Term (product term): meaning and examples

A term, or product term, is an AND of literals, such as A AND NOT B. A sum of products is an OR of terms, and a term is true only on the rows where every one of its literals is true.

In the algebraic notation that writes AND as multiplication and OR as addition, A AND NOT B is the product AB′, hence “product term”. A term with every variable of the function present is a minterm, true on exactly one row; a term with fewer literals covers several rows, two for each variable it omits. Terms are the dual of clauses: a clause is an OR of literals and is falsified only when all its literals are false, while a term is an AND of literals and is satisfied only when all of them are true.

The artefact on this page is (A AND NOT B) OR (B AND C), an OR of the two terms A AND NOT B and B AND C. The converter reports it already in disjunctive normal form, 2 terms, 4 literals, no law applied, and derives the conjunctive form (A OR B) AND (A OR C) AND (NOT B OR C), 3 clauses and 6 literals, by one distribution and one complement step. The satisfiable check holds with the witness A = FALSE, B = TRUE, C = TRUE, a row where the second term is true.

Terms are what a Karnaugh map produces: each group of cells becomes one term, and the derived expression is the OR of the groups. In hardware a term is one AND gate feeding the final OR, so the term count of a sum of products is a gate count. The term is confused with a clause (the OR counterpart), with a minterm (a term that is as long as possible), and with “term” in the everyday sense of a word or expression.

Example

f := (A AND NOT B) OR (B AND C) · output

Exact checks

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

Always false (contradiction)? No Counterexample: A = F, B = T, C = T · Exhaustively checked and solver cross-checked · 4 rows

Can be true? Yes Example assignment: A = F, B = T, C = T · Exhaustively checked and solver cross-checked · 4 rows

Truth table (8 rows)

Term (product term): meaning and examples
#	A	B	C	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE
2	FALSE	TRUE	FALSE	FALSE
3	FALSE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	TRUE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	FALSE
7	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when at least one of the following holds (both may hold):
1. A is true and B is false
2. both B and C are true
In every other case, f is false.
About the operators:
- OR is inclusive: it is true when at least one part is true, including when several are.
- AND is true only when all of its parts are true.
- NOT reverses a value: true becomes false and false becomes true.

Intermediatef is true exactly when at least one of the following holds (both may hold):
1. A is true and B is false
2. both B and C are true
In every other case, f is false.

Advancedf is true exactly when at least one of the following holds (both may hold) [inclusive disjunction ∨]:
1. A is true and B is false [conjunction ∧]
2. both B and C are true [conjunction ∧]
In every other case, f is false.
Formal notation: (A ∧ ¬B) ∨ (B ∧ C)

Normal forms

Conjunctive normal form (CNF)

(A OR B) AND (A OR C) AND (NOT B OR C)

3 clauses, 6 literals · complete · Distribution ×1, Complement ×1

Disjunctive normal form (DNF)

(A AND NOT B) OR (B AND C)

2 terms, 4 literals · complete

The common slip

Assuming a term must mention every variable. B AND C is a term of a three-variable function even though it omits A; it covers two rows, one for each value of A. Only a minterm has to mention them all.

Questions

Why does a sum of products have fewer terms when its terms are shorter?

Each literal dropped from a term doubles the rows it covers, so shorter terms cover more of the function each. The Karnaugh map grouping rule is exactly this trade: a larger group is a shorter term.

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.