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My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite.

I understand that for the set of powers to be countably infinite, there must exist a bijection between the set and $\mathbb{N}$.

I'm having some trouble trying to figure out how to properly prove this! Please help!

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  • $\begingroup$ Did you get anywhere with this? $\endgroup$ – Rubicon Apr 5 '16 at 15:31
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$y\mapsto 10^y$ is your desired bijection: It is onto by definition, and readily verified to be one-to-one.

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  • $\begingroup$ Hagen, could you explain this a little more? How would you go about showing the bijection. Obviously you need to show the injection and surjection, but how...specific to this example? I know this would be simple for you, but I can't quite get there. None of our course literature covers questions like this. @badatmaths have you had any success developing your solution any further? $\endgroup$ – Rubicon Apr 5 '16 at 14:08

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