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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.
De Morgan's laws: In propositional logic and Boolean algebra, De Morgan's laws, [15] [16] [17] also known as De Morgan's theorem, [18] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician.
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.
This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples. More generally, ∧ (¬ x i) = ¬ ∨ x i. The left side is true if and only if ∀i.¬x i, and the right side if and only if ¬∃i.x i.
[29] [1] Examples of statements might include: Wikipedia is a free online encyclopedia that anyone can edit. London is the capital of England. ... De Morgan's Theorem (2)
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.
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]
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.