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A Dedekind cut is a partition of the rational numbers into two sets A and B, such that each element of A is less than every element of B, and A contains no greatest element. The set B may or may not have a smallest element among the rationals.
Learn how to construct real numbers as pairs of rational subsets, called Dedekind cuts, and perform arithmetic operations and order relations on them. See examples, definitions, and properties of real numbers as cuts.
1. Notes on Dedekind cuts De nition 1.1. A subset LˆQ of the rationals is called a Dedekind cut if (I) Lis proper (i.e. L6= ;;L6= Q); (II) Lhas no maximal element; (III) for all elements a;b2Q with a<b, b2L=)a2L. Example 1.2. (i) If a2Q, the open interval L a:= (1 ;a) \Q is a Dedekind cut that we take to represent the rational number a. (ii ...
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Dedekind cut, in mathematics, concept advanced in 1872 by the German mathematician Richard Dedekind that combines an arithmetic formulation of the idea of continuity with a rigorous distinction between rational and irrational numbers. Dedekind reasoned that the real numbers form an ordered.
- The Editors of Encyclopaedia Britannica
get what you want using something called Dedekind cuts. The basic problem with the rational numbers is that the rational number system has holes in it – missing numbers. The beauty of Dedekind cuts is that it gives a formal way to talk about these holes purely in terms of rational numbers. 1
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A Dedekind cut, or simply a cut, is a subset $\alpha \subset \mathbb{Q}$ with the following three properties: Nonempty and proper: $\alpha \neq \varnothing$ and $\alpha\neq \mathbb{Q}$ Closed downward: If $a \in \alpha$, $y \in \mathbb{Q}$, and $b \le a$, then $b \in \alpha$.
Feb 9, 2018 · Dedekind cuts are subsets of rationals that define real numbers as complete and ordered. Learn the definition, properties and references of Dedekind cuts from PlanetMath, a free online mathematics encyclopedia.
