countable or uncountable set of functions I was working through some first year past papers(maths exams) and i found these two questions. I think the first statement is true and the second statement is false, but i do not really now how to formalize this in any way.
Let S = {0, 1}. The set of functions f : S → N is countable.
xix Let S = {0, 1}. The set of functions f : N → S is countable
Any ideas?thanks.

your right, the first set is countable, it is essentially $\mathbb N ^{|S|}$.  My favorite injection from this set to $\mathbb N$ is the one that sends $(a_1,a_2,\dots,a_{|S|})$ to the number $2^{a_1}3^{a_2}5^{a_3}\dots p_{|S|}^{a_{|S|}}$
The second set is $|S|^\mathbb N$ and is not countable. You can show that $2^\mathbb N$ is not countable by using Cantor's theorem for example, or bijecting it with the real numbers via binary representation.
