Disjunctive Normal Form/Examples

Examples of Disjunctive Normal Form

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.