Disjunctive Normal Form/Examples
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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.