# When is epsilon a part of the result of computing the right quotient?

The right quotient of two languages is defined as

$$L_1/L_2=\{ x \mid xy \in L_1 \text{ and } y \in L_2 \}.$$

Example:
$L_1=\{ab,aab,baa\}$ and $L_2=\{b\}.$

I read somewhere that $L_1/L_2=\{a,aa,\varepsilon\}$ but I cannot understand why $\varepsilon$ is contained in the quotient.

• $\varepsilon$ is not in $L_1/L_2$. Dec 7 '12 at 17:28
• I see that my title edit was not perfect, but it was likely too nondescriptive to begin with. Perhaps we need to include a proper question in the post too? Dec 7 '12 at 17:36

In words, your definition says "The right quotient, written $L_1 / L_2$, is defined to be the set of all strings $x$ such that $xy$ is a string in $L_1$ for some string $y$ in $L_2$." According to this definition, the empty string is a member of the right quotient if and only if there is a common string in the two languages. In symbols, $\epsilon \in L_1 / L_2 \iff L_1 \cap L_2 \neq \emptyset$. In particular, this is not the case in your example, hence, the empty string is not in your right quotient.