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The question is:

Consider $ \mathbb{R^2} $ with the usual metric and let

$E = \{ (t, \sin t) : t > 0 \} $ . Identify $E'$ explicitly.

Thank you so much !

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    What does $E'$ denote?2012-10-16
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    complement of E2012-10-16
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    Okay. Well, the set $E$ is just the graph of the sine function for $t>0$, so the complement of $E$ in $\mathbb{R}^2$ is everything not on this graph2012-10-16
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    Thank you so much. I wonder is it just a graph of F(t) = sine t?2012-10-16
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    Yes, $E$ is the graph of $F(t)$ for $t>0$.2012-10-16
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    Can I say complement of cosine is cosine t? or just shaded everything other than the line cosine t in graph?2012-10-16
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    I don't know what "the complement of cosine is cosine t" means, so I'm not sure what you're asking. The complement of a set $A$ is everything outside of $A$ (but still in the universe you care about; for your problem the universe is $\mathbb{R}^2$). The set $E$ in the problem you posted above is a portion of the graph of the sine function; the complement of $E$ is the set of points that do not lie on this portion of the graph (that is, these are the points outside of $E$).2012-10-16
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    To get curly braces in $\LaTeX$, use `\{` and `\}`.2012-10-16

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$R'$ contains all points $(x,y)\in\mathbb R^2$ that are not of the form $(t,\sin t)$ for some $t>0$. That is $$ E'=\{(x,y)\in\mathbb R^2\mid x\le 0\lor y\ne\sin x\}.$$