Minterm expansion: the canonical sum of products (Σm)
Minterm expansion rewrites a formula as the OR of one AND term per truth-table row where it is true: the canonical sum of products, written Σm(…).
The formula is rewritten as the OR of one AND term (minterm) for each truth-table row where it is true.
A minterm is an AND of every variable, each either plain or negated, so it is true on exactly one row. OR-ing the minterms of the true rows reproduces the function, and the list of their row numbers is the compact name Σm(1, 2) that the engine accepts as input. The example on this page, A XOR B = Σm(A, B; 1, 2), lists rows 1 (A false, B true) and 2 (A true, B false).
The expansion is canonical: two formulas are equivalent exactly when they have the same minterms over the same variable order. That is why the truth table generator prints the Σm form, why the Karnaugh map labels each cell with its minterm number, and why the normal-form converter offers a canonical method that produces exactly this expansion, beside its algebraic method and its prime-implicant fallback.
The convention for row numbers is the engine's: the first variable is the most significant bit and row 0 is all FALSE, the same as the truth table pages.
Statement
A XOR B = Σm(A, B; 1, 2) = (NOT A AND B) OR (A AND NOT B) Proof by truth table: A XOR B = Σm(A, B; 1, 2) = (NOT A AND B) OR (A AND NOT B)
| # | A | B | lhs | mid1 | rhs |
|---|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | TRUE | TRUE | TRUE |
| 2 | TRUE | FALSE | TRUE | TRUE | TRUE |
| 3 | TRUE | TRUE | FALSE | FALSE | FALSE |
The same law in four notations
A XOR BA XOR BA ⊕ BA ⊕ BA \oplus BΣm(A, B; 1, 2)Σm(A, B; 1, 2)Σm(A, B; 1, 2)Σm(A, B; 1, 2)\sum m_{(A, B)}(1, 2)(NOT A AND B) OR (A AND NOT B)(!A && B) || (A && !B)(¬A ∧ B) ∨ (A ∧ ¬B)(A'·B) + (A·B')\left(\overline{A} \cdot B\right) + \left(A \cdot \overline{B}\right)The common slip
Numbering rows with the wrong variable order. Σm(A, B; 1) and Σm(B, A; 1) are different functions; the order in the list is part of the notation.
Questions
How many minterms can a function have?
Up to 2 to the power of the number of variables, one per row. A function with n variables that is true on half the rows has 2 to the n minus 1 minterms, which is why the canonical form is rarely the shortest and why the simplifier and the map exist.
Related
Guides
Lesson that teaches this
Tools
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Logical equivalence checker: two expressions, exact verdict, counterexample
Practice
See also
- Minterm: meaning and examples
- Maxterm expansion: the canonical product of sums (ΠM)
- Σm(1, 2, 4, 7): canonical sum of products for three variables
- Minterms vs maxterms
- Lesson 10: Normal forms: DNF, CNF, minterms, and maxterms (advanced, 15 min)
Laws
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.