Cardinality of the set of all real functions of real variable
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How does one compute the cardinality of the set of functions $f:mathbb{R} to mathbb{R}$ (not necessarily continuous)?
elementary-set-theory cardinals
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up vote
35
down vote
favorite
How does one compute the cardinality of the set of functions $f:mathbb{R} to mathbb{R}$ (not necessarily continuous)?
elementary-set-theory cardinals
You have used the assumption that the Cardinality of power set of R is equal to the Cardinality of functions from real to {0,1}. How to we prove that as well?
– user39246
Sep 2 '12 at 10:59
@krishnanem: Please ask new questions in a new thread, rather than as an answer to a previous question. Furthermore, this question was asked and answered several times before.
– Asaf Karagila♦
Sep 2 '12 at 11:17
@AsafKaragila: I have moved this non-answer to a comment so that krishnanem can pose it as a new question and can read your comment.
– robjohn♦
Sep 2 '12 at 20:39
@robjohn: I think Michael's comment would have been a more suitable choice over mine.
– Asaf Karagila♦
Sep 2 '12 at 21:44
@AsafKaragila: The point is to have krishnanem post the comment as a new question. Your comment solely addresses that point. Michael's also addresses the misplaced question and might encourage a reply.
– robjohn♦
Sep 3 '12 at 3:18
add a comment |
up vote
35
down vote
favorite
up vote
35
down vote
favorite
How does one compute the cardinality of the set of functions $f:mathbb{R} to mathbb{R}$ (not necessarily continuous)?
elementary-set-theory cardinals
How does one compute the cardinality of the set of functions $f:mathbb{R} to mathbb{R}$ (not necessarily continuous)?
elementary-set-theory cardinals
elementary-set-theory cardinals
edited Sep 2 '12 at 11:05
Asaf Karagila♦
300k32421751
300k32421751
asked Jan 17 '11 at 23:26
Benji
2,10631823
2,10631823
You have used the assumption that the Cardinality of power set of R is equal to the Cardinality of functions from real to {0,1}. How to we prove that as well?
– user39246
Sep 2 '12 at 10:59
@krishnanem: Please ask new questions in a new thread, rather than as an answer to a previous question. Furthermore, this question was asked and answered several times before.
– Asaf Karagila♦
Sep 2 '12 at 11:17
@AsafKaragila: I have moved this non-answer to a comment so that krishnanem can pose it as a new question and can read your comment.
– robjohn♦
Sep 2 '12 at 20:39
@robjohn: I think Michael's comment would have been a more suitable choice over mine.
– Asaf Karagila♦
Sep 2 '12 at 21:44
@AsafKaragila: The point is to have krishnanem post the comment as a new question. Your comment solely addresses that point. Michael's also addresses the misplaced question and might encourage a reply.
– robjohn♦
Sep 3 '12 at 3:18
add a comment |
You have used the assumption that the Cardinality of power set of R is equal to the Cardinality of functions from real to {0,1}. How to we prove that as well?
– user39246
Sep 2 '12 at 10:59
@krishnanem: Please ask new questions in a new thread, rather than as an answer to a previous question. Furthermore, this question was asked and answered several times before.
– Asaf Karagila♦
Sep 2 '12 at 11:17
@AsafKaragila: I have moved this non-answer to a comment so that krishnanem can pose it as a new question and can read your comment.
– robjohn♦
Sep 2 '12 at 20:39
@robjohn: I think Michael's comment would have been a more suitable choice over mine.
– Asaf Karagila♦
Sep 2 '12 at 21:44
@AsafKaragila: The point is to have krishnanem post the comment as a new question. Your comment solely addresses that point. Michael's also addresses the misplaced question and might encourage a reply.
– robjohn♦
Sep 3 '12 at 3:18
You have used the assumption that the Cardinality of power set of R is equal to the Cardinality of functions from real to {0,1}. How to we prove that as well?
