Consider the DFA which accepts the language L = {x | x contains the substring aaab} . For any state q in the automata, let S(q) be the set of strings defined by S(q) = {x | the computation of M on x ends at q} . For example, S(qaaab) = L, and a regular expression for S(qa) is (b a)[(b ab)b a] . Give regular expressions for S(q) and for S(qaaa). Let x, y S(qa) be any two strings that end up at the state qa when run on M. Show that x L y.