Countable set

In mathematics, a set X is said to be enumerable or countable if there exists a one-to-one mapping from the set of natural numbers onto X. By the definition, an enumerable set has the same cardinality as the set of natural numbers.

Enumerable sets are subject to many useful properties. Inductive proofs rely upon enumeration of induction variables.

Examples of enumerable 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 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	$0 \over {1}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\over{1}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\over{1}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\over{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\over{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\over{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\over{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\over{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\over{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vdots}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vdots}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vdots}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vdots}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddots}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \{ a_0, a_1, \ldots, a_{n-1} \}}

with cardinality n, this function will enumerate all elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cup \mathbb{N}} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \mapsto \begin{cases} a_i, & i < n \\ i-n, & \mbox{otherwise} \end{cases}}

Counterexamples

The set of real numbers is not enumerable, which we will prove by contraction. 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.

See also

• cardinality
• arity
• set theory
• The set of integers, Z
• The set of rational numbers, Q
• The set of real numbers, R