– user39246
Sep 2 '12 at 10:59
You have used the assumption that the Cardinality of power set of R is equal to the Cardinality of functions from real to {0,1}. How to we prove that as well?
– user39246
Sep 2 '12 at 10:59
@krishnanem: Please ask new questions in a new thread, rather than as an answer to a previous question. Furthermore, this question was asked and answered several times before.
– Asaf Karagila♦
Sep 2 '12 at 11:17
@krishnanem: Please ask new questions in a new thread, rather than as an answer to a previous question. Furthermore, this question was asked and answered several times before.
– Asaf Karagila♦
Sep 2 '12 at 11:17
@AsafKaragila: I have moved this non-answer to a comment so that krishnanem can pose it as a new question and can read your comment.
– robjohn♦
Sep 2 '12 at 20:39
@AsafKaragila: I have moved this non-answer to a comment so that krishnanem can pose it as a new question and can read your comment.
– robjohn♦
Sep 2 '12 at 20:39
@robjohn: I think Michael's comment would have been a more suitable choice over mine.
– Asaf Karagila♦
Sep 2 '12 at 21:44
@robjohn: I think Michael's comment would have been a more suitable choice over mine.
– Asaf Karagila♦
Sep 2 '12 at 21:44
@AsafKaragila: The point is to have krishnanem post the comment as a new question. Your comment solely addresses that point. Michael's also addresses the misplaced question and might encourage a reply.
– robjohn♦
Sep 3 '12 at 3:18
@AsafKaragila: The point is to have krishnanem post the comment as a new question. Your comment solely addresses that point. Michael's also addresses the misplaced question and might encourage a reply.
– robjohn♦
Sep 3 '12 at 3:18
add a comment |
4 Answers
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up vote
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accepted
All you need is a few basics of cardinal arithmetic: if $kappa$ and $lambda$ are cardinals, none of them zero, and at least one of them is infinite, then $kappa+lambda = kappalambda = max{kappa,lambda}$. And cardinal exponentiation satisfies some of the same laws as regular exponentiation; in particular, $(kappa^{lambda})^{nu} = kappa^{lambdanu}$.
The cardinality of the set of all real functions is then
$$|mathbb{R}|^{|mathbb{R}|} =mathfrak{c}^{mathfrak{c}} = (2^{aleph_0})^{2^{aleph_0}} = 2^{aleph_02^{aleph_0}} = 2^{2^{aleph_0}} = 2^{mathfrak{c}}.$$
In other words, it is equal to the cardinality of the power set of $mathbb{R}$.
With a few extra facts, you can get more. In general, if $kappa$ is an infinite cardinal, and $2leqlambdaleqkappa$, then $lambda^{kappa}=2^{kappa}$. This follows because:
$$2^{kappa} leq lambda^{kappa} leq (2^{lambda})^{kappa} = 2^{lambdakappa} = 2^{kappa},$$
so you get equality throughout. The extra information you need for this is to know that if $kappa$, $lambda$, and $nu$ are nonzero cardinals, $kappaleqlambda$, then $kappa^{nu}leq lambda^{nu}$.
In particular, for any infinite cardinal $kappa$ you have $kappa^{kappa} = 2^{kappa}$.
add a comment |
up vote
29
down vote
I guess that you know that $|mathbb{N}| = |mathbb{N}timesmathbb{N}|$ and thus $|mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$
This means that $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$. Since $fcolonmathbb{R}tomathbb{R}$ is an element of $P(mathbb{R}timesmathbb{R})$ you have that $mathbb{R}^mathbb{R}$ (all the functions from $mathbb{R}$ to itself) is of cardinality less or equal to the one of $P(mathbb{R}timesmathbb{R})$ which in turn means that $|mathbb{R}^mathbb{R}|le |P(mathbb{R})|$.
