$vertmathbb{Q}^mvert = vertmathbb{N}vert$. Proof [duplicate]
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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
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
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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.
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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
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
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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.)
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– Mees de Vries
Dec 12 '18 at 10:47
add a comment |
$begingroup$
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
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
$endgroup$
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
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
asked Dec 12 '18 at 10:43
Egor EpishinEgor Epishin
54
54
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.
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
add a comment |
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
$endgroup$
$begingroup$
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
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@AsafKaragila Yes, I agree. Which one of the three should I chose?
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– 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
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
$endgroup$
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.
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
answered Dec 12 '18 at 10:46
Dietrich BurdeDietrich Burde
79.1k647103
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
add a comment |
$begingroup$
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
add a comment |
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