Countability of Sets of Functions and Generalization to Infinite Sets

[simmonj7]

Homework Statement

Determine whether or not the set is countable or not. Justify your answer.

The set Bn of all functions f:{1,2,...,n}$\rightarrow$N,

where N is the natural numbers.

Homework Equations

1.)A countable union of countable sets is countable

2.)A finite product of countable sets is countable

The Attempt at a Solution

In the solution, a theorem is used that is not in my book.


It goes something like this Cardinality(A)=c and f:A$\rightarrow$B, then the set of functions is B^a.

I was wondering if anyone could help me figure out what information I was supposed to derive this from?

Thank you.

 

[micromass]

Maybe you can find a bijection between the set of all functions

$$\{1,2\}\rightarrow \mathbb{N}$$

and $\mathbb{N}\times \mathbb{N}$. Generalize.