Let $X_n$ be a random variable taking on one of three values $a,b$ or $c$ over time. That is, for each $n \in \mathbb{N}$, we have $X_n \in \{a,b,c\}$.
Also, for each $n \in \mathbb{N}$, let $F_n$ be either $G$ or $H$, described by the following "rule" where the value of $F_n$ depends on both the time $n$ and the outcome of the random variable $X$ at time $n-1$: \begin{equation*} F_n(n,X_{n-1})= \begin{cases} G, & n=1\textrm{ or }X_{n-1}=a \\ H, & n>1\textrm{ and }X_{n-1}\in\{b,c\} \end{cases} \end{equation*} where note that we have "fixed" $F_1=G$.
My question is how to write this "rule" formally as a function? I believe it would make sense to write \begin{equation*} F_n:\mathbb{N} \times \{a,b,c\} \to \{G,H\}. \end{equation*} Now in set theory, a function is a set of ordered pairs. So our function would look something like \begin{equation*} F=\{(1,?,G),(2,a,G),(2,b,H),(2,c,H),(3,a,G),(3,b,H),(3,c,H),...\} \end{equation*} The problem of course is what to do when $n=1$, because $X_{n-1}$ is undefined. (Notice I left a question mark inside one of the tuples.) One obvious solution is to "define" $X_0=a$, but it's an ad hoc fix, since it only makes sense for the random variable $X_n$ to start taking values at time $n=1$. How could this function be written formally? Thank you for your thoughts and suggestions!