[stiglitz]

You can always "take an element from each set" over and over, but for this mapping to be shown to be a 1:1, you need to be more specific.

Example: the set of all natural numbers (0, 1, ...) is the same "size" as the set of all even numbers, because you can map any number N to the number 2N. It's obvious that any natural number N is uniquely accounted for by this mapping, since you can double any natural number (with a unique even result), and it's obvious that any even number is uniquely accounted for, because any even number divided by 2 is a (unique) natural number.

But if your mapping is defined as "take an arbitrary rational number and arbitrary irrational number, over and over forever", then there's no guarantee that, given an arbitrary irrational number, your mapping has defined a unique associated rational number.