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I'm having trouble designing a DFA that accepts substrings of both 01 or 10. So far, I have constructed separate DFAs that accept the substrings "01" and "10" respectively.

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What I'm stuck with is how to get the union of these two DFAs while preserving all the transitions and states. May I please get some help with this? Thank you in advanced.

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1 Answer 1

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If you think about it, there are only 3 kinds of string not in your language: $$\varepsilon, 00...0 \text{ and } 11...1.$$ So when you read a string, left to right, there are only four states you need to keep track of to decide if you want to accept

  1. I haven't read any characters yet,
  2. I have read a bunch of $0$s,
  3. I have read a bunch of $1$s, or
  4. none of the above.

If you read the last character and are in 1.-3. then you have read one of the strings mentioned above and reject, otherwise accept.

Translating that thought into a DFA you get (try doing it yourself first):

enter image description here

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