Let C be a countable set. Prove C^n is countable for all n.
I was thinking to show that the Cartesian product of any countable sets is countable by writing them out in a table and going through the elements diagonally to count them, similar to a proof I have seen to show that the rational numbers are countable.
this proves C^2 is countable since C is countable.
then using induction I can say that C^n+1 is countable because it = C^n (which is countable from the induction premise) x C and is therefore the cartesian product of two countable sets.
May 18th 2012, 02:50 PM
Re: countable sets
Originally Posted by stuffthings
Let C be a countable . Prove C^n is countable for all n.
Suppose that is the set of natural numbers.
Map by .
Show that is an injection.
Thus is countable. Now use induction.