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4.2 SOP ANDPOS FORMS 133

SOP Term Binary Decimal m i SOP Term Binary Decimal mi

AB C D 0000 0 rr> 0 AB C D 1000 8 m 8
A B C D 0001 1 m i AB C D 1001 9 m g
A B C D 0010 2 m 2 AB C D 1010 10 m io
A I C D 0011 3 m 3 A B C D 1011 11 m n
A B C D 0100 4 m 4 AB C D 1100 12 m i2
A B C D 0101 5 m s A B C D 1101 13 m i3
A B C D 0110 6 m e A B C D 1110 14 m i4
A B C D 0111 7 m ? A B C D 1111 15 m
!5
FIGURE 4.1
Minterm code table for four variables.

demonstrated by expanding Eq. (4.1) as follows:

f(A, B, C)=AB + BC + ABC
= AB(C + O + (A + A)BC + ABC
= ABC + ABC + ABC + ABC + ABC
= ni2 + m?, + ms + mj + m^
,4,7). (4.3)

Note that the OR law X+X = 1 has been applied twice and that the two identical minterms
ABC are combined according to the OR law X + X = X.
The canonical truth table for Eqs. (4.3), shown in Fig. 4.2, is easily constructed from
the minterm code form. However, the truth table can also be constructed directly from the
original reduced form given by Eqs. (4.1). Notice that a logic 1 is placed in the / column
each time an AB = 01 occurs, each time a BC occurs, and for ABC. Thus, construction

AB C f

00 0 0
00 1 0
01 0 1 m 2
01 1 1 m 3
1 0 0 1 m.
4
10 1 0
1 1 0 0
1 1 1 1 m 7
FIGURE 4.2
Truth table for Eq. (4.3).