$14\mathrm{.}2$: Characterizing Strings with a Behavior

Let M be a machine with state set $\{1,2,3\},$ alphabet $\{$a$,$ b$\},$ final state set $\{2\},$ and a transition function where $\backslash$delta $($i$,$ a$)=2$ and $\backslash$delta $($i$,$ b$)=3$ for all states i$.$ Then for any string w$,\backslash$delta $*(1,$ w$)=2$ if and only if w is in the language a$*.$