Now, since $|P(mathbb{R})| = |2^mathbb{R}|$ which is the set of all functions from $mathbb{R}$ to ${0,1}$, and clearly every function from $mathbb{R}$ into ${0,1}$ is in particular a function from $mathbb{R}$ into itself, we have:
$$|P(mathbb{R})| = |2^mathbb{R}| le |mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})| = |P(mathbb{R})|$$
So all in all we have that $|mathbb{R}^mathbb{R}| = |P(mathbb{R})| = |2^mathbb{R}|$.
2
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
2
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
1
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
2
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
|
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3
down vote
This answer is based on, but differs slightly from, user Asaf Karaglia's above.
First, observe that by definition, ${text{all real functions of real variable}}:= {f: ; f: mathbb{R}tomathbb{R}} := mathbb{R}^mathbb{R}$.
The question is about $|{text{all real functions of real variable}}|$, so examine an arbitrary real function of real variable: $f,colon,mathbb{R}tomathbb{R}.$
By inspection, $f,colon,mathbb{R}tomathbb{R} := {(r, f(r)) : r in mathbb{R}} quad subseteq quad P(mathbb{R} times mathbb{R})$.
Thus, $color{green}{|mathbb{R}^{mathbb{R}}| le |P(mathbb{R}timesmathbb{R})|}$.
Before continuing, let's try to simplify $|P(mathbb{R}timesmathbb{R})|$. Observe that $|mathbb{R}| = |mathbb{R}^k| , forall , k in mathbb{N}$. Its proof by mathematical induction requires the induction hypothesis of $|mathbb{R}| = |mathbb{R}^2|$, one proof of which is : $|mathbb{N}| = |mathbb{N}timesmathbb{N}| implies |mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$.
Verily, $mathbb{R} neq mathbb{R}^2$. Howbeit, for infinite sets $A,B$: $|A| = |B| Longrightarrow require{cancel} cancel{Longleftarrow} |P(A)| = |P(B)|$.
(The converse is discussed here.)
Thus, $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})| implies color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$. Now scrutinise $|P(mathbb{R})|$:
● $color{#A9057D}{|P(mathbb{R})| = |2^{mathbb{R}}|}$, where $2^{mathbb{R}} := {f : ; f: mathbb{R} to {0,1}}$,
● Every $f: mathbb{R} to {0,1}$ is a particular case of a function from $mathbb{R}$ to $mathbb{R}$, thus $color{#EC5021}{2^{mathbb{R}} subsetneq mathbb{R}^mathbb{R}}$.
Altogether, $color{#A9057D}{|P(mathbb{R})| =} color{#EC5021}{|2^mathbb{R}| le} color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$
$implies |P(mathbb{R})| qquad qquad quad leq |mathbb{R}^mathbb{R}| leq |P(mathbb{R})| implies color{#A9057D}{underbrace{|P(mathbb{R})|}_{= |2^mathbb{R}|}} = |mathbb{R}^mathbb{R}| $.
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
4
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
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2
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This is irrelevent here, still it is 'relevent'. The cardinality of set of all continuous function from $mathbb{R}$ to $mathbb{R}$ $(C(mathbb{R},mathbb{R}))$ is $2 ^ mathbb{N_0} = mathfrak{c}$ because any such function is determined by its value on rationals. hence #$(C(mathbb{R},mathbb{R}))$ = # $mathbb{R}^mathbb{Q}$ which has cardinality $2^mathbb{N_0}$.
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
37
down vote
accepted
All you need is a few basics of cardinal arithmetic: if $kappa$ and $lambda$ are cardinals, none of them zero, and at least one of them is infinite, then $kappa+lambda = kappalambda = max{kappa,lambda}$. And cardinal exponentiation satisfies some of the same laws as regular exponentiation; in particular, $(kappa^{lambda})^{nu} = kappa^{lambdanu}$.
The cardinality of the set of all real functions is then
$$|mathbb{R}|^{|mathbb{R}|} =mathfrak{c}^{mathfrak{c}} = (2^{aleph_0})^{2^{aleph_0}} = 2^{aleph_02^{aleph_0}} = 2^{2^{aleph_0}} = 2^{mathfrak{c}}.$$
In other words, it is equal to the cardinality of the power set of $mathbb{R}$.
