What do $x\in[0,1]^n$ and $x\in\left\{ 0,1\right\}^n$ mean? $x\in[0,1]^n$ 
$x\in\{0,1\}^n$ 
Thank you in advance.
 A: Remember:


*

*If $A$ is a arbitrary set $A^n:= \underbrace{A\times\ldots \times A}_{n\mbox{ times}}=\{(a_1,\ldots ,a_n): a_1\in A, \ldots,a_n\in A\}$,

*$[0,1]:= \{x\in\mathbb{R}: 0\leq x \leq 1 \}$,

*$\{0,1\}:= \{x\in\mathbb{R}: x=0 \mbox{ or } x=1 \}$.


Then 
\begin{align}
[0,1]^n:= 
& \underbrace{[0,1]\times\ldots \times [0,1]}_{n\mbox{ times}}
\\
=&
\{(x_1,\ldots ,x_n): x_1\in [0,1], \ldots,x_n\in [0,1]\}
\\
=
&
\{(x_1,\ldots ,x_n): 0\leq x_1\leq 1, \ldots,0\leq x_n\leq 1\}
\end{align}
and 
\begin{align}
\{0,1\}^n:= 
& \underbrace{\{0,1\}\times\ldots \times \{0,1\}}_{n\mbox{ times}}
\\
=&
\{(x_1,\ldots ,x_n): x_1\in \{0,1\}, \ldots,x_n\in \{0,1\}\}
\\
=
&
\{(x_1,\ldots ,x_n): x_1=0 \mbox{ or } x_1=1, \ldots,x_n=0 \mbox{ or } x_n=1\}
\end{align}
A: If $x=(x_1,\ldots,x_n)\in [0,1]^n$, then $0\leq x_i\leq 1$ for all $i$. If instead $x=(x_1,\ldots,x_n)\in \{0,1\}^n$, then $x_i=0$ or $x_i=1$ for all $i$.
For example, if $n=2$, then $[0,1]^2$ is the (filled) square in $\mathbb{R}^2$ with corners $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$, while $\{0,1\}^2$ is just the corners.
A: $x \in [0,1]^n$ means that $x$ looks like $x=(x_1, x_2, ..., x_n)$ where for each $i$,  $x_i$ lands in the range $0 \leq x_i \leq 1$. That is, for each $i$, $x_i \in [0,1]$. Whereas $x \in \{0,1\}^n$ means that $x$ looks like $x=(x_1, x_2, ..., x_n)$ where for each $i$, $x_i$ is either $0$ or $1$. That is, for each $i$, $x_i \in \{0,1\}$.
A: The first entry is an interval $ [0, 1] $ in the Reals raised to the power n. E. g. for n=2, you can think of this as the unit square. I.e. $ x \in [0, 1]^2 $ iff it is in the unit square
The second entry {0, 1} is simply the set containing only 0 and 1. Picturing our n=2 square from the fist example, this set describes only the 4 corners of the square. Therefore $ x \in \{0, 1\}^2 = \{(0,0), (0,1), (1,0), (1,1) \} $ iff it is one of the vertices of the square. 
Hopefully you can see how this extends to n in general.
