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3.2 POS

Product of sums is a less common method to encode binary logic. However, it is not much more difficult. Let us reuse truth table 1. Instead of representing the rows with 1 as the output, now we represent all the rows with an output of 0.

This yields table 3.

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

Table 3: False product terms for truth table 1.

The product terms are intentionally inverted from the output. This yields the following equation:

W = (X ′Y′Z + X ′YZ ′ +XY ′Z )′
(2)

This expression is currently the negation of a sum of products. We can now apply de Morgan’s law, so the expression becomes the following:

W = (X′Y′Z)′(X ′YZ ′)′(XY ′Z)′
(3)

Next, we apply de Morgan’s law for each product term:

′ ′ ′ ′ W = (X + Y + Z )(X + Y +Z )(X + Y + Z )
(4)

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