Product of Sums
Product-of-sums expresses a function as an AND of OR terms, read from the truth-table rows that output 0.
The form
A product-of-sums, or POS, expression is an AND (the product) of several OR terms (the sums). The function is 0 when any one of the sum terms is 0, so every sum term must be satisfied for the output to be 1.
Reading it from a truth table
For every row where the output is 0, write a sum that is false only for that row: include each input complemented if it is 1 in that row and plain if it is 0. ANDing these sums reproduces the function. Such full sums are called maxterms.
Worked example
| A | B | OUT |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
The output is 0 in the first and last rows, giving OUT = (A OR B) AND (NOT A OR NOT B). This is again the XOR function, expressed in POS form.
Relationship to SOP
- SOP is built from the 1-rows; POS is built from the 0-rows.
- De Morgan's laws convert one form into the other.
- POS maps to a two-level OR-AND network: OR gates feeding one AND gate.
- Which form is smaller depends on the specific function.
When to prefer POS
If a function outputs 0 for only a few input combinations, the POS form has few sum terms and is compact, whereas SOP would need many product terms. Designers pick whichever form yields fewer gates after minimization with algebra or a Karnaugh map.