The question i'm stuck on right now is:
For an arbitrary $S \subseteq \mathbb{R}$, show that $(0,1) \cup S$ is equinumerous with $\mathbb{R}$.
My idea was to break this up into two cases, S being countable, and S being uncountable and see what happens. If S is countable, I can pictorially see the bijection, but I have no idea how to write it in an explicit form, and not sure how to cover the uncountable case.