# injective function from N to Q+

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?

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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.

Ekesh
November 10, 2018 21:19 PM