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1. Dedekind Cuts.
dedekind cuts. If we choose a rational number q, we can use this to split the rationals in two sets, one larger than
http://hemsidor.torget.se/users/m/mauritz/math/num/real.htm

Extractions: sets are larger than all numbers in the other set we can use this to define a cut of the rationals. One of the two sets is then, Q and the other is the complement set to this set. We then write the cut as, c where the other part of the set is implicitly defined. We can omit the ' Q ' because the cut is per definition over the rationals. Such a cut can now be of three kinds, either, as the first one we looked at, a cut where the upper set has a lowest rational

2. 2.15.1 Dedekind Cuts
2.15.1 dedekind cuts. A real number is represented by a cut ,
http://www.dgp.toronto.edu/people/mooncake/thesis/node61.html

Extractions: Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations A real number is represented by a cut . Every cut has the property that for all As presented, the cut represents . Disallowing this special cut gives a representation for all non-negative real numbers. In general, Most numbers have a representation that cannot be written out directly since the representation is an infinite set. Operations on reals are inherited from the corresponding operations on rationals. For example, a binary operation on two real numbers, represented by cuts X and Y , is given by: Difficulties are encountered when generalizing this to negative real numbers. If a cut is simply redefined to be a subset of , then the product of two cuts is not a cut if the multiplicands correspond to negative numbers. See [ ] for further details concerning this representation and associated methods.

3. Dedekind Cuts Of Partial Orderings
dedekind cuts of partial orderings dedekind cuts are a clever trick for defining the reals given the rationals.
http://www.cap-lore.com/MathPhys/Cuts.html

Extractions: We may take any partial ordering and consider such cuts. The result is always a lattice. There is another different unique (within isomorphism) lattice associated with any partial ordering. There is for any partial ordering some unique smallest lattice in which it is embedded. The lattice may contain new elements but the new ordering, restricted to the old PO will contain no new orderings. This construction is also found in security considerations. The orange book provides a theory of security classifications that implicitly defines a lattice. In a particular computer system it is likely that some of the lattice values will be unused. This may cause some confusion. It should not any more than noting that the boolean or command of the CPU need not in an application produce all possible values in order to make the set of all possible values a useful concept with which to reason. It is the same with the lattice of security classifications. When the partial ordering is finite and total the cuts add nothing of interest.

4. Dedekind Cuts
dedekind cuts. The first construction of the Real numbers from the Rationalsis due to the German mathematician Richard Dedekind (1831 1916).
http://www.gap-system.org/~john/analysis/Lectures/A3.html

5. Some Early History Of Set Theory
MT2002 Analysis Previous page (Some definitions of the concept of continuity),Contents, Next page (dedekind cuts). Some Early History of Set Theory.
http://www.gap-system.org/~john/analysis/Lectures/A2.html

6. PlanetMath: Dedekind Cuts
The purpose of dedekind cuts is to provide a sound logical foundation for the real number system.
http://www.planetmath.org/encyclopedia/DedekindCuts.html

Extractions: Dedekind cuts (Definition) The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind's motivation behind this project is to notice that a real number , intuitively, is completely determined by the rationals strictly smaller than and those strictly larger than . Concerning the completeness or continuity of the real line, Dedekind notes in [ ] that If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above. He further notes that the completeness property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as

7. Theorem: R Via Dedekind Cuts
Theorem R via dedekind cuts. The proof is not done, sorry. Context Context.Interactive Real Analysis, ver. 1.9.3 (c) 19942000, Bert G. Wachsmuth.
http://www.shu.edu/projects/reals/infinity/proofs/r_dedek.html

8. Dedekind CutsThe First Construction Of The Real Numbers From The Rationals Is Du
Biography of Richard Dedekind (18311916) was his redefinition of irrational numbers in terms of dedekind cuts which, as we mentioned above, first came to him as
http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/A3.html

9. D
D. De Morgan's laws; De Morgan, Augustus (18061871); decreasing function;dedekind cuts; Dedekind's Theorem; density principle; derivative
http://www.shu.edu/projects/reals/gloss/index_d.html

10. Dedekind Cuts By Todd Trimble
Subject dedekind cuts Author todd trimble trimble@math.luc.edu Date 3 Oct 98 185819 0400 (EDT) The idea is that

11. 2.15 Real Representations
next up previous notation contents Next 2.15.1 dedekind cuts Up2 Numbers Previous 2.14 Variants 2.15 Real Representations. To
http://www.dgp.toronto.edu/people/mooncake/thesis/node60.html

Extractions: Next: 2.15.1 Dedekind Cuts Up: 2 Numbers Previous: 2.14 Variants To perform computations with the aid of digital computers we must build the reals out of a discrete system. Mathematicians have historically built up the reals with different approaches; this section details some of these approaches. Some of these approaches lead to mechanical algorithms which may be contrasted with the interval approach. Readers interested solely in the interval approach should proceed to the next chapter.

