0

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?

Spenser
  • 19,469
  • 1
    The two sets are indeed countably infinite. But I think you are being asked to give an explicit bijection. Suppose we decide to let $f(\sqrt{2})=1$. Then we could let $f(1)=2$, $f(2)=3$, and so on. – André Nicolas Apr 01 '16 at 15:49

1 Answers1

1

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 $g(1)=\sqrt{2}$.

Spenser
  • 19,469
  • Okay, then if I had something similar but the union was between the set of natural numbers and the set containing $\sqrt2$ and$\pi$ would I let $f(\sqrt2)=1$ and $f(\pi)=2$ and $f(n)=n+2$ ? –  Apr 01 '16 at 15:51
  • @user319635 Yes, that would work. – Spenser Apr 01 '16 at 15:52
  • Awesome, thank you. If it were the other way around and I was to go from the set of natural numbers to the set of natural numbers excluding some values how would I go about this? –  Apr 01 '16 at 15:54
  • Is it different if instead of $N$ union $\sqrt2$ is is $N$ excluding $\sqrt2$? –  Apr 01 '16 at 15:56
  • @user319635 Since $\sqrt{2}\notin\Bbb N$ I am not sure to understand what you mean. – Spenser Apr 01 '16 at 15:58
  • Oh exactly, I should have thought of a different example. Let me change that, If it was $N$ excluding some value in $N$, like 2? –  Apr 01 '16 at 15:59
  • Or even better, what if it was $N$ excluding all the odd numbers? –  Apr 01 '16 at 16:01
  • 1
    Define $f:\Bbb N\to\Bbb N-{2}$ by $f(1)=1$, $f(2)=3$, $f(3)=4$, $f(4)=5$, and so on. – Spenser Apr 01 '16 at 16:01
  • 1
    Define $f:\Bbb N\to\Bbb N-\Bbb N_{odd}$ by $f(n)=2n$. – Spenser Apr 01 '16 at 16:02