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

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Theorem

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


This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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


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.

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

$\blacksquare$


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