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De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
English: De Morgan's Laws graphical representation. Español: Representación gráfica de las leyes de De Morgan. ... This diagram was created with Adobe Illustrator.
(i.e. an involution that additionally satisfies De Morgan's laws) In a De Morgan algebra, the laws ¬x ∨ x = 1 (law of the excluded middle), and; ¬x ∧ x = 0 (law of noncontradiction) do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra.
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Usage on de.wikipedia.org De-morgansche Gesetze; Usage on de.wikibooks.org Digitale Schaltungstechnik/ Schaltalgebra/ De Morgan; Usage on es.wikipedia.org Leyes de De Morgan; Usage on fr.wikibooks.org Électronique numérique : logique/Fonctions logiques élémentaires; Usage on hi.wikipedia.org डिमॉर्गन नियम
The principle of inclusion–exclusion, combined with De Morgan's law, can be used to count the cardinality of the intersection of sets as well. Let A k ¯ {\displaystyle {\overline {A_{k}}}} represent the complement of A k with respect to some universal set A such that A k ⊆ A {\displaystyle A_{k}\subseteq A} for each k .
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The result is the same as if we shaded that region which is both outside the x circle and outside the y circle, i.e. the conjunction of their exteriors, which is what the left hand side of the law describes. The second De Morgan's law, (¬x) ∨ (¬y) = ¬(x ∧ y), works the same way with the two diagrams interchanged.