$vertmathbb{Q}^mvert = vertmathbb{N}vert$. Proof [duplicate]












0












$begingroup$



This question already has an answer here:




  • Produce an explicit bijection between rationals and naturals?

    7 answers



  • Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?

    2 answers



  • Is $mathbb Q times mathbb Q $ a denumerable set? [duplicate]

    1 answer




To start with I'm not completely understand what $mathbb{Q}^m$ is. Is it the number of all elements in $mathbb{Q}$ that was taken to some power $m$?



To show that two sets have the same cardinality we must introduce a map $f: mathbb{N} longrightarrow mathbb{Q}^m$ that would be a bijection, but I got stuck with it. Can someone please show me a proof or give a hint?






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This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 1




    $begingroup$
    The set $mathbb Q^m$ is the set of all sequences $(q_1, ldots, q_m)$ where each $q_i$ is a rational number. (You may be more familiar with the set $mathbb R^m$ -- the idea is the same.)
    $endgroup$
    – Mees de Vries
    Dec 12 '18 at 10:47
















0












$begingroup$



This question already has an answer here:




  • Produce an explicit bijection between rationals and naturals?

    7 answers



  • Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?

    2 answers



  • Is $mathbb Q times mathbb Q $ a denumerable set? [duplicate]

    1 answer




To start with I'm not completely understand what $mathbb{Q}^m$ is. Is it the number of all elements in $mathbb{Q}$ that was taken to some power $m$?



To show that two sets have the same cardinality we must introduce a map $f: mathbb{N} longrightarrow mathbb{Q}^m$ that would be a bijection, but I got stuck with it. Can someone please show me a proof or give a hint?






elementary-set-theory







share|cite|improve this question









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marked as duplicate by Asaf Karagila♦ elementary-set-theory
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  • 1




    $begingroup$
    The set $mathbb Q^m$ is the set of all sequences $(q_1, ldots, q_m)$ where each $q_i$ is a rational number. (You may be more familiar with the set $mathbb R^m$ -- the idea is the same.)
    $endgroup$
    – Mees de Vries
    Dec 12 '18 at 10:47














0












0








0





$begingroup$



This question already has an answer here:




  • Produce an explicit bijection between rationals and naturals?

    7 answers



  • Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?

    2 answers



  • Is $mathbb Q times mathbb Q $ a denumerable set? [duplicate]

    1 answer




To start with I'm not completely understand what $mathbb{Q}^m$ is. Is it the number of all elements in $mathbb{Q}$ that was taken to some power $m$?



To show that two sets have the same cardinality we must introduce a map $f: mathbb{N} longrightarrow mathbb{Q}^m$ that would be a bijection, but I got stuck with it. Can someone please show me a proof or give a hint?






elementary-set-theory







share|cite|improve this question









$endgroup$





This question already has an answer here:




  • Produce an explicit bijection between rationals and naturals?

    7 answers



  • Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?

    2 answers



  • Is $mathbb Q times mathbb Q $ a denumerable set? [duplicate]

    1 answer




To start with I'm not completely understand what $mathbb{Q}^m$ is. Is it the number of all elements in $mathbb{Q}$ that was taken to some power $m$?



To show that two sets have the same cardinality we must introduce a map $f: mathbb{N} longrightarrow mathbb{Q}^m$ that would be a bijection, but I got stuck with it. Can someone please show me a proof or give a hint?





This question already has an answer here:




  • Produce an explicit bijection between rationals and naturals?

    7 answers



  • Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?

    2 answers



  • Is $mathbb Q times mathbb Q $ a denumerable set? [duplicate]

    1 answer







elementary-set-theory




elementary-set-theory






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share|cite|improve this question




share|cite|improve this question










asked Dec 12 '18 at 10:43









Egor EpishinEgor Epishin

54




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marked as duplicate by Asaf Karagila♦ elementary-set-theory
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 Dec 12 '18 at 12:09


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 1




    $begingroup$
    The set $mathbb Q^m$ is the set of all sequences $(q_1, ldots, q_m)$ where each $q_i$ is a rational number. (You may be more familiar with the set $mathbb R^m$ -- the idea is the same.)
    $endgroup$
    – Mees de Vries
    Dec 12 '18 at 10:47














  • 1




    $begingroup$
    The set $mathbb Q^m$ is the set of all sequences $(q_1, ldots, q_m)$ where each $q_i$ is a rational number. (You may be more familiar with the set $mathbb R^m$ -- the idea is the same.)
    $endgroup$
    – Mees de Vries
    Dec 12 '18 at 10:47








1




1




$begingroup$
The set $mathbb Q^m$ is the set of all sequences $(q_1, ldots, q_m)$ where each $q_i$ is a rational number. (You may be more familiar with the set $mathbb R^m$ -- the idea is the same.)
$endgroup$
– Mees de Vries
Dec 12 '18 at 10:47




$begingroup$
The set $mathbb Q^m$ is the set of all sequences $(q_1, ldots, q_m)$ where each $q_i$ is a rational number. (You may be more familiar with the set $mathbb R^m$ -- the idea is the same.)
$endgroup$
– Mees de Vries
Dec 12 '18 at 10:47










1 Answer
1






active

oldest

votes


















0












$begingroup$

We have an explicit bijection between $Bbb{N}$ and $Bbb{Q}$, see



Produce an explicit bijection between rationals and naturals?,



and an bijection between $Bbb{Q}$ and $Bbb{Q}^m$, see



Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?



