Since $\Bbb N$ and ${\sqrt2}$ are each countable sets, I see that the union is also countable. From this and the fact that $\Bbb N$ union ${\sqrt2}$ is infinite we know there exists a bijection to $\Bbb N$
I understand why such a bijection exists, but i'm not sure how to create the actual bijection itself?
If we let an element in the union be the smallest element we can show that it is impossible for two elements to be the smallest element in that set so that it would be one to one. and then because it is infinite, it is also onto. Is this correct?
Define $$f:\Bbb N\cup\{\sqrt{2}\}\to\Bbb N$$ by $f(\sqrt{2})=1$ and $f(n)=n+1$ for all $n\in\Bbb N$.
Its inverse is the map $$g:\Bbb N\to\Bbb N\cup\{\sqrt{2}\}$$ defined by $g(n)=n-1$ for $n\geq 2$ and $g(1)=\sqrt{2}$.