Let $\sigma_i$ be a state in the state set $S$ of machine $M$. Let $G\left(\sigma_i\right)$ be the $\sigma_i$-reachable set, and let $G^{\prime}\left(\sigma_i\right)$ consist of the elements of $S$ not contained in $G\left(\sigma_i\right)$. Show that: $(a)$ If $G^{\prime}\left(\sigma_i\right) \neq O$ and $G\left(\sigma_i\right) \cap G\left(G^{\prime}\left(\sigma_i\right)\right)=0$, † then $G\left(\sigma_i\right)$ and $G^{\prime}\left(\sigma_i\right)$ are two
$\dagger$ The set $R_1 \cap R_2$, called the intersection of the sets $R_1$ and $R_2$, consists of all the elements included in both $R_1$ and $R_2$. $O$ denotes an empty set.
isolated submachines of $M$. (b) If $G^{\prime}\left(\sigma_i\right) \neq O$ and $G\left(\sigma_i\right) \cap G\left(G^{\prime}\left(\sigma_i\right)\right) \neq O$, then $G\left(\sigma_i\right)$ and $G^{\prime}\left(\sigma_i\right)$ are persistent and transient submachines, respectively. (c) If $G^{\prime}\left(\sigma_i\right)=0$, then $M$ contains no isolated submachines.