Two true or false questions:
$\mathbb{Q}^+$ means the positive rational numbers (no 0)
$\mathbb{N}$ means all natural numbers
- Every function $f\colon \mathbb{Q}^+ \to \mathbb{N}$ is not one-to-one
- Every function $f\colon \mathbb{N} \to \mathbb{Q}^+$ is not onto
The textbook says each of these questions are false, but doesn't explain why.
The first one kind of makes sense to me, because it seems like $\mathbb{Q}^+$ has a bigger cardinality than $\mathbb{N}$. However, if that was the case, wouldn't #2 be true? I think of $\mathbb{Q}^+$ as... infinite in two dimensions (1,2,3,4,5.... AND 1.1, 1.01, 1.001, 1.0001....).
Can anyone help me get some intuitive grasp one why these two questions are false?