Say that: A={a,b} and B is equal to or a subset of {a,b}* then A*=B* implies A is equal to or a subset of B. I have a hard time with this since B equal to or subset of {a,b}* means that we don't really know what B contains. A* = B* makes sense as I would assume that B* would have the same elements as A* but the implication to me reads like A{a,b} is equal to or a subset of B{?}. For the implication to be true, I know I would need to prove that {a,b} is in the set B, or at least contains the same elements as A. I am not sure how to approach this because I am already thinking the possibility exists that {a,b} is not in B so A would not be a subset. Not sure what I am missing here.
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Define A = {a, b} and let B be some subset of {a, b}*.
Assume A* = B*. A* contains the strings a and b. The only way A* = B* can be true is if B* also contains the strings a and b. But B* can only contain the strings a and b if B contains the strings a and b. That is, {a, b} is a subset of B. But A = {a, b}.
So, A* = B* implies that A is a subset of B.
Patrick87
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That was the same conclusion I came to as well – Wendell Best May 31 '21 at 16:59