TITLE: Are regular languages necessarily deterministic context-free languages?
QUESTION [6 upvotes]: The original problem
Suppose M is DCFL (Deterministic Context Free Language) and N is a regular language. Answer the following questions and justify your answers.
a) Is M-N necessarily context-free?
b) Is N-M necessarily context-free?
My attempt
Since DCFLs are closed under complementation, if N is DCFL, then both a and b are true, but are regular languages necessarily DCFLs?
Thanks in advance.

REPLY [5 votes]: 1) Any regular language can be represented by a deterministic finite automaton.  Any (deterministic) finite automaton, can be interpreted as a (deterministic) pushdown automata where the stack stays empty.  Therefore any regular language is a DCFL.
2) That DCFLs are closed under complement is certainly not enough to justify that the difference of two DCFLs is a DCFL.  Since $M-N = M\cap\overline{N}$ you would also need closure under intersection.  Unfortunately, DCFLs are not closed under union nor intersection.
Hint: prove that (D)CFLs are closed under intersection with a regular language by considering a (deterministic) PDA for a DCFL, a DFA for some regular language, and building a (deterministic) PDA for the intersection.