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I'm a bit confused on how to use homomorphims to prove irregularity or to prove that a language is not context free. This is what I'm currently thinking:

Example 1:
Let $L = \{ a^{i}b^{j}c^{k} : i = j = k \}$ and $h: \{a, b, c\} \rightarrow \{a, b, \epsilon\}$ be defined as follows: $$h(x) = \begin{cases} \epsilon & x = c\\ x & o.w.\end{cases}$$ Then it follows that $h(L) = \{a^ib^j: i=j \} = \{a^{n}b^{n}: n \ge 0 \}$. Now because $\{a^nb^n: n \ge 0 \}$ is known to be irregular, then so is $L$.

Example 2:
We know that $P = \{0^p : p \text{ is a primes}\}$ is irregular (by the pumping lemma). Defined $h: \{0\} \to \{1\}$ as $h(1)=0$. It follows that $h(P) = ${1^p : p \text{ is a primes}}$ and therefore is irregular.

I wrote these as answers to an exercise but I've been told that they are incorrect. I don't understand why?

Kaveh
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Xsy
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    Questions only admit "yes/no" answers help neither you nor future visitors. Please read related meta discussions here and here and adjust your question accordingly, e.g. by formulating a specific question about a single element of your answer you are uncertain about. If you just want general feedback, you are welcome to visit us in [chat]. – Raphael May 26 '15 at 09:49
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    The question I linked as duplicate contains hints on how to structure proofs with closure properties. You can find more examples here, and also using the site search. Working from these answers, you should be able to find the flaw in your work (if any). – Raphael May 26 '15 at 09:50
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    If all you're interested in is getting us to check your proof, your question is off-topic anyway. – David Richerby May 26 '15 at 11:03
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    You should edit your question so that it asks more than Am I right or wrong, which makes it off-topic. If you had the reasons given for refusing your answer you could present it as a problem. It is not always easy, but we do not want the site swamped with this kind of question (we do get an awful lot). To me it seems that your answers are technically correct, but your notations are not, which may have caused the rejection. For example you should not have the ϵ in {a,b,ϵ}∗ . Also you should write just h(L), not L(h(L)). Your set notation varies with comma or semi-colon. Can you ask what's wrong – babou May 26 '15 at 11:47
  • Did the comment help you? – babou May 26 '15 at 14:17
  • Yes thank you, your comment helped. I apologize for the question, I usually roam in the math section more, and from my experience this type of questions are fine there, guess not here though. Noted for next time. Also, I cannot explain why were my answers rejected because no reason was given, and I'm unable to contact the staff at the moment to ask why, which is why I asked here. – Xsy May 26 '15 at 15:06
  • I think there is not a suitable question here. The only possible question I see is asking about how to use the homomorphism to prove that a language is not regular and in that case the question linked by Raphael answers it. If you want to know why someone told you your solution is not acceptable asked that person. – Kaveh Jun 05 '15 at 02:19

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