Given two finite sets, it is simple to compare their sizes. For example, wewould say that the set of corners of a pentagon is larger than the set of playersin a string quartet, simply because the first set has five elements, while thesecond has only four.But can we compare the sizes of infinite sets in any meaningful way? Wehave encountered many different infinite sets at various points in this book,such as N, Z,Q, R, C, N×N,Q×R×C, and so on. How can we comparethese with each other?There is a way to do this using functions. To set this up, let us begin withan elementary observation about finite sets. If S is a set of size n, say S = {s1,s2, . . . ,sn}, then the function f : S→{1,2, . . . ,n} defined by

f (s1) = 1, f (s2) = 2, . . . , f (sn) = n is a bijection. Thus we can say

S has size n ⇔ there is a bijection from S to {1,2, . . . ,n}. We now extend this notion to arbitrary sets.