De Morgan's Laws (Logic)/Conjunction/Formulation 1
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Theorem
- $p \land q \dashv \vdash \neg \paren {\neg p \lor \neg q}$
This can be expressed as two separate theorems:
Forward Implication
- $p \land q \vdash \neg \left({\neg p \lor \neg q}\right)$
Reverse Implication
- $\neg \left({\neg p \lor \neg q}\right) \vdash p \land q$
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.
$\begin{array}{|ccc||cccccc|} \hline p & \land & q & \neg & (\neg & p & \lor & \neg & q) \\ \hline \F & \F & \F & \F & \T & \F & \T & \T & \F \\ \F & \F & \T & \F & \T & \F & \T & \F & \T \\ \T & \F & \F & \F & \F & \T & \T & \T & \F \\ \T & \T & \T & \T & \F & \T & \F & \F & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.4$: Relations between Truth-Functions
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $5$ Further Proofs: Résumé of Rules: Exercise $1 \ \text {(h)}$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.3$