College

Let \( L_1 \) and \( L_2 \) be any two context-free languages, and let \( R \) be any regular language. Then, which of the following is correct?

A. \( L_1 \cup L_2 \) is context-free.

B. \( L_1 \) is context-free.

C. \( L_1 - R \) is context-free.

D. \( L_1 \cap L_2 \) is context-free.

Answer :

The complement of a context-free language is not necessarily context-free, but the intersection of a context-free language with the complement of another context-free language is guaranteed to be context-free. The correct statement is (d): L1 intersect complement of L2 is context-free.

Let's analyze the given options:

a) L1 intersect L2 is context-free: This is not necessarily true. The intersection of two context-free languages is not guaranteed to be context-free.

b) L1 - L2 is context-free: This is not necessarily true. The difference of two context-free languages is not guaranteed to be context-free.

c) L1 intersect complement of R is context-free: This is not necessarily true. The complement of a regular language is always regular, but the intersection with a context-free language may not be guaranteed to be context-free.

d) L1 intersect complement of L2 is context-free: This is correct. The complement of a context-free language is not guaranteed to be context-free, but the intersection of a context-free language with the complement of another context-free language is indeed context-free.

In conclusion, option (d) L1 intersect complement of L2 being context-free is the correct statement based on the given options.