Cartesian Product

**Ruun** · June 3rd 2009, 08:25 AM

Hi!

I'm trying to self-study some math and I've decided to "forget" almost but high school math and start from the beggining.

I have a seriously easy question about Cartesian Product.

It is true that $(\mathbb{R}^{2} \times \mathbb{R}=\mathbb{R} \times \mathbb{R}^{2}=\mathbb{R}^3)$ ?

I've thougth that the first equality is false since the first one is $\mathbb{R}^2 \times \mathbb{R}= \{(x,y) : x \in \mathbb{R}^2$ and $y \in \mathbb{R} \}$ and in the second one $\mathbb{R} \times \mathbb{R}^2 = \{(x,y) : x \in \mathbb{R}$ and $y \in \mathbb{R}^2 \}$ but then I thought that both are equal to $\mathbb{R}^{3}$ .

Thanks in advice

**HallsofIvy** · June 3rd 2009, 10:20 AM

Originally Posted by Ruun

Hi!

I'm trying to self-study some math and I've decided to "forget" almost but high school math and start from the beggining.

I have a seriously easy question about Cartesian Product.

It is true that $(\mathbb{R}^{2} \times \mathbb{R}=\mathbb{R} \times \mathbb{R}^{2}=\mathbb{R}^3)$ ?

I've thougth that the first equality is false since the first one is $\mathbb{R}^2 \times \mathbb{R}= \{(x,y) : x \in \mathbb{R}^2$ and $y \in \mathbb{R} \}$ and in the second one $\mathbb{R} \times \mathbb{R}^2 = \{(x,y) : x \in \mathbb{R}$ and $y \in \mathbb{R}^2 \}$ but then I thought that both are equal to $\mathbb{R}^{3}$ .

Thanks in advice

Strictly speaking, no, they are not the same.
$R^2\times R= \{((x,y),z)\}$
$R\times R^2= \{(x, (y,z))\}$
$R^3= \{(x, y, z)\}$

However, there is an obvious correspondence between any two of them. Further, with the various operations (pointwise addition, etc.) that we can define on them, that correspondence becomes an isomorphism.

**Ruun** · June 3rd 2009, 10:33 AM

Right, solved, thank you!

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