First a definition for those who need it: If $S$ is a set and $S_0$ is a subset of $S$ then the characteristic function of $S_0$ as a subset of $S$ is the function $f$, with domain $S$ defined by $f(s) = 1$ if $s \in S_0$ and $f(s) = 0$ if $s \notin S_0$.
The definition is clear and I understand the proof of the theorem that for any set $S$ the set of any characteristic function with domain $S$ has the same cardinality as the power set of $S$. But I'm failing to see what purpose it will play in what sorts of problems. Can anyone give an example of a problem for which the solution includes introducing the characteristic function of a set?
Edit I believe it is also known as the Indicator function (as per Wikipedia).
Edit 2 I mostly mean applications to problems in elementary set theory.
$\endgroup$ 22 Answers
$\begingroup$One of the uses of indicator functions is that they make it possible to translate statements about sets to statements about functions. This is sometimes very useful.
Other applications can be found in integration theory. Here we approximate the function we want to integrate by step-functions, which can be defined in terms of indicator functions.
Another application can be found in the theory of commutative C*-algebras, which are always isomorphic to some space of continuous functions on some topological space. Once you're familiar with the idiom of C*-algebras, it is easy to see that a projection in a commutative C*-algebra precisely corresponds to an indicator function on a clopen subset.
$\endgroup$ 2 $\begingroup$Any analysis problem! For instance, you define the Riemann integral using characteristic functions of appropriate subsets of $\mathbb R$.
$\endgroup$
