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In order to show that $\mathbb{N} \times \mathbb{N}$ is countable, we can define a bijection $f : \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ like this one:

  • $0 \rightarrow (0, 0)$
  • $1 \rightarrow (1, 0)$
  • $2 \rightarrow (0, 1)$
  • $3 \rightarrow (2, 0)$
  • $4 \rightarrow (1, 1)$
  • $5 \rightarrow (0, 2)$
  • $6 \rightarrow (3, 0)$
  • $7 \rightarrow (2, 1)$
  • $8 \rightarrow (1, 2)$
  • ...

I need to prove that a set is countable, I know how to use this proof method, but I can't remember its name. Any hints?

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  • $\begingroup$ You are just establishing a bijection between $\mathbb{N}\times \mathbb{N}$ and $\mathbb{N}$. You can call this enumeration of $\mathbb{N}\times \mathbb{N}$. $\endgroup$ – baharampuri May 16 '15 at 10:04
  • $\begingroup$ @ChristianBlatter: fixed that $\endgroup$ – hey hey May 16 '15 at 14:27
  • $\begingroup$ @baharampuri: in my context this is only one of a few possible enumerations. This is why I needed a name for it. $\endgroup$ – hey hey May 16 '15 at 14:28
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Aha, I found it! The $f$ function I'm using is Cantor's pairing function.

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