elementary set theory - Prove that $mathbb{Q}$ and $mathbb{Q} cup {pi, e}$ have the same cardinality

Prove that $\mathbb{Q}$ and $\mathbb{Q} \cup \{\pi, e\}$ have the same cardinality.

I know I must show that there exists a bijection between these two sets but I'm having a difficult time trying to come up with a function that relates them. Any suggestions? Thanks.

Answer

Hint: Let $f: \mathbb{Q} \to \mathbb{Q} \cup \{\pi, e\}$ be the function

$$f(x) = \begin{cases} \pi&\text{if } x = 0 \\ e &\text{if } x = 1 \\ x-2 &\text{if } x \in \{2, 3, 4, 5, 6, \ldots\} \\ x &\text{otherwise}. \end{cases}$$