Prescribing V shapes to the real numbers
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Can I prescribe a V shape to every number on the real number line so that the point of the V is in contact with this number, and no two V shapes intersect? (You may choose any specifically shaped "V" for each number, change the width of the V's, change the length of the legs of the V's, place V's in in the upper half-plane or lower half-plane)
Here's a picture describing the situation
real-analysis general-topology
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
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add a comment |
$begingroup$
Can I prescribe a V shape to every number on the real number line so that the point of the V is in contact with this number, and no two V shapes intersect? (You may choose any specifically shaped "V" for each number, change the width of the V's, change the length of the legs of the V's, place V's in in the upper half-plane or lower half-plane)
Here's a picture describing the situation
real-analysis general-topology
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
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Is it allowed to change the length of the $V$?
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– Guacho Perez
Dec 23 '18 at 22:48
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@GuachoPerez Doesn't matter - it's impossible either way.
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– Patrick Stevens
Dec 23 '18 at 22:49
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@GuachoPerez You may change the length of the the V provided that you don't shrink the V to a single point.
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– Andrew Lizarraga
Dec 23 '18 at 23:47
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@RobArthan The "V shapes" are pairs of line segments
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– Andrew Lizarraga
Dec 24 '18 at 0:33
add a comment |
$begingroup$
Can I prescribe a V shape to every number on the real number line so that the point of the V is in contact with this number, and no two V shapes intersect? (You may choose any specifically shaped "V" for each number, change the width of the V's, change the length of the legs of the V's, place V's in in the upper half-plane or lower half-plane)
Here's a picture describing the situation
real-analysis general-topology
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
$endgroup$
Can I prescribe a V shape to every number on the real number line so that the point of the V is in contact with this number, and no two V shapes intersect? (You may choose any specifically shaped "V" for each number, change the width of the V's, change the length of the legs of the V's, place V's in in the upper half-plane or lower half-plane)
Here's a picture describing the situation
real-analysis general-topology
real-analysis general-topology
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
asked Dec 23 '18 at 22:37
Andrew LizarragaAndrew Lizarraga
425
425
$begingroup$
Is it allowed to change the length of the $V$?
$endgroup$
– Guacho Perez
Dec 23 '18 at 22:48
$begingroup$
@GuachoPerez Doesn't matter - it's impossible either way.
$endgroup$
– Patrick Stevens
Dec 23 '18 at 22:49
$begingroup$
@GuachoPerez You may change the length of the the V provided that you don't shrink the V to a single point.
$endgroup$
– Andrew Lizarraga
Dec 23 '18 at 23:47
$begingroup$
@RobArthan The "V shapes" are pairs of line segments
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:33
add a comment |
$begingroup$
Is it allowed to change the length of the $V$?
$endgroup$
– Guacho Perez
Dec 23 '18 at 22:48
$begingroup$
@GuachoPerez Doesn't matter - it's impossible either way.
$endgroup$
– Patrick Stevens
Dec 23 '18 at 22:49
$begingroup$
@GuachoPerez You may change the length of the the V provided that you don't shrink the V to a single point.
$endgroup$
– Andrew Lizarraga
Dec 23 '18 at 23:47
$begingroup$
@RobArthan The "V shapes" are pairs of line segments
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:33
$begingroup$
Is it allowed to change the length of the $V$?
$endgroup$
– Guacho Perez
Dec 23 '18 at 22:48
$begingroup$
Is it allowed to change the length of the $V$?
$endgroup$
– Guacho Perez
Dec 23 '18 at 22:48
$begingroup$
@GuachoPerez Doesn't matter - it's impossible either way.
$endgroup$
– Patrick Stevens
Dec 23 '18 at 22:49
$begingroup$
@GuachoPerez Doesn't matter - it's impossible either way.
$endgroup$
– Patrick Stevens
Dec 23 '18 at 22:49
$begingroup$
@GuachoPerez You may change the length of the the V provided that you don't shrink the V to a single point.
$endgroup$
– Andrew Lizarraga
Dec 23 '18 at 23:47
$begingroup$
@GuachoPerez You may change the length of the the V provided that you don't shrink the V to a single point.
$endgroup$
– Andrew Lizarraga
Dec 23 '18 at 23:47
$begingroup$
@RobArthan The "V shapes" are pairs of line segments
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:33
$begingroup$
@RobArthan The "V shapes" are pairs of line segments
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:33
add a comment |
1 Answer
1
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oldest
votes
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I'm afraid not. Every V shape (when filled in) contains a rational point, and there are only countably many of those, but there are uncountably many reals, so at least one point is shared between two filled-in V's. Now show that those two V's intersect at an edge.
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
$endgroup$
1
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I'm afraid not. Every V shape (when filled in) contains a rational point, and there are only countably many of those, but there are uncountably many reals, so at least one point is shared between two filled-in V's. Now show that those two V's intersect at an edge.
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
$endgroup$
1
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
add a comment |
$begingroup$
I'm afraid not. Every V shape (when filled in) contains a rational point, and there are only countably many of those, but there are uncountably many reals, so at least one point is shared between two filled-in V's. Now show that those two V's intersect at an edge.
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
$endgroup$
1
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
add a comment |
$begingroup$
I'm afraid not. Every V shape (when filled in) contains a rational point, and there are only countably many of those, but there are uncountably many reals, so at least one point is shared between two filled-in V's. Now show that those two V's intersect at an edge.
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
$endgroup$
I'm afraid not. Every V shape (when filled in) contains a rational point, and there are only countably many of those, but there are uncountably many reals, so at least one point is shared between two filled-in V's. Now show that those two V's intersect at an edge.
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
answered Dec 23 '18 at 22:43
Patrick StevensPatrick Stevens
28.7k52874
28.7k52874
1
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
add a comment |
1
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
1
1
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
$begingroup$
@RobArthan I took it that Patrick Stevens took the problem in $mathbb{R}^2$, and then noted that $mathbb{Q}^2$ is dense in $mathbb{R}^2$. So if the $V$'s were instead solid triangles (this doesn't change the problem) then each triangle will contain a point in $mathbb{Q}^2$ (a rational point).
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:50
add a comment |
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$begingroup$
Is it allowed to change the length of the $V$?
$endgroup$
– Guacho Perez
Dec 23 '18 at 22:48
$begingroup$
@GuachoPerez Doesn't matter - it's impossible either way.
$endgroup$
– Patrick Stevens
Dec 23 '18 at 22:49
$begingroup$
@GuachoPerez You may change the length of the the V provided that you don't shrink the V to a single point.
$endgroup$
– Andrew Lizarraga
Dec 23 '18 at 23:47
$begingroup$
@RobArthan The "V shapes" are pairs of line segments
$endgroup$
– Andrew Lizarraga
Dec 24 '18 at 0:33