Sets to compare

Below is a list of sets whose sizes we are interested in comparing.

The original list:
$\mathbb{Z}$, the set of all integers.
$\mathbb{N}$, the set of positive integers (also called "natural numbers").
$\mathcal{P}(\mathbb{N})$, the power set of $\mathbb{N}$.
$\mathbb{R}$, the set of all real numbers (points on a number line).
[0,1], the closed interval from 0 to 1, defined as {x : x is a real number and 0 $\leq$ x $\leq$ 1}.
$\mathbb{Q}$, the set of all rational numbers
$\mathbb{C}$, the set of all complex numbers

(0,1), the open interval from 0 to 1, defined as {x : x is a real number and 0 < x < 1}.

$I_n$, the integers from 1 to n.
$(1, \infty)$, aka {x : x is a real number and 1 < x }.
The set of nonnegative integers, $\mathbb{N} \cup \{0\}$.
$\mathcal{P}(I_n)$, the power set of $I_n$.