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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.
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.
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]
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.
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).
Thus 0 and 1 are dual, and ∧ and ∨ are dual. The duality principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged. One change not needed to make as part of this interchange was to complement. Complement is a self-dual operation.
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In general, the triple (i,u,n) is called De Morgan Triplet iff. i is a t-norm, u is a t-conorm (aka s-norm), n is a strong negator, so that for all x,y ∈ [0, 1] the following holds true: u(x,y) = n( i( n(x), n(y) ) ) (generalized De Morgan relation). [1] This implies the axioms provided below in detail.