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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.
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).
(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.
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 .
In classical logic and many modal logics, every formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inwards, and eliminating double negations.
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.
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Negated propositions are comparably weak, in that the classically valid De Morgan's law, granting a disjunction from a single negative hypothetical, does not automatically hold constructively. The intuitionistic propositional calculus and some of its extensions exhibit the disjunction property instead, implying one of the disjuncts of any ...