With a few extra facts, you can get more. In general, if $kappa$ is an infinite cardinal, and $2leqlambdaleqkappa$, then $lambda^{kappa}=2^{kappa}$. This follows because:
$$2^{kappa} leq lambda^{kappa} leq (2^{lambda})^{kappa} = 2^{lambdakappa} = 2^{kappa},$$
so you get equality throughout. The extra information you need for this is to know that if $kappa$, $lambda$, and $nu$ are nonzero cardinals, $kappaleqlambda$, then $kappa^{nu}leq lambda^{nu}$.
In particular, for any infinite cardinal $kappa$ you have $kappa^{kappa} = 2^{kappa}$.
add a comment |
up vote
37
down vote
accepted
All you need is a few basics of cardinal arithmetic: if $kappa$ and $lambda$ are cardinals, none of them zero, and at least one of them is infinite, then $kappa+lambda = kappalambda = max{kappa,lambda}$. And cardinal exponentiation satisfies some of the same laws as regular exponentiation; in particular, $(kappa^{lambda})^{nu} = kappa^{lambdanu}$.
The cardinality of the set of all real functions is then
$$|mathbb{R}|^{|mathbb{R}|} =mathfrak{c}^{mathfrak{c}} = (2^{aleph_0})^{2^{aleph_0}} = 2^{aleph_02^{aleph_0}} = 2^{2^{aleph_0}} = 2^{mathfrak{c}}.$$
In other words, it is equal to the cardinality of the power set of $mathbb{R}$.
With a few extra facts, you can get more. In general, if $kappa$ is an infinite cardinal, and $2leqlambdaleqkappa$, then $lambda^{kappa}=2^{kappa}$. This follows because:
$$2^{kappa} leq lambda^{kappa} leq (2^{lambda})^{kappa} = 2^{lambdakappa} = 2^{kappa},$$
so you get equality throughout. The extra information you need for this is to know that if $kappa$, $lambda$, and $nu$ are nonzero cardinals, $kappaleqlambda$, then $kappa^{nu}leq lambda^{nu}$.
In particular, for any infinite cardinal $kappa$ you have $kappa^{kappa} = 2^{kappa}$.
add a comment |
up vote
37
down vote
accepted
up vote
37
down vote
accepted
All you need is a few basics of cardinal arithmetic: if $kappa$ and $lambda$ are cardinals, none of them zero, and at least one of them is infinite, then $kappa+lambda = kappalambda = max{kappa,lambda}$. And cardinal exponentiation satisfies some of the same laws as regular exponentiation; in particular, $(kappa^{lambda})^{nu} = kappa^{lambdanu}$.
The cardinality of the set of all real functions is then
$$|mathbb{R}|^{|mathbb{R}|} =mathfrak{c}^{mathfrak{c}} = (2^{aleph_0})^{2^{aleph_0}} = 2^{aleph_02^{aleph_0}} = 2^{2^{aleph_0}} = 2^{mathfrak{c}}.$$
In other words, it is equal to the cardinality of the power set of $mathbb{R}$.
With a few extra facts, you can get more. In general, if $kappa$ is an infinite cardinal, and $2leqlambdaleqkappa$, then $lambda^{kappa}=2^{kappa}$. This follows because:
$$2^{kappa} leq lambda^{kappa} leq (2^{lambda})^{kappa} = 2^{lambdakappa} = 2^{kappa},$$
so you get equality throughout. The extra information you need for this is to know that if $kappa$, $lambda$, and $nu$ are nonzero cardinals, $kappaleqlambda$, then $kappa^{nu}leq lambda^{nu}$.
In particular, for any infinite cardinal $kappa$ you have $kappa^{kappa} = 2^{kappa}$.
