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

In set theory, a one-to-one correspondence is the fundamental method used to compare the sizes, or cardinality, of two sets.12

If two sets can be put into a one-to-one correspondence—meaning every element in the first set is paired with exactly one unique element in the second set, and vice versa—they are considered to have the same size. For finite sets, this process aligns with our intuitive understanding of counting; for example, a set is defined as having a cardinality of n if its elements can be paired with the set {1, 2, 3, ..., n}.12

This concept is particularly significant when applied to infinite sets. Georg Cantor demonstrated that two infinite sets can have the same cardinality if a one-to-one correspondence exists between them, even if one set is a proper subset of the other. Furthermore, by showing that certain sets cannot be put into a one-to-one correspondence with others, Cantor proved that some infinite sets are strictly larger than others, leading to the development of transfinite numbers.2