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I have a problem with the following question on Dedekind's cut.

Is the set $\{t\in \mathbb{Q}: -t\not\in r\}$, where $r$ is a real number (a cut), a Dedekind cut? Why or why not?

The definition of Dedekind's cut here is: nonempty, not $\mathbb{Q}$, contains all rational number smaller than it, and does not contain a largest element.

Thank you.

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  • $\begingroup$ What have you tried? About which property of a Dedekind cut are you unsure here? $\endgroup$ – Michael Greinecker Sep 30 '12 at 14:43
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    $\begingroup$ Hint: What if $r$ is rational? $\endgroup$ – celtschk Sep 30 '12 at 14:51
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Hint: Consider a cut $r$ which is the set of rationals less than zero. So we are to be consider $\{t \in \mathbb{Q}\colon -t \not< 0\}$. How does the condition simplify?

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Hint: If $r$ represents an irrational number, then yes, if a rational number, then not (in the given interpretation of Dedekind-cut).

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