2. Truth table to minterm notation

Let's imagine that we've been given a truth table for a particular expression. We know that there are four inputs (A, B, C, and D), producing sixteen possible combinations ranging from 0000 to 1111.

We also know the output value (Q) for each set of inputs.

Can we work out what the simplest form of the original expression is?

The first step is to label each of the expressions that produced an output of 1. The input set '0000' outputs '0', as does '0001'. So we can ignore those.

The first set to output '1' is '0010', or not-A AND not-B AND C AND not-D. Or, in boolean algebra, $\overline A. \overline B.C. \overline D $

This expression is called a minterm. Let's fill in the minterms for each of the Q=1 entries:

Putting all of these minterms into a list produces a complete (but very complicated) expression for the truth table. Remember that in Boolean algebra, + means OR. The complete expression is as follows:

$$Q = \overline A. \overline B.C. \overline D + \overline A. \overline B.C.D + \overline A.B.C. \overline D + \overline A.B.C.D + A. \overline B. \overline C. \overline D + A.B. \overline C. \overline D + A.B. \overline C.D $$

We want to simplify this expression as much as possible. This is where karnaugh maps come in.