DeMorgan’s Laws
DeMorgan’s laws express some fundamental equivalences that obtain between the Boolean Truth-functional_connectives.
First Law
The negation of a conjunction is logically equivalent to the disjunction of the negations of the original conjuncts.
$$ \lnot (P \land Q) \leftrightarrow \lnot P \lor \lnot Q $$
The equivalence is demonstrated with the following truth-table
| \(P\) | \(Q\) | $ \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 ### Truth conditions |
The negation of a disjunction is equivalent to the conjunction of the negation of the original disjuncts.
$$ \lnot (P \lor Q) \leftrightarrow \lnot P \land \lnot Q $$
| \(P\) | \(Q\) | $ \lnot (P \lor Q)$ | $ \lnot P \land \lnot Q$ |
|---|---|---|---|
| T | T | F | F |
| T | F | F | F |
| F | T | F | F |
| F | F | T | T |
