For example, if I have a non-context free language of B, is there such a context free language A such that A is a subset of B? I have been thinking of examples but unable to think of any valid ones.
I thought I got it when I said that A = {A = {w | w is of even length and w ∈ {a, b, c}}, which is context-free, and B = {ww | w ∈ {a,b,c}}, which is not context free. However, I realized that there are some strings A can produce that B cannot, and therefore, A is not a subset of B.
Does anyone know of any examples that could be valid in my situation?
Any finite set of strings is a context-free language. (Indeed, it is a regular language.) So any finite subset of a language is context-free, regardless of what the language is.
Another trivial case is the language L = L1 ∪ L2 where the alphabet of L1 is Σ1 and the alphabet of L2 is Σ2 and Σ1 ∩ Σ2 = ∅. Now L is context-free only if both L1 and L2 are context-free. (This is not the case if the alphabets are not disjoint.) So if exactly one of L1 and L2 is context-free, then it is a subset of the non-context-free language L.
If neither of those are interesting enough for you, then the language { a* } (where a is a symbol) is a subset of { ww | w ∈ {a, b}* }. Another subset of the same classic non-context-free language is { ww | w ∈ {a, b}* ∧ w = wR } (that is, the language of all duplicated even-length palindromes) which is context-free because it is exactly the same as { wwR | w ∈ {a, b}* ∧ w = wR }, as a result of the second condition.