Correspondence

A correspondence is a relation between two sets such that each member in one set corresponds to $n$ members in the other set, where $n$ commonly equals $1$.

Lists - the beginning

Consider the task of counting the number of integers between 14 and 103 inclusive. We could simply list those integers and count them. However, we can renumber those integers so that they correspond to the counting numbers (positive integers), starting with 1. In this correspondence, 14 corresponds to 1 (for the 1st integer in the list), 15 with 2, 16 with 3, etc. The relationship between the members of each pair is that the second is 13 less than the first. So, we know that 103 corresponds to the 103 - 13 = 90th integer in the list. Thus, the list is 90 integers long.

Note that $13 = 14 - 1$, or 1 less than the first integer in the list. If we start our list with $n$ and end with $m$ (i.e. m and n inclusive), the number of integers in the list is

$m - (n -1) = m - n + 1.$

One-to-One Correspondence

Main article: Bijection

A one-to-one correspondence, or bijection, is a function which is both injective (or one-to-one) and surjective (or onto). A function has a two-sided inverse exactly when it is a bijection between its domain and range.

One-to-one correspondences are useful in a variety of contexts. In particular, bijections are frequently used in combinatorics in order to count the elements of a set whose size is unknown. Bijections are also very important in set theory when dealing with arguments concerning infinite sets.