Set Theory Question...

I've been thinking about this problem for awhile and I cannot seem to come up with anything...

two languages X and Y where X* = Y* but X is not a subset of Y and Y is not a subset of X.

Moved from the Lounge...

What does X* mean in terms of set operations? What does it do to the set?

X* is an alphabet with infinitely many strings in it.
An example would be, if X = {a, b} then, X* = {a, b, aa, bb, ab, ba, aab, ...}

So, I am looking for two such alphabets in the case that they aren't subsets of each other, and their *'s = each other...

My understanding is that such is not possible. By the definition of set equality, two sets are equal if and only if they are subsets of each other. I'm actually somewhat confused by this. Would the empty set count?

He wants the generating sets not to be equal but the generated languages to be equal. If I understand, {a, b, ab} and {a, b, ba} works. Or, even more simply, {a, aa} and {a, aaa}.

I think I've figured out that the two languages
e = empty string

A = {e, x}
B = {x, xx}

These are not subsets of each other...but

A* = {e, x, xx, xxx,...}
and
B* = {e, x, xx, xxx,...} so A* = B*. (under the rule that any alphabet using the Kleene star contains e, unless otherwise stated.)

Does this look correct to you guys?

Is it just me or does this seem like it would fit better in the homework thread?

-MBirchmeier