Let $F$ be the set of all functions $f : ${0,1}$ -> \Bbb N$. Prove that $F$ is countable.
Let $G$ be the set of all functions $g$ : $\Bbb N$ -> ${0,1}$. Prove that $G$ is uncountable.
My poor attempt (a):
The left hand grid shows the elements $ ${0,1}$ \times \Bbb N$ and the right hand grid shows the natural numbers. By taking the max$(a,b)$ where $a \in ${0,1}$ $, $b$ $\in$ $\Bbb N$, there is a bijection from $ ${0,1}$ \times \Bbb N$ to $\Bbb N$, namely the function $g$ : $ ${0,1}$ \times \Bbb N$ -> $\Bbb N$. Each natural number appears exactly once in the right hand grid and each pair $ ${0,1}$ \times \Bbb N$ appears exactly once in the left hand grid and maps to exactly one natural number. So countably infinite.
(b) I don't really know. I was thinking of forming an injective function from $\Bbb R$ to $ \Bbb N \times ${0,1}$ $. Any help ?