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Logic/Laws_and_theorems.md/DeMorgan's_Laws.md
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@ -6,26 +6,31 @@ tags: [logic, laws]
# DeMorgan's Laws # DeMorgan's Laws
DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](Truth-functional%20connectives.md): DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](/Logic/Propositional_logic/Truth-functional_connectives.md).
## First Law ## First Law
> The negation of a conjunction is logically equivalent to the disjunction of the negations of the original conjuncts. > The negation of a conjunction is logically equivalent to the disjunction of the negations of the original conjuncts.
$$ $$
\sim (P \& Q) \equiv \sim P \lor \sim Q \lnot (P \land Q) \leftrightarrow \lnot P \lor \lnot Q
$$ $$
The equivalence is demonstrated with the following truth-table The equivalence is demonstrated with the following truth-table
![demorgan-1.png](/img/demorgan-1.png) | $P$ | $Q$ | $ \lnot (P \land Q)$ | $ \lnot P \lor \lnot Q$ |
| --- | --- | -------------------- | ----------------------- |
| T | T | F | F |
| T | F | T | T |
| F | T | T | T |
| F | F | T | T |
## Second Law ## Second Law
> The negation of a disjunction is equivalent to the conjunction of the negation of the original disjuncts. > The negation of a disjunction is equivalent to the conjunction of the negation of the original disjuncts.
$$ $$
\sim (P \lor Q) \equiv \sim P \& \sim Q \lnot (P \lor Q) \leftrightarrow \lnot P \land \lnot Q
$$ $$
![demorgan-2.png](/img/demorgan-2.png) ![demorgan-2.png](/img/demorgan-2.png)