Help with Dedekind cuts I am reviewing what Dedekind cuts are for my quiz tomorrow. I had posted a question before about Dedekind cuts and I thought that was the only problem but there were these two problems as well for this unit I am having trouble understanding. I know what they are and could find it if it was in a real line. but i am having difficulty understanding these 2 problems
For two subsets $X, Y$  of ${\mathbb{O}}$, define the subset $X + Y$ of ${\mathbb{O}}$ by $X + Y = \{x + y |x \in X$ and $y \in Y\}$.
Let $(A_1, A_2)$ and $(B_1, B_2)$ be Dedekind cuts of ${\mathbb{O}}$. Let $C_1 = A_1 + B_1$  (in the above sense) and let $C_2  = {\mathbb{O}} \diagdown  C_1$.
Prove that  $(C_1, C_2)$ is a Dedekind cut of ${\mathbb{O}}$
and
Let $(A'_1, A'_2)$ be a Dedekind cut of ${\mathbb{O}}$ that represents the same real number as $(A_1, A_2)$. Let $C'_1 = A'_1 + B_1$ and $C'_2 = {\mathbb{O}} \diagdown  C'_1$. Prove that $(C'_1, C'_2)$ represents the same real number as $(C_1, C_2)$.
thank you
 A: By Dedekind cut, I take you to mean a pair $(A,B)$ such that:

(i) $A,B\neq\emptyset,$
(ii) $A\cap B=\emptyset,$
(iii) $A\cup B=\Bbb Q,$
(iv) $y\in A$ whenever $y\in\Bbb Q$ and there is some $x\in A$ with $y<x,$ and
(v) $A$ has no greatest element.

(Let me know if that doesn't match your definition.)
Since $A_1,B_1$ non-empty sets of rationals, then so is $C_1$. On the other hand, since $A_2,B_2$ non-empty, then $C_1$ can't be all of $\Bbb Q$, so $C_1,C_2$ partition the rationals. Since $A_1,B_1$ have no greatest element, then neither can $C_1$. Suppose $y\in\Bbb Q$, and that $y<x$ for some $x\in C_1$. Since $C_1=A_1+B_1,$ then $x=a+b$ for some $a\in A_1,b\in B_1$. Take $a'\in\Bbb Q$ such that $a'<a$--so $a'+b<a+b=x$--and $y<a'+b$. (Why can we do this?) Since $(A_1,A_2)$ is a Dedekind cut and $a\in A_1$, then $a'\in A_1$. Put $b'=y-a'$, so that $a'+b'=y<a'+b$, whence $b'<b$, so $b'\in B_1$, and so $y=a'+b'\in C_1$, as desired.

In the second part, they seem to be asking you to show that $A+B$ is well-defined as an operation on left sides of Dedekind cuts. I'm not quite sure what they mean by different Dedekind cuts representing the same real number, though. I was under the impression that each real number corresponded to a unique Dedekind cut.