Is $mathbb Q times mathbb Q $ a denumerable set?



Now compose the two bijections.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 10:52










  • $begingroup$
    @AsafKaragila Yes, I agree. Which one of the three should I chose?
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:08










  • $begingroup$
    Since we can have multiple duplicate targets, why not all of them?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 12:09










  • $begingroup$
    Ah, I didn't know that multiple assigments are possible, thank you!
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:11


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

We have an explicit bijection between $Bbb{N}$ and $Bbb{Q}$, see



Produce an explicit bijection between rationals and naturals?,



and an bijection between $Bbb{Q}$ and $Bbb{Q}^m$, see



Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?



Is $mathbb Q times mathbb Q $ a denumerable set?



Now compose the two bijections.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 10:52










  • $begingroup$
    @AsafKaragila Yes, I agree. Which one of the three should I chose?
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:08










  • $begingroup$
    Since we can have multiple duplicate targets, why not all of them?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 12:09










  • $begingroup$
    Ah, I didn't know that multiple assigments are possible, thank you!
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:11
















0












$begingroup$

We have an explicit bijection between $Bbb{N}$ and $Bbb{Q}$, see



Produce an explicit bijection between rationals and naturals?,



and an bijection between $Bbb{Q}$ and $Bbb{Q}^m$, see



Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?



Is $mathbb Q times mathbb Q $ a denumerable set?



Now compose the two bijections.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 10:52










  • $begingroup$
    @AsafKaragila Yes, I agree. Which one of the three should I chose?
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:08










  • $begingroup$
    Since we can have multiple duplicate targets, why not all of them?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 12:09










  • $begingroup$
    Ah, I didn't know that multiple assigments are possible, thank you!
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:11














0












0








0





$begingroup$

We have an explicit bijection between $Bbb{N}$ and $Bbb{Q}$, see



Produce an explicit bijection between rationals and naturals?,



and an bijection between $Bbb{Q}$ and $Bbb{Q}^m$, see



Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?



Is $mathbb Q times mathbb Q $ a denumerable set?



Now compose the two bijections.






share|cite|improve this answer









$endgroup$



We have an explicit bijection between $Bbb{N}$ and $Bbb{Q}$, see



Produce an explicit bijection between rationals and naturals?,



and an bijection between $Bbb{Q}$ and $Bbb{Q}^m$, see



Does There exist a continuous bijection $mathbb{Q}to mathbb{Q}times mathbb{Q}$?



Is $mathbb Q times mathbb Q $ a denumerable set?



Now compose the two bijections.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 12 '18 at 10:46









Dietrich BurdeDietrich Burde

79.1k647103




79.1k647103












  • $begingroup$
    If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 10:52










  • $begingroup$
    @AsafKaragila Yes, I agree. Which one of the three should I chose?
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:08










  • $begingroup$
    Since we can have multiple duplicate targets, why not all of them?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 12:09










  • $begingroup$
    Ah, I didn't know that multiple assigments are possible, thank you!
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:11


















  • $begingroup$
    If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 10:52










  • $begingroup$
    @AsafKaragila Yes, I agree. Which one of the three should I chose?
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:08










  • $begingroup$
    Since we can have multiple duplicate targets, why not all of them?
    $endgroup$
    – Asaf Karagila♦
    Dec 12 '18 at 12:09










  • $begingroup$
    Ah, I didn't know that multiple assigments are possible, thank you!
    $endgroup$
    – Dietrich Burde
    Dec 12 '18 at 12:11
















$begingroup$
If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
$endgroup$
– Asaf Karagila♦
Dec 12 '18 at 10:52




$begingroup$
If your answer is just links to other questions, then perhaps the question is just a duplicate, and instead of writing an answer you should close it as such?
$endgroup$
– Asaf Karagila♦
Dec 12 '18 at 10:52












$begingroup$
@AsafKaragila Yes, I agree. Which one of the three should I chose?
$endgroup$
– Dietrich Burde
Dec 12 '18 at 12:08




$begingroup$
@AsafKaragila Yes, I agree. Which one of the three should I chose?
$endgroup$
– Dietrich Burde
Dec 12 '18 at 12:08












$begingroup$
Since we can have multiple duplicate targets, why not all of them?
$endgroup$
– Asaf Karagila♦
Dec 12 '18 at 12:09




$begingroup$
Since we can have multiple duplicate targets, why not all of them?
$endgroup$
– Asaf Karagila♦
Dec 12 '18 at 12:09












$begingroup$
Ah, I didn't know that multiple assigments are possible, thank you!
$endgroup$
– Dietrich Burde
Dec 12 '18 at 12:11




$begingroup$
Ah, I didn't know that multiple assigments are possible, thank you!
$endgroup$
– Dietrich Burde
Dec 12 '18 at 12:11



-

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