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 finitely many countable sets is countable.

Examples of countably infinite sets

Integers

The set of integers is countable (countably infinite). Indeed, the function

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

is a one-to-one correspondence between all natural numbers and all integers:

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

Union of two countable sets

The union of the set of natural numbers and any finite set is countable. For instance, given the finite set

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

of n elements, the function

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

shows that $A\cup \mathbb {N}$ is countable.

More generally, consider two countably infinite sets:

$A=\{a_{0},a_{1},a_{2},\ldots \}$ and $B=\{b_{0},b_{1},\ldots \}$

then

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

is a one-to-one correspondence between $\mathbb {N}$ and $A\cup B$.
(Note that in the example of the integers the same method has been used: Let A be the positive integers and B be the negative integers.)
This situation is addressed in the example of Hilbert's hotel.

Rational numbers

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

Rational numbers

The set of rational numbers is not countable. The proof is a proof by contradiction, an indirect proof:

Suppose that the set of rational numbers is countably infinite, then the interval of rational numbers r with $0<r<1$ is (as a subset) also countable, and the interval can be written as a sequence:

$\{r\mid 0\leq r<1,r\in \mathbb {R} \}=\{r_{i}\mid i\in \mathbb {N} \}$

Since any real number between 0 and 1 can be written as a decimal number the sequence r_i can be written as an infinitely long 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.