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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.
(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.
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician.He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the underlying principles of which he formalized. [1]
In logic, a rule of replacement [1] [2] [3] is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a ...
The second De Morgan's law, (¬x) ∨ (¬y) = ¬(x ∧ y), works the same way with the two diagrams interchanged. The first complement law, x ∧ ¬x = 0, says that the interior and exterior of the x circle have no overlap. The second complement law, x ∨ ¬x = 1, says that everything is either inside or outside the x circle.
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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