Some of the things I've learned every day since Oct 10, 2016
103: Dedekind Cuts
January 29, 2017Posted by on
The Dedekind cut is a method of constructing from . A cut is first defined to be a partition of into 2 non-empty subsets such that
- if and , then
- if , then there is a such that (meaning has no maximum element)
A cut can be equivalently determined solely by alone, rather than the pair .
Cuts can then be used to construct by defining any to be the cut where is the set of all members of such that . That is, is simply defined as the subset of rationals smaller than itself, and is the set of all such subsets.