De Morgan's Laws (Logic)/Disjunction of Negations

Formulation 1

 * $\neg p \lor \neg q \dashv \vdash \neg \left({p \land q}\right)$

Formulation 2

 * $\left({\neg p \lor \neg q}\right) \iff \left({\neg \left({p \land q}\right)}\right)$

Proofs
By the tableau method of natural deduction:

Proof by Truth Table
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all models.

$\begin{array}{|ccccc||cccc|} \hline \neg & p & \lor & \neg & q & \neg & (p & \land & q) \\ \hline T & F & T & T & F & T & F & F & F \\ T & F & T & F & T & T & F & F & T \\ F & T & T & T & F & T & T & F & F \\ F & T & F & F & T & F & T & T & T \\ \hline \end{array}$

Also see

 * De Morgan's Laws (Set Theory) for a set theoretic application of these laws.
 * De Morgan's Laws (Predicate Logic) for a predicate logic application of these laws.