Cardinality of Natural and positive Rational numbers

Question

I'm trying to prove that $\mathbb{Q}^{+}$ has the same cardinality with $\mathbb{N}$ by using the theorem Schroder-Bernstein.

So I just have to prove that $\mathbb{Q}^{+}\preceq \mathbb{N}$ and $\mathbb{N}\preceq \mathbb{Q}^{+}$.

For the first one we have to find a function $f:\mathbb{Q^{+}}\rightarrow \mathbb{N}$ one-to-one.Is this the same like showing that there exists $f:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$ one-to-one ???Because $\mathbb{Q^{+}}$ it's of the form $\left \{ \frac{p}{q},p,q\in\mathbb{N} \right \}$.

Also the same idea for $\mathbb{N}\preceq \mathbb{Q^{+}}$,is it true that is the same with finding an one-to-one function $f:\mathbb{N}\rightarrow \mathbb{N}\times\mathbb{N}$???

Any advise will be helpful.

Answer 1

You need an injective map from $\Bbb Q^+ \to \Bbb N$. Yes, it is enough to find an injection $\Bbb {N \times N \to N}$ because you have an injection $\Bbb {Q^+ \to N \times N}$ by taking the lowest term representation of each rational. Then you need an injection $\Bbb {N \to Q^+}$ but the identity supplies that.