Show that $\mathbb{R}$ cross $\mathbb{R}$ is equinumerous to $\mathbb{R}$

Question

I need help showing that $\mathbb{R}$ cross $\mathbb{R}$ is equinumerous to $\mathbb{R}$.

I know that you need to show a bijection, however need help on that part.

Answer 1

By Cantor-Schröder-Bernstein ("if we have injections both ways, then we can build a bijection"), you just need an injection $\mathbb{R} \times \mathbb{R} \to \mathbb{R}$. Since $\mathbb{R}$ bijects with $\mathcal{P}(\mathbb{N})$ (power-set of the naturals), that's equivalently an injection $\mathcal{P}(\mathbb{N}) \times \mathcal{P}(\mathbb{N}) \to \mathcal{P}(\mathbb{N})$. (I'll take $\mathbb{N}$ to exclude $0$, for convenience.)

But the latter is easy: given two sets $A, B \subseteq \mathbb{N}$, construct the set $\{2^a, 3^b: a \in A, b \in B \} \subseteq \mathbb{N}$. This uniquely specifies $A$ and $B$.