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Let $\Sigma = \{a,b,c,\ldots,x,y,z\}$ be the Latin alphabet, consisting of 26 letters. Consider the language $L$ of all words $\alpha$ over $\Sigma$ satisfying the following constraints:

  • If $\alpha$ contains $a$ then it contains exactly $4$ many $a$s.
  • If $\alpha$ contains $b$ then it contains exactly $8$ many $b$s.
  • ...
  • If $\alpha$ contains $`$ then it contains exactly $2^{27}$ many $z$s.

How do I prove that $L$ is regular?

Note: I did research and I have found that we can use the pumping lemma.

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    $\begingroup$ It seems unlikely that you can show that the language is regular by using the pumping lemma since it is a necessary but not sufficient condition for a language to be regular. Usually the pumping lemma is shown not to hold for some language $L$, thus proving that $L$ cannot be regular. Try to write your language as a finite union of regular languages. $\endgroup$
    – Steven
    Jan 12 at 23:23
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You cannot use the pumping lemma to show that a language is regular. The pumping lemma gives a property of regular languages: if $L$ is regular then $L$ can be "pumped". You can use this to show that $L$ is not regular: if $L$ cannot be pumped, then it is not regular. But you cannot use it to show that $L$ is regular: if $L$ can be pumped, it doesn't follow that $L$ is regular; moreover, there are examples of pumpable languages which are not regular. There are extensions of the pumping lemma which can be used to show that a language is regular, but they are not so useful in practice.

In your case, every word in $L$ contain $0$ or $4$ many $a$s, $0$ or $8$ many $b$s, and so on. In particular, it contains at most $4$ many $a$s, at most $8$ many $b$s, and so on. Therefore every word in $L$ has length at most $4+8+\cdots+2^{27}$. Consequently, $L$ is finite. Every finite language is regular.

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