What is the cardinality of the set of real numbers in union with the set containing the empty set.

Question

I'm trying to figure out the cardinality of $|\mathbb{R}\cup \left \{\varnothing\right \}|$. I think it is $\mathfrak{c}$ because clearly the set has at least as many elements as $\mathbb{R}$ however, I'm having trouble finding an injection from the set to $\mathbb{R}$.

Answer 1

Hint: this is Hilbert's hotel with an uncountable set of rooms that you can leave reserved: map $\emptyset$ to $0$ and then shuffle the numbers in the sequence $\langle 0, 1/1, 1/2, 1/3, \ldots\rangle$ along one place.

Answer 2

You could use the fact that $\operatorname{arctan}:\Bbb R \rightarrow (-\pi/2,\pi/2)$ is a bijection and continue it to a bijection $\Bbb R \sqcup * \rightarrow (-\pi/2,\pi/2]$. This yields an injection $\Bbb R \sqcup * \rightarrow \Bbb R$.