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?
elementary-set-theory cardinals
asked Dec 9 '18 at 19:44
TsarNTsarN
465
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add a comment |
$begingroup$
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
asked Dec 9 '18 at 19:44
TsarNTsarN
465
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First step: this duplicate.
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– Dietrich Burde
Dec 9 '18 at 19:49
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The $Bbb {N leftrightarrow Q}$ part is covered here
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– Ross Millikan
Dec 9 '18 at 19:53
add a comment |
$begingroup$
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
asked Dec 9 '18 at 19:44
TsarNTsarN
465
$endgroup$
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
asked Dec 9 '18 at 19:44
TsarNTsarN
465
asked Dec 9 '18 at 19:44
TsarNTsarN
465
asked Dec 9 '18 at 19:44
TsarNTsarN
465
asked Dec 9 '18 at 19:44
TsarNTsarN
465
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
add a comment |
$begingroup$
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
add a comment |
2 Answers
2
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oldest
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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.
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
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add a comment |
$begingroup$
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).
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
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add a comment |
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2 Answers
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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.
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
$endgroup$
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.
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
answered Dec 9 '18 at 19:47
Ross MillikanRoss Millikan
294k23198371
294k23198371
add a comment |
add a comment |
$begingroup$
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).
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
$endgroup$
add a comment |
$begingroup$
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).
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
$endgroup$
add a comment |
$begingroup$
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).
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
$endgroup$
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).
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
edited Dec 9 '18 at 20:06
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
answered Dec 9 '18 at 19:49
GenericMathematicianGenericMathematician
863
863
add a comment |
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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