45 lines
1.4 KiB
Markdown
45 lines
1.4 KiB
Markdown
|
---
|
||
|
|
tags:
|
||
|
|
- theorems
|
||
|
|
- logic
|
||
|
|
- propositional-logic
|
||
|
|
---
|
||
|
|
|
||
|
|
# DeMorgan's Laws
|
||
|
|
|
||
|
|
DeMorgan's laws express some fundamental equivalences that obtain between the
|
||
|
|
Boolean
|
||
|
|
[connectives](Truth-functional_connectives.md).
|
||
|
|
|
||
|
|
## 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 |
|