Injective function from $\mathbb{R}^2$ to $\mathbb{R}$?

Solution 1:

Let's construct an injective function $f : (0,1) \times(0,1) \to (0,1)$. Since there exist bijections between $\mathbb{R}$ and $(0, 1)$, the proposed function $f$ is sufficient to prove the existence of an injective function from $\mathbb{R}^2$ to $\mathbb{R}$.

Let the decimal representation of $x$ be $0.x_1x_2x_3\cdots$, and that of $y$ be $0.y_1y_2y_3\cdots$. Let $f(x, y)$ be $0.x_1y_1x_2y_2x_3y_3\cdots$.

To make this function well-defined, we should avoid decimal representations that end with infinite successive $9$s. Once this is taken care of, it's easy to show that this function is injective.

Solution 2:

Hint: Take the interval $[0, 1]$ and think how we might try to map pairs of numbers from that interval 1-1 into the interval by 'interleaving' binary representations. In other words if $a = \cdot a_1a_2a_3a_4\ldots$ and $b = \cdot b_1b_2b_3b_4\ldots$, then send $\langle a, b \rangle$ to $\cdot a_1b_1a_2b_2a_3b_3\dots$ ....