TITLE: Proving OP(L) is regular
QUESTION [0 upvotes]: i am currently working on a proof to show that if L is regular then OP(L) is also regular without the use of pumping lemma. For the sake of this question, i am using {0,1} as alphabets.
OP(L) = {w| wz ∉ L for every z ∈ {0,1}* - {∈} }
At the moment, i have a hard time starting the proof. What i understand is that, If L is regular then we can define a DFA that can decide L. And then we can define another machine for OP(L) with every transition from M along with ∈ transitions. 
Am i on the right track? I could really use some hints.

REPLY [0 votes]: Hint. Use the fact that regular languages are closed under complement.
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