Why do the natural numbers and even numbers both have cardinality aleph-null?

In mathematics, the natural numbers and the set of even numbers both have the cardinality aleph-null (${\aleph }_0$) because they can be placed into a one-to-one correspondence, a concept fundamental to set theory.12

A set is defined as having cardinality ${\aleph }_0$ if its elements can be paired with the set of natural numbers $\{1,2,3,\dots \}$. For the set of even numbers, this pairing is achieved by the mapping $f(n)=2n$, where every natural number $n$ is associated with a unique even number $2n$ (e.g., $1\to 2$, $2\to 4$, $3\to 6$, and so on).13

Because every element in the set of natural numbers has a distinct partner in the set of even numbers and vice versa, the two sets are considered equivalent in size, despite the even numbers being a proper subset of the natural numbers. This property, which initially surprised many mathematicians, is a defining characteristic of infinite sets and was systematically explored by Georg Cantor.412