How can the counting numbers be paired one-to-one with the even numbers?

In mathematics, the counting numbers (or natural numbers) can be paired one-to-one with the even numbers by using a simple rule that maps each counting number $n$ to its double, $2n$.123

This correspondence is expressed as:
{1, 2, 3, ...} $\to$ {2, 4, 6, ...}2

By applying the function $f(n)=2n$, every counting number is matched with a unique even number, and every even number is matched with a unique counting number, with no elements left over on either side. This demonstrates that the set of even numbers is countably infinite, possessing the same cardinality as the set of all counting numbers, a concept known as aleph-null (${\aleph }_0$).2431

While our intuition might suggest that the set of even numbers should be smaller because it is a proper subset of the counting numbers, this paradoxical result illustrates that infinite sets do not follow the same rules as finite sets regarding size and inclusion.154