Why is {w | no prefix of w starts with b} = {w | the first character of w is a} ∪ {e}? \begin{align} L &amp;= \{ w \in \{a, b\}^* \mid \text{ no prefix of $w$
 starts with $b$}\} \\ &amp;= \{w \in \{a, b\}^* \mid \text{ the first
 character of $w$ is  $a$} \} \cup \{e\} \end{align}


Why is it in union with an empty string? If an empty string from $b$ can also be a prefix.
The unique prefix of the empty word $e$ is $e$, and it does not start with $b$. Therefore $e$ satisfies the condition "no prefix of $e$ starts with $b$" and hence $e$ belongs to $L$.