Define addition and multiplication with Dedekind cuts

Given two Dedekind cuts, $X, Y \in \mathbb{R}$, define addition and multiplication of real numbers. (I was thinking that it should match up with the already established properties like identity and associativity. I tried to work it out using things like union or intersection, but nothing seemed to work.)

No other functions are allowed.

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Try addition. :) – Hurkyl Oct 9 '12 at 23:08
@Hurkyl I mean in terms of set theory with use of Dedekind cuts. – Casquibaldo Oct 9 '12 at 23:15
You have to use rational number arithmetic somewhere.... – Hurkyl Oct 9 '12 at 23:24

Recall that an interpretation of a Dedekind cut (of rational numbers) is as identifying a real number $x$ by decomposing the rational numbers into the set of all things less than $x$ and the set of all things greater than $x$. (insert your favorite way to deal with cuts identifying rational numbers)