Cardinality of ${f:mathbb{R}rightarrow mathbb{R}: (forall x in mathbb{R} setminus mathbb{Q})(f(x)-x in...
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Let $S$ be ${f:mathbb{R}rightarrow mathbb{R}: (forall x in mathbb{R} setminus mathbb{Q})(f(x)-x in mathbb{N})}$. Determine the cardinality of S!
My attempt:
I honestly have no clue, I know that $forall x in mathbb{R} setminus mathbb{Q} ,f(x) = x+n, n in mathbb{N}$ but other than that no useful conclusions. I can't even seem to intuitively guess this. Of course, $k(S) leq k(mathbb{R}^{mathbb{R}})$, but other than that I'm lost here and would appreciate any hints!
I've tried to build different injections from $P(mathbb R)$,$mathbb{R}^{mathbb{R}}$ and $P(mathbb{R} setminus mathbb{Q})$ to $S$ but have failed miserably!
elementary-set-theory
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
asked Nov 22 at 16:14
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716420
add a comment |
up vote
0
down vote
favorite
Let $S$ be ${f:mathbb{R}rightarrow mathbb{R}: (forall x in mathbb{R} setminus mathbb{Q})(f(x)-x in mathbb{N})}$. Determine the cardinality of S!
My attempt:
I honestly have no clue, I know that $forall x in mathbb{R} setminus mathbb{Q} ,f(x) = x+n, n in mathbb{N}$ but other than that no useful conclusions. I can't even seem to intuitively guess this. Of course, $k(S) leq k(mathbb{R}^{mathbb{R}})$, but other than that I'm lost here and would appreciate any hints!
I've tried to build different injections from $P(mathbb R)$,$mathbb{R}^{mathbb{R}}$ and $P(mathbb{R} setminus mathbb{Q})$ to $S$ but have failed miserably!
elementary-set-theory
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
asked Nov 22 at 16:14
Collapse
716420
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $S$ be ${f:mathbb{R}rightarrow mathbb{R}: (forall x in mathbb{R} setminus mathbb{Q})(f(x)-x in mathbb{N})}$. Determine the cardinality of S!
My attempt:
I honestly have no clue, I know that $forall x in mathbb{R} setminus mathbb{Q} ,f(x) = x+n, n in mathbb{N}$ but other than that no useful conclusions. I can't even seem to intuitively guess this. Of course, $k(S) leq k(mathbb{R}^{mathbb{R}})$, but other than that I'm lost here and would appreciate any hints!
I've tried to build different injections from $P(mathbb R)$,$mathbb{R}^{mathbb{R}}$ and $P(mathbb{R} setminus mathbb{Q})$ to $S$ but have failed miserably!
elementary-set-theory
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
asked Nov 22 at 16:14
Collapse
716420
Let $S$ be ${f:mathbb{R}rightarrow mathbb{R}: (forall x in mathbb{R} setminus mathbb{Q})(f(x)-x in mathbb{N})}$. Determine the cardinality of S!
My attempt:
I honestly have no clue, I know that $forall x in mathbb{R} setminus mathbb{Q} ,f(x) = x+n, n in mathbb{N}$ but other than that no useful conclusions. I can't even seem to intuitively guess this. Of course, $k(S) leq k(mathbb{R}^{mathbb{R}})$, but other than that I'm lost here and would appreciate any hints!
I've tried to build different injections from $P(mathbb R)$,$mathbb{R}^{mathbb{R}}$ and $P(mathbb{R} setminus mathbb{Q})$ to $S$ but have failed miserably!
elementary-set-theory
elementary-set-theory
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
asked Nov 22 at 16:14
Collapse
716420
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
asked Nov 22 at 16:14
Collapse
716420
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
edited Nov 22 at 16:17
Asaf Karagila♦
301k32422753
301k32422753
asked Nov 22 at 16:14
Collapse
716420
asked Nov 22 at 16:14
Collapse
716420
asked Nov 22 at 16:14
Collapse
716420
716420
add a comment |
add a comment |
1 Answer
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If $fin S$, define $g_f$ to be $g_f(x)=f(x)-x$. Then $g_finBbb{N^{Rsetminus Q}}$, define $h_f=frestrictionBbb Q$. Note, moreover, that that $fmapsto (g_f,h_f)inBbb{N^{Rsetminus Q}times R^Q}$ is a bijection.
What is the cardinality of $Bbb{N^{Rsetminus Q}times R^Q}$?
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
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
up vote
1
down vote
accepted
If $fin S$, define $g_f$ to be $g_f(x)=f(x)-x$. Then $g_finBbb{N^{Rsetminus Q}}$, define $h_f=frestrictionBbb Q$. Note, moreover, that that $fmapsto (g_f,h_f)inBbb{N^{Rsetminus Q}times R^Q}$ is a bijection.
What is the cardinality of $Bbb{N^{Rsetminus Q}times R^Q}$?
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
add a comment |
up vote
1
down vote
accepted
If $fin S$, define $g_f$ to be $g_f(x)=f(x)-x$. Then $g_finBbb{N^{Rsetminus Q}}$, define $h_f=frestrictionBbb Q$. Note, moreover, that that $fmapsto (g_f,h_f)inBbb{N^{Rsetminus Q}times R^Q}$ is a bijection.
What is the cardinality of $Bbb{N^{Rsetminus Q}times R^Q}$?
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If $fin S$, define $g_f$ to be $g_f(x)=f(x)-x$. Then $g_finBbb{N^{Rsetminus Q}}$, define $h_f=frestrictionBbb Q$. Note, moreover, that that $fmapsto (g_f,h_f)inBbb{N^{Rsetminus Q}times R^Q}$ is a bijection.
What is the cardinality of $Bbb{N^{Rsetminus Q}times R^Q}$?
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
If $fin S$, define $g_f$ to be $g_f(x)=f(x)-x$. Then $g_finBbb{N^{Rsetminus Q}}$, define $h_f=frestrictionBbb Q$. Note, moreover, that that $fmapsto (g_f,h_f)inBbb{N^{Rsetminus Q}times R^Q}$ is a bijection.
What is the cardinality of $Bbb{N^{Rsetminus Q}times R^Q}$?
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
answered Nov 22 at 16:17
Asaf Karagila♦
301k32422753
301k32422753
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
add a comment |
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
The cardinality of that set is $2^c$ ? (where $c=k(mathbb{R})$?
– Collapse
Nov 22 at 16:23
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
Yes. It is.${}{}$
– Asaf Karagila♦
Nov 22 at 16:24
add a comment |
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