Definition:Disjunctive Normal Form

Definition
A propositional formula $P$ is in disjunctive normal form it consists of a disjunction of:
 * $(1):\quad$ conjunctions of literals

and/or:
 * $(2):\quad$ literals.

Examples

 * $\paren {\neg p \land q \land r} \lor \paren {\neg q \land r} \lor \paren {\neg r}$

is in DNF.


 * $\paren {\neg p \land q \land r} \lor \paren {\paren {p \lor \neg q} \land r} \lor \paren {\neg r}$

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


 * $\paren {\neg p \land q \land r} \lor \neg \paren {\neg q \land r} \lor \paren {\neg r}$

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.

Also defined as
Some sources include parentheses as appropriate within both the conjunctions and disjunctions in a standard format, for example:
 * $\paren {\paren {\paren {\neg p \land q} \land r} \lor \paren {\neg q \land r} } \lor \paren {\neg r}$

but this is usually considered unnecessary in light of the Rule of Distribution.

Also known as
This is often found referred to in its abbreviated form DNF.

Also see

 * Definition:Conjunctive Normal Form
 * Definition:Negation Normal Form


 * Existence of Disjunctive Normal Form of Statement