# Definition:Disjunctive Normal Form

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## Definition

A propositional formula $P$ is in **disjunctive normal form** if and only if it consists of a disjunction of:

- $(1):\quad$ conjunctions of literals

and/or:

- $(2):\quad$ 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.

## Also defined as

Some sources include parentheses as appropriate within both the conjunctions and disjunctions in a standard format, for example:

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

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 **DNF**.

## Also see

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Exercises, Group $\text{III}$