Product of maxterms (ΠM)

Also called: canonical POS, canonical product of sums, pi notation, capital pi notation, standard POS, maxterm list

The canonical POS: an AND of one maxterm for each row where F = 0. ΠM(0, 3) means M0 · M3. It lists the 0s of the truth table.

The product of maxterms, or canonical POS, writes a function as the AND of one maxterm for every row where F = 0. The shorthand lists those rows: F = ΠM(1, 2, 6) means F = M1 · M2 · M6.

Why it works. An AND is 0 as soon as any factor is 0.

On a listed row, that row's maxterm is 0, so F = 0.
On any other row, every maxterm in the product is 1, so F = 1.

So the product reproduces the truth table exactly, just as the sum of minterms does from the 1s.

Σm lists the 1s; ΠM lists the 0s. Every row is one or the other, so for n inputs the two lists split the rows 0 to 2ⁿ − 1 between them. Converting is just taking the other rows; see canonical form conversion.

Like every canonical form, it is unique for each function and always easy to write, but rarely cheap: each maxterm has n literals. It is the starting point for finding a minimal pos.

With don't-cares, list only the required 0s in ΠM and carry the don't-cares separately: F = ΠM(…) · d(…). See dont care condition.


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

Worked examples

Example

From a list to an expression

Write F(A, B, C) = ΠM(1, 2, 6) in full.

1.
1 = 001 → M1 = .
2.
2 = 010 → M2 = .
3.
6 = 110 → M6 = .
4.
F = . The table above is 0 exactly on rows 1, 2 and 6.

Example

From a truth table to ΠM

F(A, B, C) is 0 on rows 3 and 7 and 1 everywhere else. Write its canonical POS, then simplify.

1.
F = ΠM(3, 7).
2.
3 = 011 → M3 = . 7 = 111 → M7 = .
3.
F = .
4.
The two maxterms differ only in A, so they combine: F = .

Common mistakes

Listing the 1-rows in ΠM. ΠM lists the rows where F = 0.
Putting don't-care rows into the ΠM list. That forces them to 0; keep them in d(…).
Writing each maxterm with minterm polarity.

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