# 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**.

## 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 $\text{III}$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**disjunctive normal form** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**disjunctive normal form**