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 $1$?


Is it correct to call $[0,1]$ a set?

Yes, although it doesn't sound natural to me if you "call $[0,1]$ a set". I'd rather call it the closed interval. You can also write as $\{x|x\in[0,1]\}$ (trivial). 
$x\in \{0,1\}$ means $x=1$ or $x=0$.
