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Hey guys I have a quiz soon and I really dont know how to prove this question. I tried my best but it is not working. Please help out with anything or hints.

Let $(A'_1, A'_2)$ be a Dedekind cut of ${\mathbb{Q}}$ that represents the same real number as $(A_1, A_2)$. Let $C'_1 = A'_1 + B_1$ and $C'_2 = {\mathbb{Q}} \diagdown C'_1$.

Prove that $(C'_1, C'_2)$ represents the same real number as $(C_1, C_2)$.

By Dedekind cut, it means 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.

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What is $C_1$ and $C_2$? –  Pedro Tamaroff Feb 26 '13 at 17:50
    
@Peter: Presumably $C_1=A_1+B_1$ and $C_2=\Bbb Q\setminus C_1$. –  Brian M. Scott Feb 26 '13 at 17:52
    
Do you know when two Dedekind cuts represent the same real number? You’ve not given us your definition of Dedekind cut $-$ there are different ones in use $-$ but if it’s what I think, this happens only when the cut represents a rational. For example, $0$ is represented by both $\langle(\leftarrow,0),[0,\to)\rangle$ and $\langle(\leftarrow,0],(0,\to)\rangle$. –  Brian M. Scott Feb 26 '13 at 17:54
    
i will fix the question again –  MathGeek Feb 26 '13 at 17:56
    
If that’s your definition of a Dedekind cut, $\langle A_1',A_2'\rangle$ and $\langle A_1,A_2\rangle$ represent the same real number if and only if $A_1=A_1'$ and $A_2=A_2'$, i.e., if and only if they’re the same cut. But that’s the definition that Cameron Buie used in his answer to your other question; are you sure that it’s the same as the one that you’re actually using? In particular, does yours include condition (v)? –  Brian M. Scott Feb 26 '13 at 18:31
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