Cardinality of Set of Characteristic Functions on Finite Set

Theorem
Let $I$ be a finite set.

The number of characteristic functions on $I$ is:
 * $2^{\card I}$

where $\card I$ denotes the cardinality of $I$.

Proof
Let $A = \set {0, 1}$.

A characteristic function of $I$ is a mapping from $I$ to $A$.

Hence the set of characteristic functions on $I$ is the indexed Cartesian space $A_I$:


 * $A^I = \ds \prod_{i \mathop \in I} A := \set {f: \paren {f: I \to A} \land \paren {\forall i \in I: \paren {\map f i \in A} } }$

Hence from Cardinality of Set of All Mappings:


 * $\card {A^I} = 2^{\card I}$