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

The natural numbers can be paired one-to-one with the even numbers by establishing a bijection—a mapping where each element of the first set corresponds to exactly one unique element of the second set.

This correspondence is demonstrated by the function $f(n)=2n$, which maps every natural number $n$ to its double. Because every even number can be expressed as $2n$ for some natural number $n$, this mapping covers the entire set of even numbers, confirming that both sets have the same cardinality, known as aleph-null (${\aleph }_0$).12

The pairing can be visualized as follows:

• {1, 2}
• {2, 4}
• {3, 6}
• ...
• {$n$, $2n$}
• ...1

This result illustrates a fundamental property of infinite sets: a set can be put into a one-to-one correspondence with one of its own proper subsets, a concept that was central to the work of Georg Cantor in developing modern set theory.31