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3.1.10 · D5Boolean Algebra & Logic Gates

Question bank — Product of sums (POS) form

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Figure — Product of sums (POS) form

The colored strip in the figure is the whole game: for row 5 () each variable looks at its own bit and dresses accordingly — the -bits () get a bar, the -bit () stays plain — so every literal reads there and their OR collapses to at that one row only.

True or false — justify

TF1. A product of sums must always contain a maxterm for every variable in each term.
False — only the canonical POS uses full maxterms. A simplified POS (e.g. from a K-map) can have short sum terms like with a variable missing.
TF2. The number of factors in a canonical POS equals the number of rows where .
False — it equals the number of rows where . POS is built from zeros; rows feed the SOP form instead.
TF3. If a function is for every input row, its canonical POS is the empty product, which equals .
True — there are no rows, so no maxterms exist; an AND of nothing is the identity for AND, namely , matching .
TF4. A single maxterm on its own is already a valid POS.
True — "product of one or more sum terms" includes a product of exactly one, so alone is a legal (single-factor) POS.
TF5. Swapping every literal in a canonical POS to its complement gives the canonical SOP of the same function.
False — complementing literals inside maxterms and turning ORs into ANDs gives the SOP of (via De Morgan), not of . The forms of use different rows entirely.
TF6. For a 3-variable function, the maxterm list and the minterm list together always cover all 8 row numbers with no overlap.
True — every row is either (a maxterm) or (a minterm), never both, so the two index sets partition .
TF7. and (same index ) are complements of each other.
True — . The minterm is only at row ; its complement is only at row , which is exactly the maxterm.

Spot the error

Figure — Product of sums (POS) form
SE1. "Maxterm for row is ." — find the mistake.
Wrong literals: you complement a variable when it is 1 in the row. Here are and is , so it must be .
SE2. " for the function that is at rows ." — spot the slip.
The student listed the rows. takes the rows, so the correct form is . They accidentally wrote the minterm list.
SE3. " is a valid sum term in a POS." — what's broken?
A sum term is a pure OR of literals; the AND () inside destroys the canonical form. It should be flattened, e.g. into .
SE4. "To convert SOP to POS, just AND the minterms instead of ORing them." — why is this wrong?
ANDing minterms does not give POS — it gives almost everywhere. The real bridge is De Morgan on : , which changes both the operation and the terms.
SE5. "Row 0 () has a maxterm because all bits are 0." — correct it.
Backwards: at row every variable is , and a plain literal is when the variable is , so no complements are needed — the maxterm is .
SE6. "A maxterm outputs at exactly one row." — fix the statement.
A maxterm outputs at exactly one row and at all the others. It's the minterm that is at exactly one row.

Why questions

WHY1. Why do we AND maxterms rather than OR them?
AND returns if any factor is . Each maxterm is at its own forbidden row, so ANDing lets any single forbidden row drive the output to while every other row keeps all factors at .
WHY2. Why must a literal be complemented exactly when its variable is in the target row?
An OR term is only when every literal in it is . A literal is at the target row only if it is when (or plain when ) — so complementing the -valued variables forces the whole OR to there.
WHY3. Why does canonical POS use the rows where and not where ?
Those rows are precisely the minterms of . Taking De Morgan on turns that minterm sum into a product of maxterms over the same rows.
WHY4. Why does De Morgan's theorem naturally connect SOP and POS?
De Morgan turns a complemented OR into an AND of complements (and vice versa). Since maxterms are complemented minterms, De Morgan is exactly the operation that flips a sum-of-minterms into a product-of-maxterms.
WHY5. Why does a POS circuit map to an AND-of-ORs (two-level) gate layout?
Each sum term is an OR gate; ANDing all sum terms is one final AND gate. So level 1 is ORs, level 2 is a single AND — exactly the schematic drawn on the right of the figure above (see Logic gates).
WHY6. Why can two different-looking POS expressions describe the same function?
Because simplification (algebra or K-map grouping) can merge or drop factors while preserving the same set of rows. Only the canonical POS is unique.

Edge cases

EC1. What is the canonical POS of a function that is for every input?
It is the product of all maxterms, one per row. Their AND is everywhere, correctly giving .
EC2. What is the canonical POS of the constant function ?
There are no rows, so the product is empty; an empty AND is , matching the function.
EC3. A single-variable function is only at . What is its POS?
Only one row: at the maxterm is . So — a degenerate one-literal sum term is still a legal maxterm.
EC4. If a function's truth table has a "don't-care" row, how does it affect the canonical POS?
A don't-care can be chosen as (include its maxterm) or (omit it), whichever yields a simpler expression. In the strict canonical form you must first commit each don't-care to a definite value.
EC5. For an -variable function, what is the largest possible number of factors in a canonical POS?
— the case where at every one of the rows, so every maxterm appears.
EC6. Can a canonical POS and canonical SOP of the same function have the same number of terms?
Yes — if exactly half the rows are (so zeros and ones), both forms have terms. It's a coincidence of the row split, not a rule.
EC7. What does a maxterm look like at a row other than its own — how many literals are there?
At any non-forbidden row, at least one literal is , so the OR evaluates to . That single is enough to keep the maxterm from tripping.

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