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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.
For example, LC is known not to prove all theorems of SmL, but it does not directly compare in strength to BD 2. Likewise, e.g., KP does not compare to SL. The list of equalities for each logic is by no means exhaustive either. For example, as with WPEM and De Morgan's law, several forms of DGP using conjunction may be expressed.
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).
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
An XOR gate is made by considering the conjunctive normal form (+) (¯ + ¯), noting from de Morgan's Law that a NOR gate is an inverted-input OR gate. This construction entails a propagation delay three times that of a single NOR gate and uses five gates.
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In classical logic each propositional formula can be converted to an equivalent formula that is in CNF. [1] This transformation is based on rules about logical equivalences: double negation elimination, De Morgan's laws, and the distributive law.