## Enumerability

A set of objects is enumerable or countable if it can be counted, or more precisely, if it can be put into a one to one correspondence with the set of natural numbers

Some sets which are 'obviously' not enumerable turn out to be, and also surprisingly, sets whih are larger or smaller than setn , containing setn as a proper subset, or a proper subset of setn , turn out to be capable of being put into a one to one correspondence with elements of setn , so are in a very real sense, the same size as setn .

The set of even numbers E can be put into a one to one correspondence with elements of setn .

Ifthen there existsand there is one such element offor each and every element ofand vice versa so the mapping

with

is a one to one mapping fromto

Much more surprisingly, the set of rational numbers (fractions) is enumerable. There are many more rational numbers that natural numbers since every natural number can be written as a fraction, but not vice versa. To demonstrate enumerability, construct the table shown below.

With natural numbers in the margins, start in the upper left hand corner and work diagonally. The first rational number isthe second isthe third isthe fourth isand so on. Every rational number appears somewhere is the table (the rational numberappears at positionin the table) so the rational numbers are enumerable.