I'm preparing an Formal language exam, One question from previous year's final is:
Prove or disprove:If L is a context free language, then there exists a language P that is generated by a pure context-free grammar and a regular language R so that $L = P \cap R$.
My solution to this question is quite simple. Assume context-free language P is equal to context free language L, and let regular language R to be $\Sigma^*$, where $\Sigma$ is the alphabet set. $\Sigma^*$ is definitely a regular language. So the answer to this question is yes.
However, I do came up with a more difficult problem. What if I add a limitation $P \neq L?$ Will this statement still hold? I current have no idea about this.