# Definition:Disjunctive Normal Form

## 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

### Arbitrary Example $1$

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

is in disjunctive normal form.

### Arbitrary Example $2$

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

is not in disjunctive normal form because there is a disjunction buried in the second conjunction.

### Arbitrary Example $3$

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

is not in disjunctive normal form because the second conjunction is negated.

### Arbitrary Example $4$

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

is in disjunctive normal form.

It is immediate that the above forms a contradiction.

### Disjunction

$p \lor q$

is in disjunctive normal form, as it is a disjunction of literals.

### Conjunction

$p \land q$

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

## Also defined as

When presenting disjunctive normal form, 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

Disjunctive normal form is often found referred to in its abbreviated form DNF or dnf.

## Also see

• Results about disjunctive normal form can be found here.