I am trying to solve the following problem:
Find languages S and T over the alphabet $\{a, b\}$ such that $ S \not\subset T $ and $ T \not\subset\ S $ ($S$ is not contained in $T$ and not equal to $T$, $T$ is not contained in $S$ and is not equal to $S$) but $S^* = T^*$.

It might be trivial, but I would appreciate some help.

  • $\begingroup$ There is no requirement for these sets to be disjoint. $\endgroup$ – Dmitry Jul 12 at 8:23
  • $\begingroup$ @Dmitry, thanks for the clarification. I think I can figure it out now. $\endgroup$ – Stephen Mwangi Jul 12 at 8:48

I had originally assumed that the sets are disjoint, but there's no such requirement in the question. I'll post my solution in case anyone might have the same problem in future.

S = {a, b, ab}
T = {a, b, ba}

S* = T* since the intersection has all the letters of the alphabet plus ab and ba can easily be obtained from a and b.

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