De Morgan's laws

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

The rules can be expressed in English as:

the negation of a conjunction is the disjunction of the negations; and
the negation of a disjunction is the conjunction of the negations;

or

the complement of the union of two sets is the same as the intersection of their complements; and
the complement of the intersection of two sets is the same as the union of their complements.

In set theory and Boolean algebra, these are written formally as

where

In formal language, the rules are written as

and

where

Applications of the rules include simplification of logical expressions in computer programs and digital circuit designs. De Morgan's laws are an example of a more general concept of mathematical duality.

The negation of conjunction rule may be written in sequent notation:

The negation of disjunction rule may be written as:

In rule form: negation of conjunction

and negation of disjunction

and expressed as a truth-functional tautology or theorem of propositional logic:

where ${\displaystyle P}$ and ${\displaystyle Q}$ are propositions expressed in some formal system.

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