What are the cardinality problems in set theory and how can they be solved?

[jav]

I have been studying set theory, and come across a few problems that I have not been able to solve. I am trying to prove the bijections exist.

Let N = Set of all natural numbers
Let B^A = Set of all functions from A to B

1) Prove |N^N x N^N| = |N^N|

2) Prove |(N^N)^N| = |N^N|

Any explanation into the inspiration behind solutions would be greatly appreciated.

It is simple to consider the identity function going the obvious direction, so I really only need to prove injection (ie. |A| < |B| for the sets in the order stated).

Thanks
Edit: Solved

 

[lippyka]

1) We can use the Cantor-Bernstein-Schroeder theorem to prove this. Assume we have two injections, f: NN x NN → NN and g: NN → NN x NN. We can construct an injection h: NN x NN → NN x NN by defining h(x,y) = (f(x,y),g(x)). This shows that |NN x NN| ≤ |NN|, and since the reverse is clearly true, |NN x NN| = |NN|.2)(NN)N is the set of all functions from N to NN. We can use a similar technique as above. Assume we have two injections, f : (NN)N → NN and g : NN → (NN)N. We can construct an injection h : (NN)N → (NN)N by defining h(f) = (f,g(f)). This shows that |(NN)N| ≤ |NN|, and since the reverse is clearly true, |(NN)N| = |NN|.