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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).
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.
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.
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.
De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = ([0, 1], max( x , y ), min( x , y ), 0, 1, 1 − x ) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold.
Davis's law, in anatomy, describes how soft tissue models along imposed demands. Corollary to Wolff's law. De Morgan's laws apply to formal logic regarding the negation of pairs of logical operators. Dermott's law: The sidereal period of major satellites tends to follow a geometric series. Named after Stanley Dermott.
Negation normal form is not a canonical form: for example, () and () are equivalent, and are both in negation normal form. 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 ...