FACTOID # 19: Cheap sloppy joes: Looking for reduced-price lunches for schoolchildren? Head for Oklahoma!
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Countable set

In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers. A set that is not countable is called uncountable. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... In mathematics, an uncountable set is a set which is not countable. ...


Note that countable set is sometimes given a more specific definition: sometimes, it is defined as a set with the same cardinality as the set of natural numbers. The difference between the two definitions is that the former defines finite sets to be countable, while the latter does not.

Contents


Definition

A set S is called countable if there exists an injective function In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... Partial plot of a function f. ...

fcolon S to mathbb{N}

If f is also bijective then S is called countably infinite or denumerable. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...


As noted above, this terminology is not universal: some authors define denumerable to mean what we have called "countable"; some define countable to mean what we have called "countably infinite".


The next result offers an alternative definition of a countable set S in terms of a surjective function: A surjective function. ...


THEOREM: Let S be a nonempty set. The following statements are equivalent:

  1. S is countable
  2. There exists an injective function fcolon S to mathbb{N}
  3. There exists a surjective function gcolon mathbb{N} to S

Gentle introduction

The elements of a finite set can be listed, say { a1, a2, ..., an }. However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble pieces if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of scrabble blocks, rather than apple pies or racecars. Thus an infinite set is still a set, insofar as it is a tool for separating out things with different properties.


Now what is a countably infinite set?


Technically, a countably infinite set is any set which, in spite of its boundlessness, can be shown equivalent to the natural numbers — nothing more, nothing less. This makes it possible to set apart elements of a countably infinite set using natural numbers as indices, and in turn puts the logic associated with them in very close proximity to the logic associated with the natural numbers themselves; and this makes such sets easily logically tractable. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...


A more formal introduction

It might then seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size.


To elaborate this we need the concept of a bijection. Do the sets { 1, 2, 3 } and { a, b, c } have the same size? In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

"Obviously, yes."
"How do you know?"
"Well it's obvious. Look, they've both got 3 elements".
"What's a 3?"

This may seem a strange situation but, although a "bijection" seems a more advanced concept than a "number", the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a ↔ 1, b ↔ 2, c ↔ 3

Since every element of { a, b, c } is paired with precisely one element of { 1, 2, 3 } (and vice versa) this defines a bijection.


We now generalise this situation and define two sets to be of the same size precisely when there is a bijection between them. For all finite sets this gives us the usual definition of "the same size". What does it tell us about the size of infinite sets?


Consider the sets A = { 1, 2, 3, ... }, the set of positive integers and B = {2,4,6,...}, the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this we need to exhibit a bijection between them. But this is easy: 1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...


As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets, a situation which is impossible for finite sets.


Likewise, the set of all ordered pairs of natural numbers is countably infinite, as can be seen by following a path like this one: An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

begin{matrix} (0,0) & rightarrow & (0,1) & & (0,2) & rightarrow & (0,3) &  & swarrow & & nearrow & & swarrow & &  (1,0) & & (1,1) & & (1,2) & & ddots &  downarrow & nearrow & & swarrow & & & &  (2,0) & & (2,1) & & ddots & & &  & swarrow & & & & & &  (3,0) & & ddots & & & & &  downarrow & & & & & & &  vdots & & & & & & & end{matrix}

The resulting mapping is like this: 0 ↔ (0,0), 1 ↔ (0,1), 2 ↔ (1,0), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), … It is evident that this mapping will cover all such ordered pairs.


Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for every possible fraction, we can come up with a distinct number corresponding to it. Since every natural number is also a fraction N/1, we can conclude that there are the same number of fractions as there are of whole numbers. In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ...


THEOREM: The Cartesian product of finitely many countable sets is countable. In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...


This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedly mapping the first two elements to a natural number. For example, (2,0,3) maps to (5,3) which maps to 41. The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...


Sometimes more than one mapping is useful. This is where you map the set which you want to show countably infinite, onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers can easily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (pq). In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...


What about infinite subsets of countably infinite sets? Do these have fewer elements than N?


