I'm trying to prove that $\mathbb{N}\times\mathbb{N}$ is equinumerous to $\mathbb{N}$ using that fact the function $f:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ defined by $f(m,n)=1/2((m+n)^2+3n+m)$ is a bijection. Pick two different points $(m,n)$ and $(m',n')$ from $\mathbb{N}\times\mathbb{N}$. I have have tried to break into cases depending on whether $m=m'$, $m\lt m'$, or $m\gt m'$. But I can't finish the whole process. Any help would be much appreciated.
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FYI, using an Approach0 search, I had just found the proposed duplicate question. There are also quite a few other basically duplicate questions here, e.g., Prove a bijection between $\mathbb{N}^2$ and $\mathbb{N}$., How to prove that $(a,b)\mapsto \frac{1}{2}((a+b)^{2}+3a+b)$ is bijective?, How would you reverse this double-variable equation?, ... – John Omielan Aug 14 '23 at 02:07
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(cont.) How does one get the formula for this bijection from $\mathbb{N}\times\mathbb{N}$ onto $\mathbb{N}$?, Countability of cartesian product of $\mathbb{N} \times \mathbb{N}$, Diophantine equation to characterize natural numbers, etc. – John Omielan Aug 14 '23 at 02:08
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@JohnOmielan Thank you very much. I've found the answer – OSCAR Aug 14 '23 at 02:36
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@Sil Thank you very much. I've found the answer – OSCAR Aug 14 '23 at 02:36