Dedekind cut , in mathematics , particularly in number systems, in memory of the German mathematician Richard Dedekind , is a special subset of ordered body of rational numbers. These subsets are used to build a complete Archimedean ordered body, specifically the real numbers.
Summary
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- 1 Definition
- 2 Examples
- 3 Cut operations
- 4 Addition
- 5 Bibliography
Definition
A subset A of Q is a Dedekind cut if it satisfies these properties:
- A is a non-empty set of Q.
- If m is in A and n is in Q such that n <m, then n is in A
- If m is in A, then there are n in A such that m <n.
Intuitively, a cut is a rational straight line that has no greater element or upper bound.
Examples
- The set of rational numbers less than 15; A 1= {x at Q / x <15}
- The set of rationals whose cubes <three, united with all negative rationals. A 2= {x in Q / x 3 <3} UQ –
Cut operations
Considering K the set of all cuts, we can define a relation of order, the addition and multiplication of elements of K, so that K is an ordered body with the Archimedean property, and finally, K, defined in this way satisfies the Dedekind’s postulate , that is: K be a full body. That is, every subset of K bounded superiorly has supreme.
Addition
We want to define the application sum +: K × K → K, which has a pair (A, B) in an element A + B of k. We define A + B: = {s + t in Q / s is in A and t is in B}. It is proved that the set A + B, thus defined, is a cut and that the sum application meets the definition of abelian group in collection K.
