Difference between $x\in[0,1]$ and $x\in \{0,1\}$? If I write 
$$
x\in [0,1] \tag 1
$$
does it mean $x$ could be ANY number between $0$ and $1$?
Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$? 
Q2:
If I instead have
$$
x\in \{0,1\} \tag 2
$$
does it mean $x$ could be only $0$ OR $1$?

$[0,1]$ is (defined as) the set $\{ x \in \mathbb{R} : 0 \leq x \leq 1 \}$, i.e. it is a set that contains every real number between $0$ and $1$ (inclusive). It contains an uncountable number of elements.
$\{0,1\}$ is a set containing 2 elements: $0$, and $1$.
