Express Set of Dyadic with union/complementation of closed intervals

Question

Let $S = \{{m\over 2^n}|m\in\mathbb Z, n\in \mathbb N\cup\{0\}\}$

1. Can $S \cap[0,1]$ be expressed as countable union, complement or intersection of closed intervals with non zero, dyadic length and dyadic endpoints ?
2. Further, can any subset of $[0,1]$ be expressed with the same rules?

As for 1, as we can do the following:
Let $\{a_n\}_n$ be an enumeration of dyadics in $[0,1]$
def $A_n = [0,a_n]\cap[a_n,1] = \{a_n\}$
Then take a union of $A_n$'s
Incase $a_n = 0$ or $1$, we can create decreasing/increasing sequence of dyadics and then take complement
Is the solution correct? Please also provide hints/counterexample for 2