De Morgan dual
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:
or
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 and are propositions expressed in some formal system.
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