F(A,B,C) = Σm(1,3,5,7) — SOP and POS
introΣm and ΠM of the same function
Answer
Σm(1,3,5,7) = ΠM(0,2,4,6)
Why this example is worth doing
Odd minterms only, which is the set where C is 1, so the function is just C. The page uses it to nail the index convention that everything else depends on: A is the most significant bit, so minterm 5 is A=1, B=0, C=1 and reads A·B̄·C. Maxterm 5 over the same variables is (Ā + B + C̄) — the bits invert. Getting that inversion wrong is the single most common error in POS work, and it is silent, because the result still looks like a plausible expression.
Try your own input in the SOP & POS canonical forms. Expand to canonical minterms and maxterms, and convert between the two.
How the answer is reached
Canonical sum of products
A′ · B′ · C + A′ · B · C + A · B′ · C + A · B · C
Canonical product of sums
(A + B + C) · (A + B′ + C) · (A′ + B + C) · (A′ + B′ + C)
Truth table
| # | A | B | C | F |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 |
| 2 | 0 | 1 | 0 | 0 |
| 3 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 1 |
| 6 | 1 | 1 | 0 | 0 |
| 7 | 1 | 1 | 1 | 1 |