THEOREM: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite set is countably infinite.


For example, the set of prime numbers is countable, by mapping the n-th prime number to n: In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct positive divisors, one and itself. ...

  • 2 maps to 1
  • 3 maps to 2
  • 5 maps to 3
  • 7 maps to 4
  • 11 maps to 5
  • 13 maps to 6
  • 17 maps to 7
  • 19 maps to 8
  • 23 maps to 9
  • etc.

What about sets being "larger than" N? An obvious place to look would be Q, the set of all rational numbers, which is "clearly" much bigger than N. But looks can be deceiving, for we assert In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...


THEOREM: Q (the set of all rational numbers) is countable.


Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto the subset of ordered triples of natural numbers (a, b, c) such that b > 0, a and b are coprime, and c ∈ {0, 1} such that c = 0 if a/b ≥ 0 and c = 1 otherwise. Coprime - Wikipedia /**/ @import /skins-1. ...

  • 0 maps to (0,1,0)
  • 1 maps to (1,1,0)
  • −1 maps to (1,1,1)
  • 1/2 maps to (1,2,0)
  • −1/2 maps to (1,2,1)
  • 2 maps to (2,1,0)
  • −2 maps to (2,1,1)
  • 1/3 maps to (1,3,0)
  • −1/3 maps to (1,3,1)
  • 3 maps to (3,1,0)
  • −3 maps to (3,1,1)
  • 1/4 maps to (1,4,0)
  • −1/4 maps to (1,4,1)
  • 2/3 maps to (2,3,0)
  • −2/3 maps to (2,3,1)
  • 3/2 maps to (3,2,0)
  • −3/2 maps to (3,2,1)
  • 4 maps to (4,1,0)
  • −4 maps to (4,1,1)
  • ...

By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers. In mathematics, an algebraic number relative to a field is any element of a given field containing such that is a solution of a polynomial equation of the form: anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree of the polynomial, every coefficient... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). ...


THEOREM: (Assuming the axiom of choice) The union of countably many countable sets is countable. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...


For example, given countable sets a, b, c ...


Using a variant of the triangular enumeration we saw above:

  • a0 maps to 0
  • a1 maps to 1
  • b0 maps to 2
  • a2 maps to 3
  • b1 maps to 4
  • c0 maps to 5
  • a3 maps to 6
  • b2 maps to 7
  • c1 maps to 8
  • d0 maps to 9
  • a4 maps to 10
  • ...

Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem. In mathematics, two sets are said to be disjoint if they have no element in common. ...


THEOREM: The set of all finite-length sequences of natural numbers is countable. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...


This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.


THEOREM: The set of all finite subsets of the natural numbers is countable. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...


If you have a finite subset, you can order the elements into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.


Further theorems about uncountable sets

Remember our example of the scrabble words. Although we can keep asking for more letters from the bag, each word we form is finitely long. The number of possible words is the same as the number of natural numbers. If we permit infinitely long words, then the number of possible "words" is greater than this. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...


In fact, with infinitely long words, the number of words is the same as the number of real numbers.


We noted earlier that there are no more fractions than there are natural numbers. The decimal expansion of a fraction is always a finitely long decimal number followed by a repeating decimal. A numeral is a symbol or group of symbols that represents a number. ...

  • 0.33333333333 ...
  • 12.648986986986986986 ...
  • 1.75

Let's say we use our decimal point to also indicate the start of the repeater:

  • ..3
  • 12.648.986
  • 1.75.

Then we can express any fraction using a finitely long decimal expansion with repeating bit. It's clear that this is the same situation as with our finitely long scrabble words, and so once again the number of possible fractions is not greater than the number of natural numbers.


Mentionable numbers

The set of all mentionable numbers is countable, where this means a correspondence with finite, and even infinite, strings in English, or any other language.


See also


  Results from FactBites:
 
Countable set - Biocrawler (1054 words)
In mathematics the term countable set is used to describe the size of a set, e.g.
The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision, both from the beginning the distinguishing faculties of mathematicians.
The set of real numbers is uncountable, and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument).
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m