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.

Let $f$ be defined as
$f(n)=n+2$ for $n\in\mathbb{N}$
and
$f(x)=x$ for $x\in \mathbb{Q}-\mathbb{N}$.
Prove that $f$ is bijective from $\mathbb{Q}$ onto $\mathbb{Q}-\{1,2\}$.
Let now $g:\mathbb{Q}\cup \{\pi,e\}\to \mathbb{Q}$ be defined as:
$g(x)=f(x)$ if $x\in \mathbb{Q}$, and $g(e)=0, g(\pi)=1$. Prove that $g$ is bijective and then you're done. 
