Every Boolean function can be written two equally valid ways: as a sum-of-products (SOP) or a product-of-sums (POS). Neither is "more correct" — they're the same function, just built from opposite starting points, and one of them is usually smaller than the other for any given function. Knowing both, and knowing which one to check first, saves you from implementing a circuit that's bigger than it needed to be.
Sum-of-products (SOP)
An SOP expression is a set of AND terms combined with OR — for example, F = AB + A′C. It's built by grouping the 1s on a K-map exactly as covered in the Karnaugh map guide: each group becomes one AND term, and the terms are OR'd together. As a circuit, this is literally what it looks like — one AND gate per term, feeding into a single OR gate.
Product-of-sums (POS)
A POS expression is the mirror image: OR terms combined with AND — for example, F = (A+B)(A′+C). It's built by grouping the 0s instead of the 1s, using the identical adjacency and grouping rules, and then applying De Morgan's theorem to each resulting term: the group's literals get complemented and the implicit AND between them becomes an OR. As a circuit, it's OR gates feeding into a single final AND gate.
A worked comparison, same function
Take F(A,B,C) = Σm(0,3,4,5,6,7) — six of the eight possible minterms are 1, with only minterms 1 and 2 as zeros.
Grouping the 1s gives three essential terms: A (covering 4,5,6,7), B′C′ (covering 0,4), and BC (covering 3,7) — an SOP of A + B′C′ + BC, three terms and five literals total.
Grouping the 0s instead — just two isolated cells, 001 and 010, too far apart to combine with each other — gives two maxterm groups, each covering a single cell. Converted via De Morgan, that's a POS of (A + B + C′)(A + B′ + C), two terms but six literals total.
Neither one is "wrong." SOP has fewer literals here; POS has fewer terms. Which one actually makes a smaller circuit depends on what you're optimizing for and how your gates are structured — which is exactly why it's worth generating both instead of assuming one automatically wins.
Which one should you use?
There's no universal answer — it depends entirely on which set (the 1s or the 0s) happens to group more efficiently for a given function, and that's specific to each problem. In practice: compute both, compare literal and gate counts, and pick whichever is actually smaller for your case, rather than defaulting to SOP out of habit just because it's the more commonly taught form.