Minterm Σm(1, 3, 5, 6) equals to maxterm

Concept

Any Boolean function of n variables can be written either as a sum of minterms (the input rows where the output is 1) or, equivalently, as a product of maxterms (the input rows where the output is 0). The two lists are complements over the full set of 2n row indices: an index that is a minterm is NOT a maxterm, and vice versa. So to convert Σm to ΠM, list every index from 0 to 2n − 1 that does NOT appear in the minterm list.

Application

1. There are 3 variables here (the largest index, 6, needs 3 bits), so the full index set is 0, 1, 2, 3, 4, 5, 6, 7.

2. The minterms given are 1, 3, 5, 6 — these are the rows where the function is 1.

3. The maxterms are the remaining indices where the function is 0: 0, 2, 4, 7.

4. Therefore Σm(1, 3, 5, 6) = ΠM(0, 2, 4, 7).

Cross-check

Count check: 4 minterms + 4 maxterms = 8 = 23 total rows — consistent. Also verify disjointness: {1, 3, 5, 6} and {0, 2, 4, 7} share no index and together cover every value 0–7, confirming the complement is exact.