the Cartesian product of a finite number of countable sets is 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 , the set consists of all countably infinite sequences of zeros and ones. Each element of can be viewed as a binary fraction and can therefore be mapped to a unique real number in and is, of course, not countable.
|Title||the Cartesian product of a finite number of countable sets is countable|
|Date of creation||2013-03-22 15:19:45|
|Last modified on||2013-03-22 15:19:45|
|Last modified by||BenB (9643)|
|Synonym||The product of a finite number of countable sets is countable|