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3.1 SOP

Sum of products is the more common encoding method. Let us consider an example. Given three terms, X, Y and Z, we can define a truth table for the output as in table 1.

XYZoutput
0 0 01
0 0 10
0 1 00
0 1 11
1 0 01
1 0 10
1 1 01
1 1 11

Table 1: An arbitrary truth table of three variables.

We can define a product term for each row of the table, yielding table 2.

XY ZW (output)product
0 0 0 1 X′Y ′Z′
0 0 1 0
0 1 0 0
0 1 1 1 X′Y Z
1 0 0 1 XY ′Z′
1 0 1 0
1 1 0 1 XY Z′
1 1 1 1 XY Z

Table 2: True product terms for truth table 1.

Note that in table 2, only the rows with an output of 1 are represented by their product terms. This is because there is no need to represent the rows with 0 as the output. Given the product terms in table 2, we can summarize and write the equation as follows:

W = X ′Y ′Z ′ + X′YZ + XY ′Z′ + XY Z′ + XY Z
(1)

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