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A set is called countable if the number of element is finite or if it has the same number of elements as the natural numbers. The term countable stems from the fact that the natural numbers are often called counting numbers. A set with more elements is called uncountable. Not all uncountable sets have the same size. The different sizes of infinite sets are investigated in the theory of cardinal numbers.
A set S is called countable if there exists a surjective function
If f is even bijective then S is called countable infinite or denumerable.
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 elements of a finite set can be listed, say {a1, a2, ..., an}. However, not all sets are finite: for instance, the set of all integers or the set of all real numbers. 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?
"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
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, ...
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 pairs of natural numbers is countably infinite through the following "triangular" mapping:
THEOREM: The Cartesian product of finitely many countable sets is countable.
This 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.
Sometimes it is useful to use more than one mapping. This is where you map the set you want to show as countably infinite to another set. You 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 through p/q maps to (p,q).
What about infinite subsets of countably infinite sets? Do these have less 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.
(Proof omitted, some version of choice is required.)
For example, the set of prime numberIn mathematics, a prime number or prime for short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number . Hence a prime number has exactly two divisors. The number 1 is neither prime nor coms is countable, by mapping the nth prime number to n:
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
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 relatively prime, and c ∈ {0, 1}, and if a = 0 then c = 0.
By a similar development, the set of algebraic numberAbstract algebra Algebra In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form : a x n + a x n minus;1 + ··· + a x + a 0 where n is a posits is countable, and so is the set of definable numberA definable number is a real number which can be unambiguously defined by some mathematical statement. Formally, a real number a is called definable if there is some logical formula φ x in set theory which contains a single free variable x and such ths.
THEOREM: The unionAbstract algebra Algebra 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. This article uses mathematical symbols. Basic definition If of countably many countable sets is countable.
For example, given countable sets a, b, c ...
Using a variant of the triangular enumeration we saw above:
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.
THEOREM: The set of all finite-length sequenceThis is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, ors of natural numbers is countable.
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 subsetIf X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes X; Y ⊇ X. Every set Y is a subset of itself. A subset of Y which is not equas of the natural numbers is countable.
If you have a finite subset, you can order the elements to get a finite sequence. There are only countably many finite sequences, so there are also only countably many finite subsets.
Further theorems about uncountable sets: