Is $\epsilon$ in every alphabet?

Question

Given a $\Sigma$ an alphabet, is $\epsilon$ in it logically?

For example, if I have a function $ f : \Sigma \to \Sigma $, can I define it for example $ f(\sigma) = \epsilon$? even if my alphabet is for example only $\Sigma = \{a,b,c\}$?

Answer 1

As Arthur Fischer wrote in the comments: $\epsilon$ is the empty string, the word of length zero. It is not contained in any alphabet (apart from some alphabet specifically defined to contain the empty string), but it is a string over any alphabet.

You should define $f:\Sigma^* \rightarrow \Sigma^*$, which is a function from the strings over the alphabet to the strings over the alphabet. If you were to leave out the stars, it'd be just from the alphabet to the alphabet, which I don't think is what you want.

(An analogue to this is regular set notation - usually, for some set $A$, $\emptyset \notin A$ unless explicitly defined otherwise, but $\emptyset \subseteq A$ for all $A$. Similarly for $\epsilon$ and $\Sigma$.)