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

Question

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 08 '11 at 09:10
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 08 '11 at 10:26
Yes, that's exactly why I thought that. However $\mathbb{R}\times\mathbb{R}$ makes more sense now. — Hannesh, May 08 '11 at 14:30

score 9 · Accepted Answer · edited May 08 '11 at 09:19

9

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

edited May 08 '11 at 09:19

t.b.

78,116

answered May 08 '11 at 09:12

Dinesh

1,737

on your last statement, you should correct the cross product to remove the 'squared' term. – davidlowryduda May 08 '11 at 09:13
I must've misheard something along the way then. Thanks! – Hannesh May 08 '11 at 09:16
@mixedmath Thanks, but I'm new to latex. @Theo Thanks for editing. – Dinesh May 08 '11 at 09:16
1

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 08 '11 at 09:20
@Theo. Yeah, good point. Thanks – Dinesh May 08 '11 at 09:23

score 0 · Answer 2 · answered Mar 10 '14 at 23:21

0

It means 2d co-ordinate space.

answered Mar 10 '14 at 23:21

user2486738

101

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

2 Answers2