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.
