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In my homework, we are given the following set $M = \{ (x, y) \in \mathbb{R}^2\, |\, x^2 + y^2 \leq 1 \}$.

Obviously, this represents the set of all points $(x, y)$ that lie within a circle of radius $1$.

However, I'm confused about the $\mathbb{R}^2$, I know that is usually means "all positive real numbers", but could it in this case mean $\mathbb{R}\times\mathbb{R}$ (Cartesian product) since we have a two dimensional set?

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    It usually means the cartesian product. Could it be that you confuse it with $\mathbb{R}^{+}$?2011-05-08
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    I suspect the OP thought $\mathbb{R}^2$ meant the set $\{ x^2 : x \in \mathbb{R} \}$, which in all fairness is not *completely* unjustified, since we can write e.g. $\mathfrak{m}^2$...2011-05-08
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    Yes, that's exactly why I thought that. However $\mathbb{R}\times\mathbb{R}$ makes more sense now.2011-05-08

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No. $\mathbb{R}^2$ is not the set of positive real numbers. I do not know of any such convection. $\mathbb{R}^2$ is $\mathbb{R} \times \mathbb{R}$.

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    on your last statement, you should correct the cross product to remove the 'squared' term.2011-05-08
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    I must've misheard something along the way then. Thanks!2011-05-08
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    @mixedmath Thanks, but I'm new to latex. @Theo Thanks for editing.2011-05-08
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    Another small LaTeX-thing: It's better to write `\mathbb{R}^2` instead of `\mathbb{R^2}`. Compare $\mathbb{R^n}$ (`\mathbb{R^n}`) to $\mathbb{R}^n$ (`\mathbb{R}^n`).2011-05-08
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    @Theo. Yeah, good point. Thanks2011-05-08
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It means 2d co-ordinate space.