Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand?

alt text

share|cite|improve this question
@Moron: Why are you so sure that those are the two things that are causing problem? There could also be an issue with $2^A$ for that matter... – Arturo Magidin Nov 10 '10 at 4:50
@Arturo: I am not sure. BinaryBro can always edit the title if there is something missing. The earlier title was too generic. – Aryabhata Nov 10 '10 at 4:58
@Moron: fair enough... – Arturo Magidin Nov 10 '10 at 4:59

$\mathcal{P}(A)$ 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).

share|cite|improve this answer

$\mathcal{P}(A)$ 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.

share|cite|improve this answer
Thanks, bro. This helps. – BinaryBro Nov 10 '10 at 2:34
Hmm... came up just as I was typing. (-: – Arturo Magidin Nov 10 '10 at 2:41
If I had known you were working on a more informative answer I probably wouldn't have posted. I guess I gave the short attention span version. :) – Jonas Meyer Nov 10 '10 at 2:46
Nah; your answer is perfectly fine. I'm just (in)famous for long writings. Just ask my old Algebraic Geometry TA. "Tree-killer", they used to call me. – Arturo Magidin Nov 10 '10 at 2:48

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.