---
tags:
  - Logic
  - propositional-logic
  - derivation-rules
  - theorems-axioms-laws
---

DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](Truth-functional%20connectives.md):

## First Law

> The negation of a conjunction is logically equivalent to the disjunction of the negations of the original conjuncts.

$$
\sim (P \& Q) \equiv \sim P \lor \sim Q
$$

The equivalence is demonstrated with the following truth-table

![demorgan-1.png](../img/demorgan-1.png)

## Second Law

> The negation of a disjunction is equivalent to the conjunction of the negation of the original disjuncts.

$$
\sim (P \lor Q) \equiv \sim P & \sim Q
$$

![demorgan-2.png](../img/demorgan-2.png)