Give an example of a language $L$ where $min(max L)neq max(min L)$












2












$begingroup$



Give an example of a language $L$ where $min(max L)neq max(min L)$.




I thought of the following language $L={a,bc, abc}$.



$$
min L={a,bc}, max L = {abc}
$$
Then:
$$
min(max L)=min ({abc})={abc}neq max(min L)=max({a,bc})={a,bc}
$$
This seems too simple so I'm wondering if it's correct.




The definitions of $max, min$:
$$
min L= {x|xin L land text{there doesn't exist a non-empty substring }y text{ of } x text{ such that } yin L }\
max L = {x|xin L land lnot exists y: xyin L, yneq epsilon}
$$






proof-verification formal-languages







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$endgroup$












  • $begingroup$
    min and max of a language are not standard operations. You should provide their definitions.
    $endgroup$
    – Peter Leupold
    Jan 28 at 12:33
















2












$begingroup$



Give an example of a language $L$ where $min(max L)neq max(min L)$.




I thought of the following language $L={a,bc, abc}$.



$$
min L={a,bc}, max L = {abc}
$$
Then:
$$
min(max L)=min ({abc})={abc}neq max(min L)=max({a,bc})={a,bc}
$$
This seems too simple so I'm wondering if it's correct.




The definitions of $max, min$:
$$
min L= {x|xin L land text{there doesn't exist a non-empty substring }y text{ of } x text{ such that } yin L }\
max L = {x|xin L land lnot exists y: xyin L, yneq epsilon}
$$






proof-verification formal-languages







share|cite|improve this question











$endgroup$












  • $begingroup$
    min and max of a language are not standard operations. You should provide their definitions.
    $endgroup$
    – Peter Leupold
    Jan 28 at 12:33














2












2








2





$begingroup$



Give an example of a language $L$ where $min(max L)neq max(min L)$.




I thought of the following language $L={a,bc, abc}$.



$$
min L={a,bc}, max L = {abc}
$$
Then:
$$
min(max L)=min ({abc})={abc}neq max(min L)=max({a,bc})={a,bc}
$$
This seems too simple so I'm wondering if it's correct.




The definitions of $max, min$:
$$
min L= {x|xin L land text{there doesn't exist a non-empty substring }y text{ of } x text{ such that } yin L }\
max L = {x|xin L land lnot exists y: xyin L, yneq epsilon}
$$






proof-verification formal-languages







share|cite|improve this question











$endgroup$





Give an example of a language $L$ where $min(max L)neq max(min L)$.




I thought of the following language $L={a,bc, abc}$.



$$
min L={a,bc}, max L = {abc}
$$
Then:
$$
min(max L)=min ({abc})={abc}neq max(min L)=max({a,bc})={a,bc}
$$
This seems too simple so I'm wondering if it's correct.




The definitions of $max, min$:
$$
min L= {x|xin L land text{there doesn't exist a non-empty substring }y text{ of } x text{ such that } yin L }\
max L = {x|xin L land lnot exists y: xyin L, yneq epsilon}
$$






proof-verification formal-languages




proof-verification formal-languages






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 28 at 12:44







Yos

















asked Jan 27 at 18:57









YosYos

1,1631823




1,1631823












  • $begingroup$
    min and max of a language are not standard operations. You should provide their definitions.
    $endgroup$
    – Peter Leupold
    Jan 28 at 12:33


















  • $begingroup$
    min and max of a language are not standard operations. You should provide their definitions.
    $endgroup$
    – Peter Leupold
    Jan 28 at 12:33
















$begingroup$
min and max of a language are not standard operations. You should provide their definitions.
$endgroup$
– Peter Leupold
Jan 28 at 12:33




$begingroup$
min and max of a language are not standard operations. You should provide their definitions.
$endgroup$
– Peter Leupold
Jan 28 at 12:33










2 Answers
2






active

oldest

votes


















3












$begingroup$

Your example is perfect.



Being simple is an advantage and not a problem ;)






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    Now with the definitions it is clear that your example is right. However, your reasoning is not correct. $bc$ can be extended to the left by $a$ to form a longer string in the language, but there is no extension to the right. The definition of $max$ only asks for extensions to the right. Therefore $bc$ is also in the $max$, and your example works like this:



    $$
    min(max L)=min ({bc, abc})={bc} neq {a,bc}=max({a,bc})=max(min L)
    $$






    share|cite|improve this answer









    $endgroup$













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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      Your example is perfect.



      Being simple is an advantage and not a problem ;)






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        Your example is perfect.



        Being simple is an advantage and not a problem ;)






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          Your example is perfect.



          Being simple is an advantage and not a problem ;)






          share|cite|improve this answer









          $endgroup$



          Your example is perfect.



          Being simple is an advantage and not a problem ;)







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 28 at 15:24









          wecewece

          2,3721923




          2,3721923























              2












              $begingroup$

              Now with the definitions it is clear that your example is right. However, your reasoning is not correct. $bc$ can be extended to the left by $a$ to form a longer string in the language, but there is no extension to the right. The definition of $max$ only asks for extensions to the right. Therefore $bc$ is also in the $max$, and your example works like this:



              $$
              min(max L)=min ({bc, abc})={bc} neq {a,bc}=max({a,bc})=max(min L)
              $$






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                Now with the definitions it is clear that your example is right. However, your reasoning is not correct. $bc$ can be extended to the left by $a$ to form a longer string in the language, but there is no extension to the right. The definition of $max$ only asks for extensions to the right. Therefore $bc$ is also in the $max$, and your example works like this:



                $$
                min(max L)=min ({bc, abc})={bc} neq {a,bc}=max({a,bc})=max(min L)
                $$






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  Now with the definitions it is clear that your example is right. However, your reasoning is not correct. $bc$ can be extended to the left by $a$ to form a longer string in the language, but there is no extension to the right. The definition of $max$ only asks for extensions to the right. Therefore $bc$ is also in the $max$, and your example works like this:



                  $$
                  min(max L)=min ({bc, abc})={bc} neq {a,bc}=max({a,bc})=max(min L)
                  $$






                  share|cite|improve this answer









                  $endgroup$



                  Now with the definitions it is clear that your example is right. However, your reasoning is not correct. $bc$ can be extended to the left by $a$ to form a longer string in the language, but there is no extension to the right. The definition of $max$ only asks for extensions to the right. Therefore $bc$ is also in the $max$, and your example works like this:



                  $$
                  min(max L)=min ({bc, abc})={bc} neq {a,bc}=max({a,bc})=max(min L)
                  $$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 29 at 11:53









                  Peter LeupoldPeter Leupold

                  63526




                  63526






























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