by Teerex
Last Updated November 10, 2018 21:20 PM

how can I show that an injection from N to Q+ is f(x)=x+1?

I know how to show there is injection from Q+ to N using the "diagonalization method". But I have no clue how to start with the reverse.Can someone help me out?

One of way of showing $f : \mathbb{N} \rightarrow \mathbb{Q}^{+}$ is injective is to show that for two arbitrary elements $a, b$ $f(a) = f(b) \implies a = b$.

Consider two arbitrary elements $a, b \in \mathbb{N}$. Then, we have

$$f(a) = a + 1, $$

and $$f(b) = b + 1.$$

It's pretty easy to verify that both $f(a)$ and $f(b)$ are in $\mathbb{Q}^{+}$ (if you want to prove this as well, just write each of them as a ratio of two integers). Then, if we have $f(a) = f(b)$, it follows that $a = b$, which is what we wanted to show.

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