = Y+XZ,
demonstrating that the SOP and POS forms for F 2 are algebraically equal, as they must
be. The minimum POS results given by Eqs. (4.15) can also be obtained by applying the
Boolean laws given in Section 3.10, but with somewhat more effort. For example, F\ is
minimized to give the result Z after three applications of the distributive law in Eqs. (3.12)
together with the AND and OR laws.
The use of third-order K-maps is further illustrated by placing the reduced SOP function

Y =ABC + AC + BC + AB (4.16)
into the third-order K-map in Fig. 4.11 a. Then by grouping adjacent minterms (shaded
loops) as in Fig. 4.1 Ib, there results the minimum expression for Eq. (4.16),

Y=A+B. (4.17)
As expected, the same results could have been obtained by grouping the adjacent maxterms
(O's) in Fig. 4.1 Ib, which is equivalent to saying "all that is NOT AB" or AB = A + B.
Other information may be gleaned from Fig. 4.11. Extracting canonical information is
as easy as reading the minterm code numbers in the lower right-hand corner of each cell.
Thus, the canonical SOP and canonical POS forms for function Y are given by

Y = JT^m(Q, 1,4,5,6,7)
= ABC + ABC + ABC + ABC + ABC + ABC

or (4.18)

as read in minterm code and maxterm code, respectively.