Definition:Disjunctive Normal Form

Definition

A logical formula $P$ is in disjunctive normal form (abbreviated DNF) if it consists of a disjunction of:

• conjunctions of literals, and/or
• literals.

Examples

$\left({\neg p \land q \land r}\right) \lor \left({\neg q \land r}\right) \lor \left({\neg r}\right)$

is in DNF.

$\left({\neg p \land q \land r}\right) \lor \left({\left({p \lor \neg q}\right) \land r}\right) \lor \left({\neg r}\right)$

is not in DNF because there is a disjunction buried in the second conjunction.

$\left({\neg p \land q \land r}\right) \lor \neg \left({\neg q \land r}\right) \lor \left({\neg r}\right)$

is not in DNF because the second conjunction is negated.

$p \lor q$

is in DNF, as it is a disjunction of literals.

$p \land q$

is in DNF, as it is a trivial (one-element) disjunction of a conjunction of literals.

See also

• Conjunctive Normal Form (CNF)
• Negation Normal Form (NNF)

Sources

• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 5$: Exercises: Group $\text{III}$