Notation help. What do $\mathcal{P}(A)$ and $\chi_{-}$ mean here? Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand? 

 A: $\mathcal{P}(A)$ is the power set of $A$; it is a set whose elements are precisely the subsets of $A$. For example, if $A=\{1,2,3\}$, then $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}$.
Given any two sets $A$ and $B$, $A^B$ represents the set of all functions from $B$ to $A$. So $2^A$, with "$2$" meaning the set $\{0,1\}$ means the set of all functions $f\colon A\to \{0,1\}$; all functions whose domain is $A$, and which take only the values $0$ and $1$. 
If $A$ is a set and $S$ is a subset, then $\chi_S$ is the characteristic function of $S$. It is a function $\chi_S\colon A\to \{0,1\}$ defined by:
$$\chi_S(a) = \left\{\begin{array}{ll}
0 &\mbox{if $a\not\in S$,}\\
1 & \mbox{if $a\in S$.}
\end{array}\right.$$
By "$\chi_{-}$" he means the function that takes each subset $S$ to the characteristic function $\chi_S$. 
There is no new notation in part (b).
A: $\mathcal{P}(A)$ is the power set of $A$, i.e., the set of subsets of $A$, and $2^A$ is the set of functions from $A$ to $\{0,1\}$.  The characteristic function $\chi_S$ of a subset $S$ of $A$ is the function $\chi_S:A\to\{0,1\}$ defined by $\chi_S(x)=1$ if $x$ is in $S$ and $\chi_S(x)=0$ if $x$ is not in $S$.  
In general it is a good idea to ask your professor to clarify notation on the homework if it is not easy to find in the notes or text.
