the Cartesian product of a finite number of countable sets is countable
Theorem 1
The Cartesian product of a finite number of countable sets is countable.
Proof: Let A1,A2,…,An be countable sets and let S=A1×A2×⋯×An. Since each Ai is countable, there exists an injective function fi:Ai→ℕ. The function h:S→ℕ defined by
h(a1,a2,…an)=n∏i=1pfi(ai)i |
where pi is the ith prime
is, by the fundamental theorem of arithmetic, a bijection between
S and a subset of ℕ and therefore S is also countable.
Note that this result does not (in general) extend to the
Cartesian product of a countably infinite collection
of countable
sets. If such a collection contains more than a finite number of sets
with at least two elements, then Cantor’s diagonal argument can be
used to show that the product
is not countable.
For example, given B={0,1}, the set F=B×B×⋯ consists of all countably infinite sequences of zeros and ones.
Each element of F can be viewed as a binary fraction and can
therefore be mapped to a unique
real number in [0,1) and [0,1) is, of course, not countable.
Title | the Cartesian product of a finite number of countable sets is countable |
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Canonical name | TheCartesianProductOfAFiniteNumberOfCountableSetsIsCountable |
Date of creation | 2013-03-22 15:19:45 |
Last modified on | 2013-03-22 15:19:45 |
Owner | BenB (9643) |
Last modified by | BenB (9643) |
Numerical id | 13 |
Author | BenB (9643) |
Entry type | Theorem |
Classification | msc 03E10 |
Synonym | The product of a finite number of countable sets is countable |
Related topic | CardinalityOfACountableUnion |
Related topic | AlgebraicNumbersAreCountable |
Related topic | CardinalityOfTheRationals |