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I've been watching lectures from Coderisland on YouTube about finite state machines, DFAs and NFAs, and in one discussion he talks about how to use the pumping lemma to show how a language is not regular. I don't know quite how to apply the lemma and want to understand if I'm doing it right. If I had something like:

w = {anbk, n =/= k}

am I correct in that I can say that:

h = {anbn + r, r > 0} is a subset of w, and thus if I show by the lemma that h is not regular, that w must not be regular since h is a subset of w.

The way I would show this is as follows:

  1. h = xyz
  2. |xy| <= n
  3. x = an-r
  4. y = ar
  5. z = bn + r
  6. xyz = an-rarbn + r
  7. xy2z = an-ra2rbn + r = an + rbn + r

Thus h cannot be regular since an + rbn + r is not of the form {anbn + r, r > 0}, and since h is not regular w must not be regular, since h is an element of w.

Have I applied it correctly? I understand how to apply it for an easy language like {anbn}, because I can apply the lemma directly to this language, but the only way I could think of for my language was to create a subset that belongs to my language, and apply the lemma to that.

If I haven't applied it correctly, is there a way to show that my language is not regular (or regular), using another lemma, or perhaps with closure properties?

This is a really awesome topic, even if I don't understand the pumping lemma fully, I'm excited to explore it further!

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A subset of a regular language could be non-regular, so the entire line of reasoning is flawed. As an example, every non-regular language over $\{0,1\}$ is a subset of the regular language $(0+1)^*$.

Regarding the language $H$, you'd do better writing your proof in paragraph-style rather than as a cryptic list. In addition, you made the following mistakes:

  1. You never mention which $h$ you take.
  2. You don't get to choose the decomposition $h = xyz$. The proof has to work for every such decomposition satisfying the properties given in the lemma ($|xy| \leq n$, where $n$ is the "pumping constant").

Several ways of showing that your original language is not regular are given in this answer.

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