# Homework Help: Dedekind cut

1. Sep 12, 2012

1. The problem statement, all variables and given/known data

Show that for any two Dedekind cuts A,B, there exists a unique cut C such that A+C=B

2. The attempt at a solution

In order to prove this, I need to prove the existence and uniqueness of such a cut.

For the existence, I started by considering a cut for which this works: C={b-a, b \in B, a \in \mathbb{Q}-A} but I am having trouble showing it is a cut.

For the uniqueness, I want to consider two cuts C and C' such that C≠C' and I wanted to show that C is not a subset of C' and C' is not a subset of C. However, could someone help me to prove this ?