Disjunctive normal form (DNF): meaning and examples

Disjunctive normal form (DNF) is an OR of terms, each term an AND of literals. Every formula has an equivalent DNF, and the conversion is a fixed sequence of laws: eliminate the derived operators, push negations inward, distribute AND over OR.

A formula is in disjunctive normal form when it is a disjunction of conjunctions of literals; a single term and a single literal count as degenerate cases. DNF is the logician's name for sum of products. It is called normal because the shape is fixed and the translation is mechanical: implications and biconditionals are rewritten, De Morgan moves every NOT onto a variable, and distribution multiplies out. The result is not unique; the canonical DNF, made of minterms, is the unique one.

The artefact on this page converts A -> (B AND C). The converter's DNF is NOT A OR (B AND C), 2 terms and 3 literals, reached with one application of implication elimination; the first term is the single literal NOT A. Its CNF is (NOT A OR B) AND (NOT A OR C), 2 clauses and 4 literals, which needs distribution as well. The 8-row table reads true on rows 0 to 3 and row 7, and the tautology check is violated with the counterexample A = TRUE, B = FALSE, C = FALSE: antecedent true, consequent false.

DNF is the form in which a condition is easiest to read as cases (“either A is false, or both B and C hold”) and easiest to test, since each term is a scenario. It is also the shape of a two-level AND-OR circuit. The term is confused with CNF, its dual, and a DNF is sometimes assumed to be short; the converter's limits of 4,096 terms and 20,000 literals exist because the distribution step can blow up.

Example

f := A -> (B AND C) · output

Exact checks

Always true (tautology)? No Counterexample: A = T, B = F, C = F · Exhaustively checked and solver cross-checked · 5 rows

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

Can be true? Yes Example assignment: A = F, B = F, C = F · Exhaustively checked and solver cross-checked · 1 rows

Truth table (8 rows)

Disjunctive normal form (DNF): meaning and examples
#	A	B	C	f
0	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	TRUE	TRUE
2	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	FALSE
5	TRUE	FALSE	TRUE	FALSE
6	TRUE	TRUE	FALSE	FALSE
7	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when the following “if … then” condition holds:
1. if: A is true
2. then: both B and C are true
In every other case, f is false.
About the operators:
- An “if … then” condition only constrains the case where its “if” part holds: when the “if” part is false, the condition is satisfied whatever the “then” part is. It does not say the reverse.
- AND is true only when all of its parts are true.

Intermediatef is true exactly when the following “if … then” condition holds:
1. if: A is true
2. then: both B and C are true
In every other case, f is false.

Advancedf is true exactly when the following “if … then” condition holds [material implication →]:
1. if: A is true
2. then: both B and C are true [conjunction ∧]
In every other case, f is false.
Formal notation: A → (B ∧ C)

Normal forms

Conjunctive normal form (CNF)

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

2 clauses, 4 literals · complete · Implication elimination ×1, Distribution ×1

Disjunctive normal form (DNF)

NOT A OR (B AND C)

2 terms, 3 literals · complete · Implication elimination ×1

The common slip

Stopping before the negations are pushed inward. NOT (A AND B) is not in DNF even though it has no OR; De Morgan must first turn it into NOT A OR NOT B. A normal form allows NOT only directly on a variable.

Questions

Why does the converter offer three methods?

The algebraic method applies laws step by step, as on this page. The canonical method reads the minterms off the truth table. The prime-implicant fallback is used when the algebraic expansion would exceed the term or literal limits. All three results are checked equivalent to the input.

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.