In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. 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.
The negation of a disjunction is the conjunction of the negations.
The rules can be expressed in formal language with two propositions P and Q as:
where:
- ¬ is the negation operator (NOT)
- is the conjunction operator (AND)
- is the disjunction operator (OR)
- ⇔ is a metalogical symbol meaning "can be replaced in a logical proof with"
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.
Read more about De Morgan's Laws: Formal Notation, History, Informal Proof, Formal Proof, Extensions
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