# What does $(x, y) \in \mathbb{R}^2$ mean?

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?

• It usually means the cartesian product. Could it be that you confuse it with $\mathbb{R}^{+}$? – t.b. May 8 '11 at 9:10
• 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. $\mathfrak{m}^2$... – Zhen Lin May 8 '11 at 10:26
• Yes, that's exactly why I thought that. However $\mathbb{R}\times\mathbb{R}$ makes more sense now. – Hannesh May 8 '11 at 14:30

No. $\mathbb{R}^2$ is not the set of positive real numbers. I do not know of any such convection. $\mathbb{R}^2$ is $\mathbb{R} \times \mathbb{R}$.
• 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). – t.b. May 8 '11 at 9:20