Question: Let X and Y be nonempty sets. Let |X| $\le$ |$\Bbb R$| and |Y| $\le$ |$\Bbb R$|. Prove that |X $\cup$ Y| $\le$ |$\Bbb R$|.
My work:
We know that there exist injections $\quad$f: X -> $\Bbb R$ and g: Y -> $\Bbb R$.
Define a function h(z) = $$ \left\{ \begin{array}{c} tan^{-1}(f(z)) \quad if z\in X \\ tan^{-1}(g(z)) + \pi /2 \quad otherwise \\ \end{array} \right. $$
So I figured we now need to prove that h is injective, because h includes the domain of X and Y and maps them to the set of real numbers. This is where I got stuck though. Conceptually, I think the place I'm stuck is in understanding how h(z) precisely represents the union of X and Y, and maps them to the reals.
Any help appreciated!