Product of sums (POS)

Also called: product-of-sums form, POS form, POS expression, conjunctive normal form, CNF

A Boolean expression written as an AND of OR terms, with bars only on single letters, such as (A + B')(A' + C).

A product of sums is an AND ("product") of sum terms, where each sum is an OR of literals. Example: . A single sum term, or a single literal, also counts.

It is the dual shape of a sum of products:

SOP: each term picks out where the output is 1; the OR collects them.
POS: each bracket rules out where the output is 0; the AND requires all of them.

Any function can be written as a POS: take one maxterm for each row where the output is 0 and AND them together. That is the canonical form, the product of maxterms.

To convert:

POS to SOP: multiply the brackets out (distributive law).
SOP to POS: use the OR-over-AND form of the distributive law, or find the SOP of the complement F′ and apply de morgans laws.

In hardware a POS is an OR-AND two-level circuit, which converts directly to NOR-NOR. Sometimes the minimal pos is cheaper than the minimal SOP, so it is worth finding both.


0	0	0	1
0	0	1	1
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

Worked examples

Example

Reading where a POS is 0

On which rows is equal to 0?

1.
A product is 0 when any bracket is 0.
2.
= 0 needs A = 0 and B = 1: rows 010 and 011.
3.
= 0 needs A = 1 and C = 0: rows 100 and 110.
4.
So F = 0 on rows 2, 3, 4, 6 and 1 on rows 0, 1, 5, 7, matching the table above.

Example

Turning an SOP into a POS

Write as a product of sums.

1.
Use OR over AND: = .
2.
Each bracket is an OR of single literals, and they are ANDed, so this is a POS.
3.
Check A = 1, B = 0, C = 0: the original is 0 + 0 = 0, and (0 + 1)(0 + 0) = 0.

Common mistakes

Reading a sum term's zero row with minterm polarity. is 0 when A = 0 and B = 1, not when A = 1 and B = 0.
Thinking multiplying out a POS gives a POS. It gives an SOP.
Assuming SOP and POS of the same function always cost the same. Often one is cheaper.

Practice Product of sums (POS)

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Learn it step by step

Product of sums (POS) is taught in Boolean Algebra, Boolean Simplification and Logic Gates.

Boolean Algebra →Boolean Simplification →Logic Gates →

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