I'm studing the definition of automaton $G=(X, E, f, \Gamma, x_o, X_m)$ where $X$ is the set of states and $E$ is the set of events.
My sources report that $\Gamma:X \rightarrow 2^E$ is the indicator function of active events for a state. I guess it should return a set of events, but what is $2^E$?
On
$\Gamma$ indeed returns a set of elements of $E$, as $2^E$ is a notation for power set, i.e. the set of all subsets of $E$.
The notation $2^E$ is ambiguous: it can mean the power set of $E$, i.e., the set of all subsets of $E$, though $\wp(E)$ is a more common notation for that. It can also mean the set of functions from the set $E$ to the set $2=\{0,1\}$.
Here it appears to be the latter. A function $h:E\to\{0,1\}$ is called the indicator function of the set $\{e\in E:h(e)=1\}$; there is a bijection between subsets of $E$ and indicator functions with domain $E$, so in practice the ambiguity of the notation $2^E$ is rarely very problematic. Note, though, that $\Gamma$ itself is not an indicator function: for each $x\in X$, $\Gamma(x)$ is apparently the indicator function of the set of events active for the state $x$. One could equally well define $\Gamma:X\to\wp(E)$ in such a way that $\Gamma(x)$ is the actual set of of events active for the state $x$, rather than the indicator function of that set; the two convey the same information.