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# Dedekind cut

(Redirected from Dedekind-completion)

In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and xa, then x is in A as well) and B is closed upwards. The cut itself is, conceptually, the "gap" defined between A and B. The original and most important cases are Dedekind cuts for rational numbers and real numbers.

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## Handling Dedekind cuts

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' for example the lower part a. For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval ( −∞, a ), in which case B must be [ a, +∞); or a pair with A the interval ( −∞, a ], in which case B must be ( a, +∞ ).

## Ordering Dedekind cuts

If a is a member of S then the set

$\{ \{ x\in S: x < a \}, \{ x\in S: x \ge a \} \}$

is a Dedekind cut we could call ( −∞, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S.

Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B. In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose.

## The cut construction of the real numbers

The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by

$A = \{ a\in\textbf{Q} : a^2 < 2 \lor a\le 0 \},$
$B = \{ b\in\textbf{Q} : b^2 \ge 2 \land b > 0 \}.$

This cut represents the real number $\sqrt{2}$ in Dedekind's construction.

## Generalization: Dedekind completions in posets

More generally, in a partially ordered set S, the set of all nonempty downwardly closed subsets (also called order ideals) is a set partially ordered by inclusion, and in the same way we embed S within a larger partially ordered set that, generally unlike the original set S, does have the least-upper-bound property. This larger poset is called the Dedekind completion of S.

## Another generalization: surreal numbers

A construction similar to Dedekind cuts is used for the construction of surreal numbers.