How can an infinite set have the same size as one of its proper subsets?

In mathematics, an infinite set can have the same size as one of its proper subsets because, unlike finite sets, infinite sets do not follow the principle that a whole must be larger than its parts.1

This phenomenon occurs because the size of a set is determined by equivalence, which is defined by the ability to create a one-to-one correspondence between the elements of two sets. For an infinite set, it is possible to pair every element of the set with a unique element of a proper subset, leaving no elements left over in either collection.12

For example, the set of all natural numbers ($\mathbb{N}$) can be paired with the set of its own squares ($n^2$) using the rule $f(n)=n^2$. Even though the squares are only a small portion of the natural numbers, every natural number has a corresponding square, and every square corresponds to a natural number, demonstrating that both sets have the same cardinality, or size. This property is a defining characteristic of infinite sets and was famously illustrated by the Infinite Hotel Paradox, where a fully occupied hotel with infinitely many rooms can still accommodate new guests by shifting existing ones.13