Definition:Conjunctive Normal Form

Definition
A propositional formula $P$ is in conjunctive normal form it consists of a conjunction of:
 * $(1):\quad$ disjunctions of literals

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

Examples

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

is in CNF.


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

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


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

is not in CNF because the second disjunction is negated.


 * $p \land q$

is in CNF, as it is a conjunction of literals.


 * $p \lor q$

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

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

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 CNF.

Also see

 * Definition:Disjunctive Normal Form
 * Definition:Negation Normal Form


 * Existence of Conjunctive Normal Form of Statement