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.

Here's an outline of a proof:


*

*Show that the sets $\{0, 1, 2, \dots\}$ and $\{-2, -1, 0, 1, 2, \dots\}$ have the same cardinality.

*Show that one of these sets has the same cardinality as $\Bbb{Q}$, and the other has the same cardinality as $\Bbb{Q} \cup \{\pi, e\}$.

