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Is the empty string always an element of an aribitrary finite alphabet?

I understand that the empty string is part of the Kleene-Star of any alphabet, but is it intrinsically part of any finite alphabet where I don't explicitly mention it?

For example, if $A=\{a,b\}$, is $\epsilon$, the empty word, in $A$? Or, would it only be in $A$ if it were specified that $A=\{\epsilon,a,b\}$?

Jeremy Jeffrey James
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    Do not confuse the alphabet and the string. The string is a sequence of characters, and you can allow a void string. But there is no "empty" character. –  Mar 09 '16 at 11:13
  • @YvesDaoust I think your comment clarifies my confusion - cheers! – Jeremy Jeffrey James Mar 09 '16 at 11:17
  • Simple example: take the languages $a^*$ and $a^+$: the empty string belongs to the first but not the the second language. – Graffitics Mar 09 '16 at 12:05

1 Answers1

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The empty string, often called $\varepsilon$ may be created using any alphabet, by creating the string which contain no characters from the alphabet. An alphabet does, by definition, only contain characters which later may be used to create strings. Kleene-star of an alphabet is a language, why it is possible for the empty string to exist in there.

In any alphabet $A$ we have that $\varepsilon\notin A$, this is because the $\varepsilon$ is not a character, but rather a string and specifically the empty string. If you create the language $\{a,b,\varepsilon\}$ then $\varepsilon$ does not represent the empty string but rather is just a character like $a$ and $b$.

Summary: The empty string is by definition not a charcter and thus it is not part of any alphabet.

Ove Ahlman
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