De Morgan's laws

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De Morgan's laws

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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 disjunction is the conjunction of the negations; andthe negation of a conjunction is the disjunction of the negations; or the complement of the union of two sets is the same as the intersection of their complements; andthe complement of the intersection of two sets is the same as the union of their complements. or not (A or B) = not A and not B; andnot (A and B) = not A or not B In set theory and Boolean algebra, these are written formally as where * A and B are sets, * A is the complement of A, * ∩ is the intersection, and * ∪ is the union. In formal language, the rules are written as and where * P and Q are propositions, * is the negation logic operator (NOT), * is the conjunction logic operator (AND), * is the disjunction logic 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.