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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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    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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    @Theo. Yeah, good point. Thanks2011-05-08
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It means 2d co-ordinate space.