Another clarification on set theory...

Let's say there is a language A.

If A = {a, b} does it also contain the empty string {e}?

If not, then A^0 would not contain the empty string e either, right?

I know that A* contains the empty string unless otherwise stated because A* starts at A^0....

I'm really confused about this notion and would greatly appreciate someone who could clear it up for me. Thanks!

I'm not up on the notation, so correct me if I'm wrong, but from my understanding of your post, A^k is the set of strings using k items from the alphabet A. If that's true, then A^0 contains only the empty string (A^0 = {e}), and A^1 contains the elements of A (A^1 = A). A itself probably doesn't contain e.