Minterms vs maxterms
A minterm names a TRUE row; a maxterm names a FALSE row. Every row is one or the other, so Σm(A, B, C; 2, 3, 6) and ΠM(A, B, C; 0, 1, 4, 5, 7) describe the same function, three minterms against five maxterms, and the Karnaugh map reduces it to two terms.
The function is TRUE on rows 2, 3 and 6 of eight. Its minterm expansion has three AND-terms, one per TRUE row, each listing every variable plain or negated; its maxterm expansion has five OR-clauses, one per FALSE row, each negating the variables that are TRUE on that row. The engine prints both lines, and the two index lists are complementary: 2, 3, 6 and 0, 1, 4, 5, 7 together are all eight rows.
Which expansion is shorter is a count of rows: three TRUE rows against five FALSE, so the minterm form wins here. A function TRUE on most rows would be the other way round. Neither canonical form is short in absolute terms; the Karnaugh map derives (NOT A AND B) OR (B AND NOT C), two prime implicants, and the gate counts the engine reports for the canonical expansion, thirty-five NAND and seventy-seven NOR, are the price of building without reducing.
The two words carry a convention that trips people: minterm m2 is the term TRUE only on row 2, NOT A AND B AND NOT C; maxterm M2 is the clause FALSE only on row 2, A OR NOT B OR C, the negation of m2. The signs inside a maxterm are the opposite of the row's values, which is why the lesson on normal forms spends a step on each.
Example
f := Σm(A, B, C; 2, 3, 6) · output
Exact checks
Truth table (8 rows)
| # | A | B | C | f |
|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | FALSE | TRUE | FALSE |
| 2 | FALSE | TRUE | FALSE | TRUE |
| 3 | FALSE | TRUE | TRUE | TRUE |
| 4 | TRUE | FALSE | FALSE | FALSE |
| 5 | TRUE | FALSE | TRUE | FALSE |
| 6 | TRUE | TRUE | FALSE | TRUE |
| 7 | TRUE | TRUE | TRUE | FALSE |
Logic diagram
Karnaugh map
- cells m2, m3
NOT A AND B - cells m2, m6
B AND NOT C
Minterms and maxterms
f(A, B, C) = Σm(2, 3, 6)f(A, B, C) = ΠM(0, 1, 4, 5, 7)Variable order: A, B, C · 8 rows
Normal forms
Conjunctive normal form (CNF)
(A OR B OR C) AND (A OR B OR NOT C) AND (NOT A OR B OR C) AND (NOT A OR B OR NOT C) AND (NOT A OR NOT B OR NOT C)5 clauses, 15 literals · complete · Minterm expansion ×1
Disjunctive normal form (DNF)
(A AND B AND NOT C) OR (NOT A AND B AND C) OR (NOT A AND B AND NOT C)3 terms, 9 literals · complete · Minterm expansion ×1
NAND-only and NOR-only
NAND-only form
NAND(NAND(NAND(NAND(NAND(NAND(NAND(A, A), B), NAND(NAND(A, A), B)), NAND(C, C)), NAND(NAND(NAND(NAND(A, A), B), NAND(NAND(A, A), B)), C)), NAND(NAND(NAND(NAND(NAND(A, A), B), NAND(NAND(A, A), B)), NAND(C, C)), NAND(NAND(NAND(NAND(A, A), B), NAND(NAND(A, A), B)), C))), NAND(NAND(NAND(A, B), NAND(A, B)), NAND(C, C)))35 gates · verified equivalent on every row
NOR-only form
NOR(NOR(NOR(NOR(NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), C), NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), NOR(C, C))), NOR(NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), C), NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), NOR(C, C)))), NOR(NOR(NOR(NOR(A, A), NOR(B, B)), NOR(NOR(A, A), NOR(B, B))), C)), NOR(NOR(NOR(NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), C), NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), NOR(C, C))), NOR(NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), C), NOR(NOR(NOR(A, NOR(B, B)), NOR(A, NOR(B, B))), NOR(C, C)))), NOR(NOR(NOR(NOR(A, A), NOR(B, B)), NOR(NOR(A, A), NOR(B, B))), C)))77 gates · verified equivalent on every row
The common slip
Building a maxterm with the signs of a minterm. For row 2 (A FALSE, B TRUE, C FALSE) the minterm is NOT A AND B AND NOT C and the maxterm is A OR NOT B OR C; using the minterm's signs in an OR gives a clause that is FALSE on row 5 instead.
Questions
Why Σ and Π?
Σ is a sum, the OR of the minterms, and Π a product, the AND of the maxterms, in the algebra notation where OR is + and AND is ·. The engine accepts minterms(…) and maxterms(…) as plain-text spellings.
Can I enter either form directly?
Yes: Σm(A, B, C; 2, 3, 6) and ΠM(A, B, C; 0, 1, 4, 5, 7) are both accepted formulas, with the variable order inside the brackets fixing which row each index names.
Related
Practice
See also
- Minterm: meaning and examples
- Maxterm: meaning and examples
- Minterm expansion: the canonical sum of products (Σm)
- Maxterm expansion: the canonical product of sums (ΠM)
- Sum of products vs product of sums
- Lesson 10: Normal forms: DNF, CNF, minterms, and maxterms (advanced, 15 min)
- CNF and DNF converter: conjunctive and disjunctive normal forms, minterms and maxterms
- Σm(1, 2, 4, 7): canonical sum of products for three variables
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