Bijection between $mathbb{Q}$ and $mathbb{N}timesmathbb{Q}$












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I want to prove that $|mathbb{Q}| = |mathbb{N}timesmathbb{Q}|$, but I have no idea how to find bijection between these sets. Can you help me?






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  • $begingroup$
    First step: this duplicate.
    $endgroup$
    – Dietrich Burde
    Dec 9 '18 at 19:49










  • $begingroup$
    The $Bbb {N leftrightarrow Q}$ part is covered here
    $endgroup$
    – Ross Millikan
    Dec 9 '18 at 19:53
















1












$begingroup$


I want to prove that $|mathbb{Q}| = |mathbb{N}timesmathbb{Q}|$, but I have no idea how to find bijection between these sets. Can you help me?






elementary-set-theory cardinals







share|cite|improve this question









$endgroup$












  • $begingroup$
    First step: this duplicate.
    $endgroup$
    – Dietrich Burde
    Dec 9 '18 at 19:49










  • $begingroup$
    The $Bbb {N leftrightarrow Q}$ part is covered here
    $endgroup$
    – Ross Millikan
    Dec 9 '18 at 19:53














1












1








1





$begingroup$


I want to prove that $|mathbb{Q}| = |mathbb{N}timesmathbb{Q}|$, but I have no idea how to find bijection between these sets. Can you help me?






elementary-set-theory cardinals







share|cite|improve this question









$endgroup$




I want to prove that $|mathbb{Q}| = |mathbb{N}timesmathbb{Q}|$, but I have no idea how to find bijection between these sets. Can you help me?






elementary-set-theory cardinals




elementary-set-theory cardinals






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asked Dec 9 '18 at 19:44









TsarNTsarN

465




465












  • $begingroup$
    First step: this duplicate.
    $endgroup$
    – Dietrich Burde
    Dec 9 '18 at 19:49










  • $begingroup$
    The $Bbb {N leftrightarrow Q}$ part is covered here
    $endgroup$
    – Ross Millikan
    Dec 9 '18 at 19:53


















  • $begingroup$
    First step: this duplicate.
    $endgroup$
    – Dietrich Burde
    Dec 9 '18 at 19:49










  • $begingroup$
    The $Bbb {N leftrightarrow Q}$ part is covered here
    $endgroup$
    – Ross Millikan
    Dec 9 '18 at 19:53
















$begingroup$
First step: this duplicate.
$endgroup$
– Dietrich Burde
Dec 9 '18 at 19:49




$begingroup$
First step: this duplicate.
$endgroup$
– Dietrich Burde
Dec 9 '18 at 19:49












$begingroup$
The $Bbb {N leftrightarrow Q}$ part is covered here
$endgroup$
– Ross Millikan
Dec 9 '18 at 19:53




$begingroup$
The $Bbb {N leftrightarrow Q}$ part is covered here
$endgroup$
– Ross Millikan
Dec 9 '18 at 19:53










2 Answers
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$begingroup$

Do you have bijections between $Bbb N$ and $Bbb {N times N}$ and between $Bbb N$ and $Bbb Q$? Usually you would when this question comes up. Compose those.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    Ross Millikan's answer is definitely the way to go for explicitly demonstrating a bijection, but it is also possible to invoke the Cantor-Bernstein-Schroeder theorem to prove this is true, simply by finding injections/surjections both ways (which I find to be easier in practice).






    share|cite|improve this answer











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      2 Answers
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      2 Answers
      2






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      1












      $begingroup$

      Do you have bijections between $Bbb N$ and $Bbb {N times N}$ and between $Bbb N$ and $Bbb Q$? Usually you would when this question comes up. Compose those.






      share|cite|improve this answer









      $endgroup$


















        1












        $begingroup$

        Do you have bijections between $Bbb N$ and $Bbb {N times N}$ and between $Bbb N$ and $Bbb Q$? Usually you would when this question comes up. Compose those.






        share|cite|improve this answer









        $endgroup$
















          1












          1








          1





          $begingroup$

          Do you have bijections between $Bbb N$ and $Bbb {N times N}$ and between $Bbb N$ and $Bbb Q$? Usually you would when this question comes up. Compose those.






          share|cite|improve this answer









          $endgroup$



          Do you have bijections between $Bbb N$ and $Bbb {N times N}$ and between $Bbb N$ and $Bbb Q$? Usually you would when this question comes up. Compose those.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 9 '18 at 19:47









          Ross MillikanRoss Millikan

          294k23198371




          294k23198371























              0












              $begingroup$

              Ross Millikan's answer is definitely the way to go for explicitly demonstrating a bijection, but it is also possible to invoke the Cantor-Bernstein-Schroeder theorem to prove this is true, simply by finding injections/surjections both ways (which I find to be easier in practice).






              share|cite|improve this answer











              $endgroup$


















                0












                $begingroup$

                Ross Millikan's answer is definitely the way to go for explicitly demonstrating a bijection, but it is also possible to invoke the Cantor-Bernstein-Schroeder theorem to prove this is true, simply by finding injections/surjections both ways (which I find to be easier in practice).






                share|cite|improve this answer











                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Ross Millikan's answer is definitely the way to go for explicitly demonstrating a bijection, but it is also possible to invoke the Cantor-Bernstein-Schroeder theorem to prove this is true, simply by finding injections/surjections both ways (which I find to be easier in practice).






                  share|cite|improve this answer











                  $endgroup$



                  Ross Millikan's answer is definitely the way to go for explicitly demonstrating a bijection, but it is also possible to invoke the Cantor-Bernstein-Schroeder theorem to prove this is true, simply by finding injections/surjections both ways (which I find to be easier in practice).







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 9 '18 at 20:06

























                  answered Dec 9 '18 at 19:49









                  GenericMathematicianGenericMathematician

                  863




                  863






























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