The Rabin/Scott algorithm, also called subset construction, and sometimes also called powerset construction is at first nothing else than a function that computes for a given nondeterministic automaton with a designated set of states a corresponding deterministic automaton.

In case of finite automata on finite words, it is easy to prove that the constructed automaton is equivalent to the given nondeterministic one. For omega-automata, this is not that simple. To determinize safety automata, you have to remove all unsafe states, and then apply the subset construction. The new acceptance condition states then that the sink state {} will never be reached (this definition is independent of the Rabin/Scott construction).

Now, what is an unsafe state? Safety automata are defined as automata with a designated set of states which are called the safe states. Their acceptance condition demands that a run is accepting if and only if it only visits safe states. Thus, unsafe states are actually useless for acceptance (since no accepting run may traverse them). We can therefore remove all unsafe states in a safety automaton, so that finally all states are safe (but this can only be done in nondeterministic safety automata).

We cannot discuss exercise sheet 8, task 3 here since there are many different versions of this exercise, so we don't know which automaton is meant here. Please ask your tutor if you think your solution is correct, and you don't understand why it is not accepted.