Product-of-Sums Simplification
Minimize by grouping the 0s instead of the 1s, giving a product-of-sums result.
Description
A minimization that yields an AND-of-ORs (POS) form instead of OR-of-ANDs. Some functions need fewer gates as POS; output may also drive AND-OR structures. Group the 0s to minimize F′, then apply DeMorgan to obtain F as a POS.
- Mark the 0-cells and group them as you would 1-cells.
- Read the groups as a minimal SOP for F′.
- Apply DeMorgan to F′ to get a minimal POS for F.
- What: A minimization that yields an AND-of-ORs (POS) form instead of OR-of-ANDs.
- Why: Some functions need fewer gates as POS; output may also drive AND-OR structures.
- How: Group the 0s to minimize F′, then apply DeMorgan to obtain F as a POS.
- Where: Designs where a POS structure maps better to the target gates.
- When: When the zeros form larger groups than the ones.
- Analogy — Sometimes it's shorter to say who is NOT invited. Grouping the 0s describes the 'not' list; DeMorgan then flips it back into a tidy positive rule (POS).
At a glance
What
A minimization that yields an AND-of-ORs (POS) form instead of OR-of-ANDs.
Why
Some functions need fewer gates as POS; output may also drive AND-OR structures.
How
Group the 0s to minimize F′, then apply DeMorgan to obtain F as a POS.
Where
Designs where a POS structure maps better to the target gates.
When
When the zeros form larger groups than the ones.
Think of it like…
Sometimes it's shorter to say who is NOT invited. Grouping the 0s describes the 'not' list; DeMorgan then flips it back into a tidy positive rule (POS).
Procedure
- Mark the 0-cells and group them as you would 1-cells.
- Read the groups as a minimal SOP for F′.
- Apply DeMorgan to F′ to get a minimal POS for F.
SOP vs POS
| Form | Structure | Group which cells |
|---|---|---|
| SOP | OR of AND terms | the 1s |
| POS | AND of OR terms | the 0s (for F′) |
Toggle cells — see the SOP cover
▶ live simulatorClick a cell to cycle 0 → 1 → don't-care (×). Minimized SOP updates live.
| 00 | 01 | 11 | 10 | |
|---|---|---|---|---|
| 00 | ||||
| 01 | ||||
| 11 | ||||
| 10 |
3 prime implicants · verified by Quine–McCluskey
The 5 Whys
- 1
Why a POS form? Some functions are cheaper as AND-of-ORs.
- 2
Why group the zeros? Grouping 0s minimizes the complement F′.
- 3
Why complement back? DeMorgan turns the minimal F′ into a minimal F (POS).
- 4
Why care about gate count either way? Area and delay still scale with terms.
- 5
Root cause: choosing SOP vs POS picks whichever grouping the function favors.
Cheat sheet
Working principle
- Group the 0s to minimize F′, then apply DeMorgan to obtain F as a POS.
- A minimization that yields an AND-of-ORs (POS) form instead of OR-of-ANDs.
Formulas & Boolean expressions
- Read the groups as a minimal SOP for F′.
- Apply DeMorgan to F′ to get a minimal POS for F.
Key facts
- Mark the 0-cells and group them as you would 1-cells.
Why it exists
- Root cause: choosing SOP vs POS picks whichever grouping the function favors.