|
In propositional logic and boolean algebra, De Morgan's laws〔Copi and Cohen〕〔Hurley〕〔Moore and Parker〕 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. or informally as: "''not (A and B)''" is the same as "''(not A) or (not B)''" The rules can be expressed in formal language with two propositions ''P'' and ''Q'' as: : and : where: * 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. ==Formal notation== 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「De Morgan's laws」の詳細全文を読む スポンサード リンク
|