Grouping 0s

Also called: grouping zeros, grouping the 0s, grouping the zeros, POS from a K-map, K-map POS, product of sums from a K-map

Finding a minimal product of sums from a K-map by grouping the 0s. Each group of 0s becomes one sum term, with each fixed variable complemented.

A K-map gives a minimal product of sums just as easily as a minimal SOP. Instead of grouping the 1s, you group the 0s.

Two ways to think about it:

Via the complement. The 0s of F are the 1s of F′. Group them with the usual rules to get a minimal SOP for F′. Then apply De Morgan's law: every product becomes a sum with each literal flipped, and F is the AND of those sums.

Directly. For each group of 0s, write a sum term. A variable that stays 0 appears plain; one that stays 1 appears primed; one that changes drops out. This is the opposite of reading a 1-group, and each sum term is a merged maxterm.

All the usual rules still apply: power-of-two rectangles, wrap-around, as large and as few as possible, and don't-cares used only when they help.

Why bother? Sometimes the POS form is cheaper than the SOP, and it maps directly onto nor nor logic. The two forms always describe the same function, so you can check one against the other.

A\BC	00	01	11	10
0	0m0	0m1	0m3	0m2
1	1m4	0m5	1m7	1m6

Worked example

Example

Minimal POS from the 0s

F = Σm(0, 1, 2, 3, 5). The map above shows F′: its 1s are the 0s of F, at m4, m6, m7.

1.
Group m4 m6 (row A = 1, columns 00 and 10): A = 1, C = 0 fixed. Sum term: .
2.
Group m6 m7 (row A = 1, columns 10 and 11): A = 1, B = 1 fixed. Sum term: .
3.
F = .
4.
Check via F′: the groups read as and , so F′ = , and De Morgan gives the same product.
5.
Multiplied out, F = : 1 for every A = 0 row and for m5. ✓

Common mistakes

Priming as in SOP. In a sum term, a variable fixed at 1 appears primed, and one fixed at 0 appears plain.
Writing the group as a product term and forgetting it describes F′, not F.
Including a 1 in a group of 0s. The same no-mixing rule applies in reverse.

Practice Grouping 0s

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Grouping 0s is taught in Karnaugh Maps and Boolean Simplification.

Karnaugh Maps →Boolean Simplification →

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