Well, what would you say is a bijection between and ?K, N. N is the set of natural numbers, and K is the set of natural numbers which are power of 2.

An element of is a function . Equivalently, it is an infinite sequence of 0's and 1's: ...p(N), {0, 1}^N. p(N) is, of course, the power set of N.

Given a set , consider the so-called characteristic function of . This function equals 1 on elements of and equals 0 otherwise. A set and its characteristic function are basically the same thing in the sense that one can be unambiguously reconstructed from the other.

This is true not only for but for any set, finite or infinite.