# GATE | GATE CS 2011 | Question 24

1) Let P be a regular language and Q be context-free language such that Q P. (For example, let P be the language represented by the regular expression p*q* and Q be {pnqn|n N}). Then which of the following is ALWAYS regular?
(A) P Q
(B) P – Q
(C) * – P
(D) * – Q

(A) A
(B) B
(C) C
(D) D

Explanation:

1. P ∩ Q would be Q, due to the given fact that Q ⊆ P, hence context free but not regular.
2. P − Q = P ∩ Q might not even be a context free language, due to the closure properties of context free languages.
3. Σ∗ − P is equivalently complement of P, hence regular. Refer to closure laws of regular languages.
4. Σ∗ − Q is equivalently complement of Q, hence it might not even be a context free language.

Refer to closure laws of CFLs.

This solution is contributed by Vineet Purswani.

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