1. ## Cartesian Product

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 $\displaystyle (\mathbb{R}^{2} \times \mathbb{R}=\mathbb{R} \times \mathbb{R}^{2}=\mathbb{R}^3)$ ?

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

2. 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 $\displaystyle (\mathbb{R}^{2} \times \mathbb{R}=\mathbb{R} \times \mathbb{R}^{2}=\mathbb{R}^3)$ ?

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

$\displaystyle R^2\times R= \{((x,y),z)\}$
$\displaystyle R\times R^2= \{(x, (y,z))\}$
$\displaystyle R^3= \{(x, y, z)\}$