Canonical & Standard Forms
Standard ways to write any function: sum of minterms or product of maxterms.
Description
Uniform expression formats: sum-of-products (SOP) and product-of-sums (POS). A canonical form is unique per function — ideal for comparison and tool input. Write one minterm per 1-row (SOP), or one maxterm per 0-row (POS).
- A minterm is an AND of all variables (true or complemented) that is 1 for exactly one row.
- F = sum (OR) of the minterms where F = 1 (the Σm form).
- A maxterm is an OR of all variables that is 0 for exactly one row.
- F = product (AND) of the maxterms where F = 0 (the ΠM form).
- What: Uniform expression formats: sum-of-products (SOP) and product-of-sums (POS).
- Why: A canonical form is unique per function — ideal for comparison and tool input.
- How: Write one minterm per 1-row (SOP), or one maxterm per 0-row (POS).
- Where: Truth-table-to-circuit conversion, PLA/ROM programming, formal equivalence checks.
- When: Right after the truth table, before minimization.
- Analogy — Minterms are like listing every guest who said YES (one exact name per yes-row); maxterms list every NO. Either complete guest list pins down the party exactly.
At a glance
What
Uniform expression formats: sum-of-products (SOP) and product-of-sums (POS).
Why
A canonical form is unique per function — ideal for comparison and tool input.
How
Write one minterm per 1-row (SOP), or one maxterm per 0-row (POS).
Where
Truth-table-to-circuit conversion, PLA/ROM programming, formal equivalence checks.
When
Right after the truth table, before minimization.
Think of it like…
Minterms are like listing every guest who said YES (one exact name per yes-row); maxterms list every NO. Either complete guest list pins down the party exactly.
Minterms (SOP)
- A minterm is an AND of all variables (true or complemented) that is 1 for exactly one row.
- F = sum (OR) of the minterms where F = 1 (the Σm form).
Maxterms (POS)
- A maxterm is an OR of all variables that is 0 for exactly one row.
- F = product (AND) of the maxterms where F = 0 (the ΠM form).
Minterm vs maxterm (2 variables)
| Row (AB) | Minterm | Maxterm |
|---|---|---|
| 00 | A′B′ (m0) | A + B (M0) |
| 01 | A′B (m1) | A + B′ (M1) |
| 10 | AB′ (m2) | A′ + B (M2) |
| 11 | AB (m3) | A′ + B′ (M3) |
Minterm = 1 for its row; maxterm = 0 for its row. They are complements.
The 5 Whys
- 1
Why canonical forms? To get one unique expression per function.
- 2
Why uniqueness? It enables direct comparison and equivalence checking.
- 3
Why SOP and POS both? Some functions are cheaper one way than the other.
- 4
Why tie them to truth-table rows? It makes conversion purely mechanical.
- 5
Root cause: anchoring expressions to minterms/maxterms removes all ambiguity from spec→form.
Cheat sheet
Working principle
- Write one minterm per 1-row (SOP), or one maxterm per 0-row (POS).
- Uniform expression formats: sum-of-products (SOP) and product-of-sums (POS).
Formulas & Boolean expressions
- F = sum (OR) of the minterms where F = 1 (the Σm form).
- F = product (AND) of the maxterms where F = 0 (the ΠM form).
Key facts
- A minterm is an AND of all variables (true or complemented) that is 1 for exactly one row.
- A maxterm is an OR of all variables that is 0 for exactly one row.
Why it exists
- Root cause: anchoring expressions to minterms/maxterms removes all ambiguity from spec→form.