$2^{\{1,2,3\}}$ explained Can someone explain me this:
$2^{\{1,2\}}$
I know this equals to:
$\{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$
Right?
So $\{1\}$ for example is an element of $2^{\{1,2\}}$,
$\{1\} \in 2^{\{1,2\}}$
But can someone explain me to what this notation equals?
$2^{\{1,2,3\}}$
Does it equal $\{\varnothing, \{1\}, \{2\}, \{1, 2, 3\}\}$?
And why is $\{\{1\}, \{2\}\}$ a subset of $2^{\{1,2,3\}}$?
Thanks! I'm very new to math at my university and I lacked the basics. But I'm studying to catch up!
 A: You're talking about powersets. A powerset  of a set is the set of all possible subsets of that set. The reason $2^S$ is used is because the number of elements in the powerset is $2$ raised to the power of the number of elements in $S$.
$2^{\{1,2,3\}}$ contains all $8$ subsets of $\{1,2,3\}$:
$\{\}$
$\{1\}$
$\{2\}$
$\{3\}$
$\{1,2\}$
$\{1,3\}$
$\{2,3\}$
$\{1,2,3\}$  
Note that each element of $2^{\{1,2,3\}}$ is a set. As you can see from the above list of the elements of $2^{\{1,2,3\}}$, two of the sets contained in the powerset are $\{1\}$ and $\{2\}$, so those two sets taken together are indeed a subset of $2^{\{1,2,3\}}$.
A: This (confusing) notation is: $2^X$ means "all subsets of $X$".  So
$$
2^{\{1,2,3\}} = \{
\\
\emptyset,
\\
\{1\},\{2\},\{3\},
\\
\{1,2\},\{1,3\},\{2,3\},
\\
\{1,2,3\}
\\
\}
$$
A: The notation $[n]=\{ 1,2, \cdots ,n \}$ denotes the set of the first $n$ natural numbers. (In particular $[3]= \{1,2,3\}$.)  This is quite natural from an enumerative point of view; how many elements does $[3]$ have ? Well, $3$. 
The notation $2^{[n]}$ to denote the set of subsets of $[n]$ is then quite natural from an enumerative point of view; how many elements does $2^{[3]}$ have ? Well, $2^3=8$. 
And of course these $8$ sets are $ \phi , \{1 \},\{2 \},\{3 \},\{1,2 \},\{1,3 \},\{2,3 \},\{1,2,3 \}$.
