Mixing
What is the right quotient of a language with itself?
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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?
formal-languages
asked Jan 29 at 9:43
YosYos
1,1631823
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add a comment |
$begingroup$
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?
formal-languages
asked Jan 29 at 9:43
YosYos
1,1631823
$endgroup$
$begingroup$
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.
$endgroup$
– Peter Leupold
Jan 29 at 12:01
add a comment |
$begingroup$
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?
formal-languages
asked Jan 29 at 9:43
YosYos
1,1631823
$endgroup$
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?
formal-languages
formal-languages
asked Jan 29 at 9:43
YosYos
1,1631823
asked Jan 29 at 9:43
YosYos
1,1631823
asked Jan 29 at 9:43
YosYos
1,1631823
asked Jan 29 at 9:43
YosYos
1,1631823
asked Jan 29 at 9:43
YosYos
1,1631823
1,1631823
$begingroup$
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.
$endgroup$
– Peter Leupold
Jan 29 at 12:01
add a comment |
$begingroup$
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.
$endgroup$
– Peter Leupold
Jan 29 at 12:01
$begingroup$
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.
$endgroup$
– Peter Leupold
Jan 29 at 12:01
$begingroup$
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.
$endgroup$
– Peter Leupold
Jan 29 at 12:01
add a comment |
1 Answer
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$begingroup$
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.)
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.)
$endgroup$
add a comment |
$begingroup$
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.)
$endgroup$
add a comment |
$begingroup$
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.)
$endgroup$
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.)
answered Jan 29 at 10:24
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MJD
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$begingroup$
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.
$endgroup$
– Peter Leupold
Jan 29 at 12:01