Problem $1.(5$ points$)$

Let $M$ be a finite automaton that accepts the language $L.$ For each one of its state $q$ we define the set $L_q$ as

$L_q=\{xin{}^{**}|{}^{**}(q_0,x)=q\},$

i$.$e$.$ the set of strings for which M$,$ after processing them starting at the initial state $q_0,$ ends up in the state $q.$ The equivalence relation defined by the L$-$indistinguishability property between strings generates a correspondent equivalence relation between states: we say that two states $p$ and $q$ are equivalent, $p-=q,$ if and only if the strings belonging to $L_p$ are pairwise L$-$indistinguishable from the strings in $L_q.$

Show that two states $p$ and $q$ are not equivalent, i$.$e$.pq,$ if and only if there exists a string zin${}^{**}$ such that only one of the two states given by ${}^{**}(p,z)$ and ${}^{**}(q,z)$ is an accepting state.