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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?

Yos
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  • You do not just take the "quotient of each string with itself" (which would indeed result in the language with only the empty string), but rather the "quotient of each string with all other strings in $L$". Therefore the result can be just about any language like in the example provided by @MJD. – Peter Leupold Jan 29 '19 at 12:01

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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.)

MJD
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