Page 113 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

94 ChaptevNine

Minterms and Maxterms

For each combination of inputs to a logical function, there is an associated
minterrn and an associated maxterrn. Consider a truth table with three inputs: a,
b, and G (Figure 9-13).

a b c m interms maxterms
000 (2 & ti & C) (a I b I G)
001 (a & ti & c) (a I b I G)
010 (2 & b & E) (a I6 I c)
011 (a & b & c) (a I b I G)
100 (a & E & E) (Z I b I c)
101 (a & E & c) (a I b I E)
110 (a & b & G) (a I b I c)
1 1 1 (a & b & G) (a I b I C)
Figure 9-1 3. Minterms and maxterms

The minterm associated with each input combination is the & (AND),

or product, of the input variables, while the maxterm is the I (OR), or sum,
of the inverted input variables. Minterms and maxterms are useful for deriving
Boolean equations from truth tables as discussed below.

Sum-of-Products and Product-of-Sums

A designer will often specify portions of a design using truth tables, and
determine how to implement these functions as logic gates later. The designer
may start by representing a function as a “black with an associated truth
table (Figure 9-14). Note that the values assigned to the output y in the truth
table shown in Figure 9-14 were selected randomly, and have no significance
beyond the purposes of this example.
There are two commonly used techniques for deriving Boolean equations
from a truth table. In the first technique, the minterms corresponding to each
line in the truth table for which the output is a logic 1 are extracted and
combined using I (OR) operators; this method results in an equation said to be

5 A “black box” is so-called because initially we don’t know exactly what’s going to be in it.