If we have a language $F$ and a regular language $D$ (a finite set) then can we say anything about the intersection of $D$ and $F$? Will the intersection of the languages be finite or regular?
This question arises from another question which says
Show that if $L$ is not a CFL and $F$ is finite then $L - F$ is not a CFL.
Now $L = (L - F) ∪ ( L ∩ F)$.
Now if I am able to prove that $L ∩ F$ is finite or regular than I can arrive at a contradiction as follows:
If $L ∩ F$ is finite or regular then union with $L-F$ will be a CFL, therefore $L$ will be a CFL, which is a contradiction.
But my problem is assuming anything about $L ∩ F$.
What will be the properties of this intersection?