I Talked to people from class and seem to not really know the idea behind these two questions:
- Is $2^{\Bbb Z}$ countable, or does it have the same cardinality as $\Bbb R$?
- Is $\Bbb Z \times \Bbb R$ countable, or does it have the same cardinality as $\Bbb R$?
Any information that can point me to better comprehension would be greatly appreciated thank you again.
$2^\mathbb{Z}$ is uncountable.
One way to prove that is to start with understanding that $2^\mathbb{N}$ can be viewed as representing the points in the unit interval: think of each function from $\mathbb{N}$ to $\{0,1\}$ as giving you the binary "decimal" representation.
Now $\mathbb{R}$ and the unit interval are equicardinal. So are $\mathbb{N}$ and $\mathbb{Z}$. So you're done.
$\mathbb{Z} \times \mathbb{R}$ uncountable because it contains the uncountable $y$ axis - points with coordinates $(0, r)$.