NAND as Disjunction of Negations/Proof by Truth Table

Theorem

$p \uparrow q \dashv \vdash \neg p \lor \neg q$

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.

$\begin{array}{|ccc||ccccc|} \hline p & \uparrow & q & \neg & p & \lor & \neg & q \\ \hline \F & \T & \F & \T & \F & \T & \T & \F \\ \F & \T & \T & \T & \F & \T & \F & \T \\ \T & \T & \F & \F & \T & \T & \T & \F \\ \T & \F & \T & \F & \T & \F & \F & \T \\ \hline \end{array}$

$\blacksquare$