**Problem 1**:
We know the difference of two context-free languages need not be context free.
Suppose we take the difference of a regular language R and a context-free language L.
There are two ways to take the difference. Is R-C necessarily context-free?
Is C-R necessarily context free? Justify your answers informally, but giving the
key ideas.

**Problem 2**:
The operation Half(L) from Exercise 4.2.8 in the text is { w | for some x with the same
length as w, wx is in L}. While regular languages are closed under Half, CFL's are not.
Give an example of a CFL L such that Half(L) is not a CFL. Prove that your language L
is a CFL by describing a CFG or PDA for L. Prove that Half(L) is not a CFL by whatever
means are appropriate, e.g., using the pumping lemma and/or applying some CFL-preserving
operation to Half(L) to produce a language known not to be a CFL.
A Hint is available.