De Morgan's Laws (Logic)/Disjunction/Formulation 1

Theorem

$p \lor q \dashv \vdash \neg \paren {\neg p \land \neg q}$

This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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

Proof

We apply the Method of Truth Tables.

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

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

$\blacksquare$