(related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
The indicator function is an example of a map of an arbitrary set X into another set with two elements, in this case, the set \{0,1\}. The definition of indicator function reveals that all maps of this kind are exactly those maps, whose carriers correspond to exactly the subsets of X.
The question, how many characteristic functions for a given set X do exist, is therefore equivalent to the question, how many subsets of S exist. But this is exactly the cardinality |\mathcal P(X)|, where \mathcal P(S) denotes the power set of S. This result can be summarized as follows: |\mathcal P(X)|=2^{|X|}. It is true for both, finite or infinite sets X. In the combinatorics, 2^{|X|} is the common notation for the cardinality of the power set \mathcal P(S).
Despite this argument, the following proposition summarizes this result for finite sets with yet another proof:
Proofs: 1
