Minterm: meaning and examples

A minterm is an AND of every variable of a function, each plain or negated, so it is true on exactly one row of the truth table. A function is the OR of the minterms of its true rows.

A function of n variables has 2 to the power n minterms, one per row, and each is named by its row number: for variables A and B in that order, m0 is NOT A AND NOT B, m1 is NOT A AND B, m2 is A AND NOT B and m3 is A AND B. The OR of the minterms of the rows where a function is true is its canonical sum of products, written Σm(…); in the engine's notation the variable order is part of the name, as in Σm(A, B; 1, 2).

The artefact on this page is the function Σm(A, B; 1, 2) itself. Its 4-row table reads false, true, true, false, true on rows 1 and 2 as the list says, and the minterm panel prints f(A, B) = Σm(1, 2) beside the dual form f(A, B) = ΠM(0, 3), whose maxterm list is the complement of the minterm list. The satisfiable check holds with the witness A = FALSE, B = TRUE, which is row 1, minterm m1. The function is A XOR B under another name.

Minterms are the vocabulary of digital design: a specification given as a truth table is a list of minterms, a Karnaugh map labels its cells with minterm numbers, and the Quine–McCluskey method begins by listing them in binary. Programmers meet the same idea when a lookup table is written as a bit mask. The term is confused with a product term in general (a minterm is the special case that mentions every variable) and with a maxterm, which is its OR-shaped dual.

Example

f := Σm(A, B; 1, 2) · output

Exact checks

Always true (tautology)? No Counterexample: A = F, B = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Can be true? Yes Example assignment: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (4 rows)

Minterm: meaning and examples
#	A	B	f
0	FALSE	FALSE	FALSE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	TRUE
3	TRUE	TRUE	FALSE

Plain-language reading

Beginnerf is true exactly when the truth-table row of (A, B) is one of the rows 1 and 2.
In every other case, f is false.
About the operators:
- Σm lists the truth-table rows (minterms) where the formula is true. Rows are numbered from 0 with the first listed input as the most significant bit.

Intermediatef is true exactly when the truth-table row of (A, B) is one of the rows 1 and 2.
In every other case, f is false.

Advancedf is true exactly when the truth-table row of (A, B) is one of the rows 1 and 2 [canonical sum of minterms (DNF)].
In every other case, f is false.
Formal notation: Σm(A, B; 1, 2)

Minterms and maxterms

f(A, B) = Σm(1, 2)f(A, B) = ΠM(0, 3)

Variable order: A, B · 4 rows

The common slip

Forgetting that minterm numbers depend on the variable order. Σm(A, B; 1, 2) and Σm(B, A; 1, 2) happen to be the same function because A XOR B is symmetric, but Σm(A, B; 1) and Σm(B, A; 1) are not; the order in the list is part of the notation.

Questions

How do I get the minterms of a formula?

Build the truth table and list the rows where the output is true. The truth table generator prints the Σm form for every output, and the canonical method of the normal-form converter writes out the minterms themselves as an expression.

Guides

Encoders and decoders

Practice

CNF and DNF exercises with answers

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. 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.