12. Dedekind Cut - Wikipedia
The Dedekind cut is named after Richard Dedekind, who invented this constructionin order to represent the real numbers as dedekind cuts of the rational numbers
http://www.wikipedia.org/wiki/Dedekind_cut

13. Real Number - Wikipedia
This sense of completeness is most closely related to the construction of the realsfrom dedekind cuts, since that construction starts from an ordered field
http://www.wikipedia.org/wiki/Real_number

14. Dedekind's Cuts
3 Oct 1998 dedekind cuts, by todd trimble

15. Math And Physics
Transformations on vector spaces Implicit Methods Code for Fields in a computerHomotopy and Homotopy groups dedekind cuts of partial orderings Stowaways See
http://www.cap-lore.com/MathPhys/

16. PlanetMath: Dedekind Cuts
dedekind cuts, (Definition). The purpose of dedekind cuts is to providea sound logical foundation for the real number system. Dedekind's
http://planetmath.org/encyclopedia/DedekindCuts.html

Extractions: Dedekind cuts (Definition) The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind's motivation behind this project is to notice that a real number , intuitively, is completely determined by the rationals strictly smaller than and those strictly larger than . Concerning the completeness or continuity of the real line, Dedekind notes in [ ] that If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above. He further notes that the completeness property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as

17. PlanetMath:
decomposition (complementary subspace) owned by rmilson. decomposition group ownedby djao. dedekind cuts owned by rmilson. Dedekind domain owned by saforres.
http://planetmath.org/encyclopedia/D/

Extractions: PlanetMath Encyclopedia (browse by subject) DAG (acyclic graph) owned by Logan Darboux's theorem (analysis) owned by mathwizard proof of Darboux's theorem owned by ariels Darboux's Theorem (symplectic geometry) owned by bwebste DCC (descending chain condition) owned by antizeus DCT (discrete cosine transform) owned by akrowne de Bruijn digraph owned by vampyr decide (decision problem) owned by Henry decision problem owned by Henry deck transformation owned by Dr_Absentius deck transformation (deck transformation) owned by Dr_Absentius decomposition (complementary subspace) owned by rmilson decomposition group owned by djao Dedekind cuts owned by rmilson Dedekind domain owned by saforres Dedekind-Hasse norm (Dedekind-Hasse valuation) owned by Henry Dedekind-Hasse valuation (Dedekind-Hasse valuation) owned by Henry Dedekind-Hasse valuation owned by Henry Dedekind infinite owned by Evandar Dedekind zeta function owned by bwebste deduction (deductions are ) owned by Henry deductions are delta 1 (deductions are ) owned by Henry deductions are owned by Henry definable owned by Timmy

18. The Reals.
As you may recall, one way of defining the reals was by using Dedekind'scuts. We do now define the the real numbers to be a dedekind cuts.
http://hemsidor.torget.se/users/m/mauritz/math/num/setreal.htm

Extractions: 2 : The cut has no largest element. Q c will thus be a Dedekind's cut. We do now define the the real numbers to be a Dedekind cuts. b, and that a=b if and only if the sets are equal. We can embed the rational numbers in the reals by, c And we can define arithmetic on the reals. We could also define a real using a Cauchy sequence . A Cauchy sequence is a sequence, x ,x ,x ,...such

19. Reals Via Dedekind Cuts
Theorem Real Numbers as dedekind cuts. The proof isnot done, sorry. To Theory Glossary Map (bgw).
http://pirate.shu.edu/projects/reals/infinity/proofs/r_dedek.html

20. Interactive Real Analysis Glossary
D. De Morgan, Augustus (18061871); De Morgan's laws; decreasing function;dedekind cuts; Dedekind's Theorem; density principle; derivative
http://pirate.shu.edu/projects/reals/gloss/index_d.html

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