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Dedekind cut
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{{For|the American record producer known professionally as Dedekind Cut|Fred Warmsley}}{{Refimprove|date=March 2011}}File:Dedekind cut- square root of two.png| thumb| right| 350px| Dedekind used his cut to construct the irrational, real numberreal numberIn mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph BertrandBOOK, Bertrand, Joseph, Trait'e d'Arithmetique,weblink 1849, page 203, An incommensurable number can be defined only by indicating how the magnitude it expresses can be formed by means of unity. In what follows, we suppose that this definition consists of indicating which are the commensurable numbers smaller or larger than it ...., BOOK, Spalt, Detlef, Eine kurze Geschichte der Analysis, 2019, Springer, 10.1007/978-3-662-57816-2, , are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B.BOOK, Dedekind, Richard, Continuity and Irrational Numbers,weblink 1872, Section IV, Whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ...., In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. See also completeness (order theory).It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the B set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.A similar construction to that used by Dedekind cuts was used in Euclid's Elements (book V, definition 5) to define proportional segments.

Definition

A Dedekind cut is a partition of the rationals mathbb{Q} into two subsets A and B such that
  1. A is nonempty.
  2. A neq mathbb{Q}.
  3. If x, y in mathbb{Q}, x < y , and y in A , then x in A . (A is "closed downwards".)
  4. If x in A , then there exists a y in A such that y > x . (A does not contain a greatest element.)
By relaxing the first two requirements, we formally obtain the extended real number line.

Representations

It is more symmetrical to use the (A,B) notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".If the ordered set S is complete, then, for every Dedekind cut (A, B) of S, the set B must have a minimal element b, hence we must have that A is the interval (−∞, b), and B the interval [b, +∞).In this case, we say that b is represented by the cut (A,B).The important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers). The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents.For example if A and B only contain rational numbers, they can still be cut at {{radic|2}} by putting every negative rational number in A, along with every non-negative number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for {{sqrt|2}}, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number.

Ordering of cuts

Regard one Dedekind cut (A, B) as less than another Dedekind cut (C, D) (of the same superset) if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut (A, B) is again less than (C, D). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (greater than, less than or equal to, equal to, and so on) can be similarly created from set relations.The set of all Dedekind cuts is itself a linearly ordered set (of sets). Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set S, which might not have had the least-upper-bound property, within a (usually larger) linearly ordered set that does have this useful property.

Construction of the real numbers

{{Cleanup|reason=Contains information outside the scope of the article|date=June 2015}}{{See also|Construction of the real numbers#Construction by Dedekind cuts}}A typical Dedekind cut of the rational numbers Q is given by the partition (A,B) with
A = { ainmathbb{Q} : a^2 < 2 text{ or } a < 0 }, B = { binmathbb{Q} : b^2 ge 2 text{ and } b ge 0 }.In the second line, ge may be replaced by > without any difference as there is no solution for x^2 = 2 in Q and b=0 is already forbidden by the first condition. This results in the equivalent expression B = { binmathbb{Q} : b^2 > 2 text{ and } b > 0 }.
This cut represents the irrational number {{sqrt|2}} in Dedekind's construction. To establish this truly, one must show that this really is a cut and that it is the square root of two. However, neither claim is immediate. Showing that it is a cut requires showing that for any positive rational x with {{math|x

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