Countable set

In mathematics, a set is said to be countable if its elements can be "numbered" using the natural numbers. More precisely, this means that there exists a one-to-one mapping from set to the set of natural numbers.

A countable set is either finite or countably infinite. A set which is not countable is called uncountable.

Any subset of a countable set is countable.

The image of a countable set (under any function) is a countable set.

The countable union (i.e., the union of a countable family) of countable sets is countable.

The Cartesian product of countable sets is countable.

Examples of countably infinite sets

The set of integers is enumerable. Indeed, the function

$f(n)=(-1)^{n}\cdot \left\lfloor {\frac {n+1}{2}}\right\rfloor$

is a bijection between the natural numbers and the integers:

n	0	1	2	3	4	5	$\cdots$
f(n)	0	-1	1	-2	2	-3	$\cdots$

The union of the set of integers with any finite set is enumerable. For instance, given the finite set

$A=\{a_{0},a_{1},\ldots ,a_{n-1}\}$

with cardinality n, this function will enumerate all elements of $A\cup \mathbb {N}$:

$i\mapsto {\begin{cases}a_{i},&i<n\\i-n,&{\mbox{otherwise}}\end{cases}}$

Interestingly, the union of any two enumerable sets is enumerable. Given $A=\{a_{0},a_{1},a_{2},\ldots \}$ and $B=\{b_{0},b_{1},\ldots \}$ which are both enumerable, the function

$i\mapsto {\begin{cases}a_{i/2},&i{\mbox{ is even}}\\b_{(i-1)/2},&i{\mbox{ is odd}}\end{cases}}$

enumerates all elements of both sets. In fact, the union of an enumerable number of enumerable sets is still enumerable. Suppose we have a collection of sets $A_{0}=\{a_{0,0},a_{0,1},a_{0,2},\ldots \},A_{1}=\{a_{1,0},a_{1,1},\ldots \},A_{2},A_{3},\ldots$. Then we can create a bijection between the whole numbers and all the elements of all the $A_{i}$ as follows:

$a_{i,j}\leftrightarrow {\frac {(i+j)^{2}+i+j}{2}}+j$

Notice that this concept is used in the proof of the enumerability of the rational numbers, given below.

The set of rational numbers is an enumerable set. Envision a table which contains all rational numbers (below). One can make a function that dovetails back and forth across the entire area of the table. This function enumerates all rational numbers.

Table of all rational numbers

	0	1	2	$\ldots$
1	$0 \over {1}$	$1 \over {1}$	$2 \over {1}$	$\ldots$
2	$0 \over {2}$	$1 \over {2}$	$2 \over {2}$	$\ldots$
3	$0 \over {3}$	$1 \over {3}$	$2 \over {3}$	$\ldots$
$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\ddots$

Counterexamples

An uncountable set is one which is not countable. One example is the set of real numbers is not enumerable, which we will prove by contradiction. Suppose you had an infinitely long list of all real numbers (below), in no particular order, expressed in decimal notation. This table, itself, is an enumeration function. We demonstrate the absurdity of such a list by finding a number which is not in the list.

Enumeration of all real numbers

Order	Real Number
0	0.32847...
1	0.48284...
2	0.89438...
3	0.00154...
4	0.32425...
...	...

Specifically, we construct a number which differs from each real number by at least one digit, using this procedure: If the ith digit after the decimal place in the ith number in the list is a five, then our constructed number will have a four in the ith place, otherwise a five. From our example list, we would construct the number 0.55544... By construction, this number is a real number, but not in our list. As a result, the enumeration function is not onto.

This is known as Cantor's diagonalization argument. It is important to note that this argument assumes that two different decimal notations represent two different numbers. This is generally true, with one notable exception. Any digit followed by an infinite series of nines is equivalent to the same digit, increased by one, followed by an infinite series of zeros. For example, 0.3999… is equivalent to 0.4000…. This argument converts individual digits to either fours or fives, thereby avoiding any complications that could arise from this detail.