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4.7 FUNCTION REDUCTION OF FIVE OR MORE VARIABLES 165

£ m(5, 6) = C 0 D = CD + CD = (C + D)(C + D). The minimum SOP and POS results
are given by

ZSOP = ABCD + ABD + ABD + CD + BC
(4.48)
(A + B + C + D}(B + C + D)(B + C + D)(A + B + D)(A + C + D).

From the results depicted in Fig. 4.34, certain conclusions are worth remembering and are
set off by the following:

• In minterm code, subfunctions of the type XY are subsets of forms of the type
X + Y.
• In maxterm code, subfunctions of the type X + Y are subsets of forms of the
type XY.

What this means is that subfunctions of the type XY can be looped out from terms of the
type X + Y to produce reduced SOP cover. For reduced POS cover, subfunctions of the
type X + Y can be looped out from terms of the type XY (there are more O's in XY than in
X + Y ). For example, in Fig. 4.34 CD is looped out of both C + D_and C + D to contribute
to minimum SOP cover. However, in Fig. 4.34b both C + D and C + D are looped out of
CD, leaving C + D to be covered by A + B + D.

4.7 FUNCTION REDUCTION OF FIVE OR MORE VARIABLES

Perhaps the most powerful application of the EV mapping method is the minimization or
reduction of functions having five or more variables. However, beyond eight variables the
EV method could become too tedious to be of value, given the computer methods available.
The subject of computer-aided minimization tools is covered in Appendix B.
Consider the function

F(A, B, C, D, E} = £]m(3, 11, 12, 19, 24, 25, 26, 27, 28, 30), (4.49)

which is to be compressed into a fourth-order K-map. Shown in Fig. 4.35 is the first-order
compression (Map Key = 2) and minimum SOP and POS cover for the five variable function
in Eqs. (4.49). The minimized results are

FSOP = BCDE + CDE + ABE + ABC
_ _ (4.50)
(A + D + E}(C + E}(B + E}(A + C + D)(fl + D),
which have gate input tallies of 5/ 17 and 6/17, respectively. Thus, the SOP result is the sim-
pler of the two. Also, since there are no don't cares involved, the two expressions are algebrai-
cally equal. Thus, one expression can be derived from the other by Boolean manipulation.
A more complex example is presented in Fig. 4.36, where the six-variable function

Z(A, B, C, D, E, F)
= ]T m(0, 2, 4, 6, 8, 10, 12, 14, 16, 20, 23, 32, 34, 36, 38, 40,
42, 44, 45, 46, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63) (4.51)