Determining the cardinality of rational set
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I have two questions, but I don't even know where I can start to solve it, can you give me a hint?
The question is like (Forgive me if MathJax is going wrong):
Determine the cardinality of these sets
a) If X = ${x in mathbb{R} | 1 leqslant x leqslant 3}$
b) Be $mathbb{Q}$ like $mathbb{Q} = { p/q | p,q in mathbb{Z} q gt 0}$
discrete-mathematics elementary-set-theory
edited Dec 3 '18 at 13:57
ervx
10.3k31338
asked Dec 3 '18 at 13:56
Will_UWill_U
11
$endgroup$
add a comment |
$begingroup$
I have two questions, but I don't even know where I can start to solve it, can you give me a hint?
The question is like (Forgive me if MathJax is going wrong):
Determine the cardinality of these sets
a) If X = ${x in mathbb{R} | 1 leqslant x leqslant 3}$
b) Be $mathbb{Q}$ like $mathbb{Q} = { p/q | p,q in mathbb{Z} q gt 0}$
discrete-mathematics elementary-set-theory
edited Dec 3 '18 at 13:57
ervx
10.3k31338
asked Dec 3 '18 at 13:56
Will_UWill_U
11
$endgroup$
1
$begingroup$
What do you know about cardinalities?
$endgroup$
– ervx
Dec 3 '18 at 13:57
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that is a beautiful question, loved when I had those! They are really enlighting. I recommend you build yourself bijective maps into either $mathbb{N}$ or $mathbb{R}$, since you know the cardinality of those
$endgroup$
– Enkidu
Dec 3 '18 at 14:11
add a comment |
$begingroup$
I have two questions, but I don't even know where I can start to solve it, can you give me a hint?
The question is like (Forgive me if MathJax is going wrong):
Determine the cardinality of these sets
a) If X = ${x in mathbb{R} | 1 leqslant x leqslant 3}$
b) Be $mathbb{Q}$ like $mathbb{Q} = { p/q | p,q in mathbb{Z} q gt 0}$
discrete-mathematics elementary-set-theory
edited Dec 3 '18 at 13:57
ervx
10.3k31338
asked Dec 3 '18 at 13:56
Will_UWill_U
11
$endgroup$
I have two questions, but I don't even know where I can start to solve it, can you give me a hint?
The question is like (Forgive me if MathJax is going wrong):
Determine the cardinality of these sets
a) If X = ${x in mathbb{R} | 1 leqslant x leqslant 3}$
b) Be $mathbb{Q}$ like $mathbb{Q} = { p/q | p,q in mathbb{Z} q gt 0}$
discrete-mathematics elementary-set-theory
discrete-mathematics elementary-set-theory
edited Dec 3 '18 at 13:57
ervx
10.3k31338
asked Dec 3 '18 at 13:56
Will_UWill_U
11
edited Dec 3 '18 at 13:57
ervx
10.3k31338
asked Dec 3 '18 at 13:56
Will_UWill_U
11
edited Dec 3 '18 at 13:57
ervx
10.3k31338
edited Dec 3 '18 at 13:57
ervx
10.3k31338
edited Dec 3 '18 at 13:57
ervx
10.3k31338
10.3k31338
asked Dec 3 '18 at 13:56
Will_UWill_U
11
asked Dec 3 '18 at 13:56
Will_UWill_U
11
asked Dec 3 '18 at 13:56
Will_UWill_U
11
11
1
$begingroup$
What do you know about cardinalities?
$endgroup$
– ervx
Dec 3 '18 at 13:57
$begingroup$
that is a beautiful question, loved when I had those! They are really enlighting. I recommend you build yourself bijective maps into either $mathbb{N}$ or $mathbb{R}$, since you know the cardinality of those
$endgroup$
– Enkidu
Dec 3 '18 at 14:11
add a comment |
1
$begingroup$
What do you know about cardinalities?
$endgroup$
– ervx
Dec 3 '18 at 13:57
$begingroup$
that is a beautiful question, loved when I had those! They are really enlighting. I recommend you build yourself bijective maps into either $mathbb{N}$ or $mathbb{R}$, since you know the cardinality of those
$endgroup$
– Enkidu
Dec 3 '18 at 14:11
1
1
$begingroup$
What do you know about cardinalities?
$endgroup$
– ervx
Dec 3 '18 at 13:57
$begingroup$
What do you know about cardinalities?
$endgroup$
– ervx
Dec 3 '18 at 13:57
$begingroup$
that is a beautiful question, loved when I had those! They are really enlighting. I recommend you build yourself bijective maps into either $mathbb{N}$ or $mathbb{R}$, since you know the cardinality of those
$endgroup$
– Enkidu
Dec 3 '18 at 14:11
$begingroup$
that is a beautiful question, loved when I had those! They are really enlighting. I recommend you build yourself bijective maps into either $mathbb{N}$ or $mathbb{R}$, since you know the cardinality of those
$endgroup$
– Enkidu
Dec 3 '18 at 14:11
add a comment |
1 Answer
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For one you'll need to make use of the fact that $omega$ (=$mathbb{N}$) is bijectively equivalent to $omegatimesomega$. A second useful fact is that every real number, except for countably many, has a unique binary expansion.
The last tool in you arsenal is the Cantor-Schröder-Bernstein theorem.
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
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add a comment |
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1 Answer
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$begingroup$
For one you'll need to make use of the fact that $omega$ (=$mathbb{N}$) is bijectively equivalent to $omegatimesomega$. A second useful fact is that every real number, except for countably many, has a unique binary expansion.
The last tool in you arsenal is the Cantor-Schröder-Bernstein theorem.
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
$endgroup$
add a comment |
$begingroup$
For one you'll need to make use of the fact that $omega$ (=$mathbb{N}$) is bijectively equivalent to $omegatimesomega$. A second useful fact is that every real number, except for countably many, has a unique binary expansion.
The last tool in you arsenal is the Cantor-Schröder-Bernstein theorem.
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
$endgroup$
add a comment |
$begingroup$
For one you'll need to make use of the fact that $omega$ (=$mathbb{N}$) is bijectively equivalent to $omegatimesomega$. A second useful fact is that every real number, except for countably many, has a unique binary expansion.
The last tool in you arsenal is the Cantor-Schröder-Bernstein theorem.
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
$endgroup$
For one you'll need to make use of the fact that $omega$ (=$mathbb{N}$) is bijectively equivalent to $omegatimesomega$. A second useful fact is that every real number, except for countably many, has a unique binary expansion.
The last tool in you arsenal is the Cantor-Schröder-Bernstein theorem.
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
answered Dec 3 '18 at 14:55
Jean-Pierre de VilliersJean-Pierre de Villiers
415
415
add a comment |
add a comment |
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$begingroup$
What do you know about cardinalities?
$endgroup$
– ervx
Dec 3 '18 at 13:57
$begingroup$
that is a beautiful question, loved when I had those! They are really enlighting. I recommend you build yourself bijective maps into either $mathbb{N}$ or $mathbb{R}$, since you know the cardinality of those
$endgroup$
– Enkidu
Dec 3 '18 at 14:11