I don't understand what an indicator function is. The definition says
I(A) = 1 if x belongs to A, where A subset of S and
I(A) = 0 if x does not belong to A.
But I am not clear. Can you give me an example?
Your notation is not usual. Here it is: $\displaystyle \chi _A (x) = \left\{ {\begin{array}{ll}
{1,} & {x \in A} \\ {0,} & {x \notin A} \\ \end{array} } \right.$
Consider: $\displaystyle X = \left\{ {a,b,c,d,e} \right\},\;A = \left\{ {a,c,e} \right\},\;\& \;B = \left\{ {b,c,d} \right\}\;$.
$\displaystyle \begin{gathered}
\chi _A (a) = 1,\chi _A (b) = 0,\chi _A (c) = 1,\chi _A (d) = 0,\chi _A (e) = 1 \hfill \\
\chi _B (a) = 0,\chi _B (b) = 1,\chi _B (c) = 1,\chi _B (d) = 1,\chi _B (e) = 0, \hfill \\
\end{gathered} $