For a language A, define the operation DF(A) = {a2...ak | a1a2... ak ∈ A} which is the language of all non-empty words of L without the first character.
Show that the class of regular languages over the alphabet Σ is closed under the operation DF(A).
Setup Let A be an arbitrary regular language over the alphabet {a, b}. Suppose that M is a DFA such that L(M) = A. Suppose M = (Q, {a, b}, δ, q0, F).
Construction Build a new NFA whose language is DF(A).
N = (Q ∪ {q'}, {a, b}, δ', q', F'}
(assume q' ∈/ Q)
I just need to define the transition function below:
δ'((q', ε))= ____________ For each possible input to the transition function, specify the output.
δ'((q, x)) = ____________ if q ∈ Q and x ∈ {a, b}
δ'((r, x)) = _____________otherwise.
I am confused as to where to begin here. What is δ', q', and F'?And how do I define the transition function? Thank you very much.
As you write in the specification of $N$, $\delta'$, $q'$ and $F'$ are the transition function, the start state and the set of final states of the new automaton $N$.
You need to begin with having an idea ;-) Bascially, you want to let the original DFA work, but want to skip its first step; then it will accept all strings that the original DFA accepted just without their first letters.
For the skipping you can use the $\epsilon$-transitions from the first part of the solution template. For letting $A$ work on the input you use the second part of the template and just copy the original transitions. I hope this is enough detail for you to write down the solution.