elementary set theory - Cross product of the reals question

Is $\Bbb {R} \times \Bbb {R} \subseteq \Bbb {R}$?

If this is the case then would it be true that $|\Bbb {R} \times \Bbb {R}| \leq |\Bbb {R}|$?

Answer

$\mathbb R \times \mathbb R \subseteq \mathbb R$ is incorrect.

It is incorrect for the same reason that in vector spaces $\mathbb R^3 \subseteq \mathbb R^2$ is incorrect. The number of components is different.

However...

The statement $|\mathbb R \times \mathbb R| \leq |\mathbb R|$ is correct. and infact, it is true that $|\mathbb R \times \mathbb R| = |\mathbb R|$

A simple proof for $|\mathbb R \times \mathbb R| \leq |\mathbb R|$ without resorting to cardinal arithmetique would be to find a function $f: \mathbb R^2 \to \mathbb R$ that is injective. Can you think of such function? how about $f(i,j)=2^i3^j$?