The sum of minterms is the canonical SOP form of a function: one minterm for every row where the function is 1, all ORed together.
The shorthand Σm lists only the row numbers. With inputs A, B, C (A is the MSB):
F = Σm(1, 2, 4) means F = m1 + m2 + m4 =
Why it always works: each minterm is 1 on its own row and 0 everywhere else. OR a set of them and the result is 1 on exactly those rows. So the sum of minterms reproduces the truth table perfectly. It isn't usually the smallest circuit, which is why the next step is simplification, but it is never wrong.
Useful facts:
- Uniqueness. Every function has exactly one minterm list, so comparing lists is a sure way to check whether two expressions are equal.
- The complement uses the rows missing from F's list. Over 3 variables, F = Σm(0, 1, 2, 3, 5) gives = Σm(4, 6, 7).
- The partner form is the product of maxterms ΠM, which lists the 0-rows. F = Σm(1, 2, 4) is the same function as ΠM(0, 3, 5, 6, 7).
- Hardware shortcut: a decoder produces every minterm, so a decoder and one OR gate build a sum of minterms directly (decoder implementation).
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 0 |