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

Jump to navigation Jump to search

## Theorem

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

This can be expressed as two separate theorems:

### Forward Implication

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

### Reverse Implication

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

## 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.

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

$\blacksquare$