Bijection between ${0,1}^*$ and the natural numbers.












3












$begingroup$


So the tasks is to show that ${0,1}^*$ is countable.



So the idea that i am having is that each number can be mapped to it's own in decimal.



$f(1001)= 9$



$f(101)=5$



But what happens with all the string which start with zeros.



For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.



Is my idea completely wrong? Any tips?






elementary-set-theory formal-languages natural-numbers







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What exactly is ${0,1}^{*}$ ?
    $endgroup$
    – Peter Melech
    Dec 29 '18 at 14:14












  • $begingroup$
    The set of all strings with the symbols 0 and 1
    $endgroup$
    – Angeld55
    Dec 29 '18 at 14:16






  • 2




    $begingroup$
    en.wikipedia.org/wiki/Bijective_numeration
    $endgroup$
    – r.e.s.
    Dec 29 '18 at 14:17
















3












$begingroup$


So the tasks is to show that ${0,1}^*$ is countable.



So the idea that i am having is that each number can be mapped to it's own in decimal.



$f(1001)= 9$



$f(101)=5$



But what happens with all the string which start with zeros.



For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.



Is my idea completely wrong? Any tips?






elementary-set-theory formal-languages natural-numbers







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What exactly is ${0,1}^{*}$ ?
    $endgroup$
    – Peter Melech
    Dec 29 '18 at 14:14












  • $begingroup$
    The set of all strings with the symbols 0 and 1
    $endgroup$
    – Angeld55
    Dec 29 '18 at 14:16






  • 2




    $begingroup$
    en.wikipedia.org/wiki/Bijective_numeration
    $endgroup$
    – r.e.s.
    Dec 29 '18 at 14:17














3












3








3





$begingroup$


So the tasks is to show that ${0,1}^*$ is countable.



So the idea that i am having is that each number can be mapped to it's own in decimal.



$f(1001)= 9$



$f(101)=5$



But what happens with all the string which start with zeros.



For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.



Is my idea completely wrong? Any tips?






elementary-set-theory formal-languages natural-numbers







share|cite|improve this question











$endgroup$




So the tasks is to show that ${0,1}^*$ is countable.



So the idea that i am having is that each number can be mapped to it's own in decimal.



$f(1001)= 9$



$f(101)=5$



But what happens with all the string which start with zeros.



For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.



Is my idea completely wrong? Any tips?






elementary-set-theory formal-languages natural-numbers




elementary-set-theory formal-languages natural-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 29 '18 at 14:15







Angeld55

















asked Dec 29 '18 at 14:11









Angeld55Angeld55

565




565








  • 1




    $begingroup$
    What exactly is ${0,1}^{*}$ ?
    $endgroup$
    – Peter Melech
    Dec 29 '18 at 14:14












  • $begingroup$
    The set of all strings with the symbols 0 and 1
    $endgroup$
    – Angeld55
    Dec 29 '18 at 14:16






  • 2




    $begingroup$
    en.wikipedia.org/wiki/Bijective_numeration
    $endgroup$
    – r.e.s.
    Dec 29 '18 at 14:17














  • 1




    $begingroup$
    What exactly is ${0,1}^{*}$ ?
    $endgroup$
    – Peter Melech
    Dec 29 '18 at 14:14












  • $begingroup$
    The set of all strings with the symbols 0 and 1
    $endgroup$
    – Angeld55
    Dec 29 '18 at 14:16






  • 2




    $begingroup$
    en.wikipedia.org/wiki/Bijective_numeration
    $endgroup$
    – r.e.s.
    Dec 29 '18 at 14:17








1




1




$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14






$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14














$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16




$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16




2




2




$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17




$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17










2 Answers
2






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5












$begingroup$

Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.



This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5












      $begingroup$

      Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.



      This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.






      share|cite|improve this answer









      $endgroup$


















        5












        $begingroup$

        Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.



        This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.






        share|cite|improve this answer









        $endgroup$
















          5












          5








          5





          $begingroup$

          Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.



          This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.






          share|cite|improve this answer









          $endgroup$



          Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.



          This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 29 '18 at 14:17









          lhflhf

          168k11172404




          168k11172404























              0












              $begingroup$

              You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.






                  share|cite|improve this answer









                  $endgroup$



                  You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 29 '18 at 14:48









                  YankoYanko

                  8,4692830




                  8,4692830






























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