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Help with a logical derivation of set theoretical statement 
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#1
Feb1111, 11:47 PM

P: 110

1. The problem statement, all variables and given/known data
This is actually from the proof of Dedekind's cut in Rudin's Principles of Mathematical analysis on the page 19. It says when [tex]\alpha\in\mathbb{R}[/tex] ([tex]\alpha[/tex] is a cut) is fixed, [tex]\beta[/tex] is the set of all [tex]p[/tex] with the following property: There exists [tex]r>0[/tex] such that [tex]pr\notin\alpha[/tex]. From the given above, I need to derive that if [tex]q\in\alpha[/tex], then [tex]q\notin\beta[/tex]. But I cannot reach this statement as my explanation for this is in the below. 2. Relevant equations 3. The attempt at a solution The draft I have done so far is that, as defining [tex]\beta[/tex] such that [tex]\beta=\left\{p\exists r\in\mathbb{Q} (r>0 \wedge pr \notin \alpha)\right\}[/tex], I derived [tex]p \notin \beta \leftrightarrow \forall r \in \mathbb{Q} (r>0 \to pr\in\alpha)[/tex]. And I'm stucked here. From the last statement, I cannot derive the conclusion I was meant to derive. If anyone gives me help, I will give thanks. 


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