What is the right quotient of a language with itself?

Question

This may be a really trivial question but I just want to make sure:

if $L$ is some language then $L/L=\{\epsilon\}$ if $\epsilon \in L$ or $L/L=\{\emptyset\}$ if $\epsilon \notin L$? Does the same go for left quotient?

Answer 1

Not necessarily. Consider the language $$L=\{{\mathtt y}, \mathtt{xy}\}.$$ The right quotient $L/L$ includes the string $\mathtt x$.

Possibly useful: finding right quotient of languages

(Also, $\{\emptyset\}$ is never correct because it is not a language at all. A language is a set of strings. $\emptyset$ is not a string.)