Page "Countable set" Paragraph 8

To understand what this means, we first examine what it does not mean.

For example, there are infinitely many odd integers, infinitely many even integers, and ( hence ) infinitely many integers overall.

However, it turns out that the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall.

This is because we arrange things such that for every integer, there is a distinct odd integer: ... − 2 → − 3, − 1 → − 1, 0 → 1, 1 → 3, 2 → 5, ...; or, more generally, n → 2n + 1.

What we have done here is arranged the integers and the odd integers into a one-to-one correspondence ( or bijection ), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.