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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)

Use this interpretation, along with the fact that you know how the reals are "supposed" to behave, to tell you which rational numbers are supposed to go into which half of a sum.

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