Today I Learned

Some of the things I've learned every day since Oct 10, 2016

103: Dedekind Cuts

The Dedekind cut is a method of constructing \mathbb{R} from \mathbb{Q}. A cut is first defined to be a partition of \mathbb{Q} into 2 non-empty subsets A, B such that

  1. if p \in A and q < p, then q \in A
  2. if p \in A, then there is a q \in A such that q > p (meaning A has no maximum element)

A cut can be equivalently determined solely by A alone, rather than the pair (A, B).

Cuts can then be used to construct \mathbb{R} by defining any r \in \mathbb{R} to be the cut where A is the set of all members q of \mathbb{Q} such that q < r. That is, r is simply defined as the subset of rationals smaller than itself, and \mathbb{R} is the set of all such subsets.

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