I am trying to describe a state space of a process in mathematical notation.
The process begins with L individuals, $X(i)$, all of a unique starting type $X_0(i)$. At each time step, a new generation of L individuals are generated, randomly choosing one of the previous generation's types. Over time types go extinct, eventually leaving just one type.
The state space can be described as each of the L individuals being able to have one of the L unique starting types. So the size of this state space is LxL.
I'm struggling to write this in mathematical notation. So far I have
$X(i) \in X_0(j) \ \text{ where } \ i,j \in \{1,2,...,L\}$
Or it could be something like:
$X(i) \in \{ X_0(j) : j \in \{1,2,..,L\}\} \forall i \in \{1,2,...,L\}$
Anyone have any ideas?