NAND as Disjunction of Negations

Theorem

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

Proof 1

\(\ds p \uparrow q\)

\(\dashv \vdash\)

\(\ds \map \neg {p \land q}\)

Definition of Logical NAND

\(\ds \)

\(\dashv \vdash\)

\(\ds \neg p \lor \neg q\)

De Morgan's Laws: Disjunction of Negations

$\blacksquare$

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 boolean interpretations.

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

$\blacksquare$

Sources

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants