How to show that $f(x,y)=\dfrac{\left(x+y\right)\left(x+y+1\right)}{2}+y$ is injective (one-to-one) from $\textbf{N}^2$ to $\textbf{N}$

Asked Nov 05 '23 at 14:41

Active Nov 05 '23 at 14:51

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How to show $$f(x,y)=\dfrac{\left(x+y\right)\left(x+y+1\right)}{2}+y$$ is injective (one-to-one) from $\mathbb{N}^2$ to $\mathbb{N}$?

edited Nov 05 '23 at 14:51

asked Nov 05 '23 at 14:41

Math 512

1

this was certainly asked before here and it is also called the Cantor pairing function – dezdichado Nov 05 '23 at 14:55
https://artofproblemsolving.com/community/c4h1078689p4726690 – dezdichado Nov 05 '23 at 14:56
Also here or here. – Jyrki Lahtonen Nov 05 '23 at 14:59
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And please check out the abridged guide to new askers for ways to make sure your questions won't attract negative attention. – Jyrki Lahtonen Nov 05 '23 at 15:01
thanks a lot!
good luck to me explaining this to a high school student !!
– Math 512 Nov 05 '23 at 15:05
You can explain how this function emerges in a countability proof $\Bbb{N}^2$ when you list pairs of numbers $(x,y)$ one diagonal at a time (sorted primarily according to the sum $x+y$ and secondarily according to $y$). – Jyrki Lahtonen Nov 06 '23 at 10:09

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