F = (A + B)·(B + C) — SOP and POS
corePOS in, both canonical forms out
Answer
Σm(2,3,5,6,7) = ΠM(0,1,4)
Why this example is worth doing
Entered as a product of sums rather than a sum of products, because half of all problem sets on this topic start from POS and most tools quietly assume SOP. Each factor is missing a variable and expands by adding X·X̄ — the dual of the SOP expansion step, and the one textbooks explain in half a sentence. The page shows both expansions side by side so the symmetry is visible rather than asserted.
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′ + A · B · C
Canonical product of sums
(A + B + C) · (A + B + C′) · (A′ + B + C)
Truth table
| # | A | B | C | F |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 | 1 |
| 3 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 1 |
| 6 | 1 | 1 | 0 | 1 |
| 7 | 1 | 1 | 1 | 1 |