How does one-to-one correspondence compare the sizes of two sets?

In mathematics, a one-to-one correspondence is a method used to compare the sizes, or cardinality, of two sets by pairing each element in the first set with exactly one unique element in the second set, and vice versa, such that no elements remain unpaired in either set.12

When two sets can be put into such a correspondence, they are considered to have the same size. This concept is particularly significant when dealing with infinite sets. While intuition might suggest that a set should be larger than its proper subset, infinite sets can often be paired with their own subsets in a one-to-one correspondence, demonstrating that they share the same infinite cardinality.314

If it is impossible to establish a one-to-one correspondence between two infinite sets—meaning one set cannot be fully paired with the other without leaving elements behind—the sets are considered to have different sizes. For instance, Georg Cantor used a diagonal argument to prove that the real numbers cannot be paired with the integers, thereby demonstrating that some infinities are strictly larger than others.134