All you need is a few basics of cardinal arithmetic: if $kappa$ and $lambda$ are cardinals, none of them zero, and at least one of them is infinite, then $kappa+lambda = kappalambda = max{kappa,lambda}$. And cardinal exponentiation satisfies some of the same laws as regular exponentiation; in particular, $(kappa^{lambda})^{nu} = kappa^{lambdanu}$.
The cardinality of the set of all real functions is then
$$|mathbb{R}|^{|mathbb{R}|} =mathfrak{c}^{mathfrak{c}} = (2^{aleph_0})^{2^{aleph_0}} = 2^{aleph_02^{aleph_0}} = 2^{2^{aleph_0}} = 2^{mathfrak{c}}.$$
In other words, it is equal to the cardinality of the power set of $mathbb{R}$.
With a few extra facts, you can get more. In general, if $kappa$ is an infinite cardinal, and $2leqlambdaleqkappa$, then $lambda^{kappa}=2^{kappa}$. This follows because:
$$2^{kappa} leq lambda^{kappa} leq (2^{lambda})^{kappa} = 2^{lambdakappa} = 2^{kappa},$$
so you get equality throughout. The extra information you need for this is to know that if $kappa$, $lambda$, and $nu$ are nonzero cardinals, $kappaleqlambda$, then $kappa^{nu}leq lambda^{nu}$.
In particular, for any infinite cardinal $kappa$ you have $kappa^{kappa} = 2^{kappa}$.
edited Jan 17 '11 at 23:43
answered Jan 17 '11 at 23:32
Arturo Magidin
259k32581901
259k32581901
add a comment |
add a comment |
up vote
29
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I guess that you know that $|mathbb{N}| = |mathbb{N}timesmathbb{N}|$ and thus $|mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$
This means that $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$. Since $fcolonmathbb{R}tomathbb{R}$ is an element of $P(mathbb{R}timesmathbb{R})$ you have that $mathbb{R}^mathbb{R}$ (all the functions from $mathbb{R}$ to itself) is of cardinality less or equal to the one of $P(mathbb{R}timesmathbb{R})$ which in turn means that $|mathbb{R}^mathbb{R}|le |P(mathbb{R})|$.
Now, since $|P(mathbb{R})| = |2^mathbb{R}|$ which is the set of all functions from $mathbb{R}$ to ${0,1}$, and clearly every function from $mathbb{R}$ into ${0,1}$ is in particular a function from $mathbb{R}$ into itself, we have:
$$|P(mathbb{R})| = |2^mathbb{R}| le |mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})| = |P(mathbb{R})|$$
So all in all we have that $|mathbb{R}^mathbb{R}| = |P(mathbb{R})| = |2^mathbb{R}|$.
2
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
2
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
1
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
2
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
|
show 2 more comments
up vote
29
down vote
I guess that you know that $|mathbb{N}| = |mathbb{N}timesmathbb{N}|$ and thus $|mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$
This means that $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$. Since $fcolonmathbb{R}tomathbb{R}$ is an element of $P(mathbb{R}timesmathbb{R})$ you have that $mathbb{R}^mathbb{R}$ (all the functions from $mathbb{R}$ to itself) is of cardinality less or equal to the one of $P(mathbb{R}timesmathbb{R})$ which in turn means that $|mathbb{R}^mathbb{R}|le |P(mathbb{R})|$.
Now, since $|P(mathbb{R})| = |2^mathbb{R}|$ which is the set of all functions from $mathbb{R}$ to ${0,1}$, and clearly every function from $mathbb{R}$ into ${0,1}$ is in particular a function from $mathbb{R}$ into itself, we have:
$$|P(mathbb{R})| = |2^mathbb{R}| le |mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})| = |P(mathbb{R})|$$
So all in all we have that $|mathbb{R}^mathbb{R}| = |P(mathbb{R})| = |2^mathbb{R}|$.
2
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
2
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
1
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
2
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
|
show 2 more comments
up vote
29
down vote
up vote
29
down vote
I guess that you know that $|mathbb{N}| = |mathbb{N}timesmathbb{N}|$ and thus $|mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$
This means that $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$. Since $fcolonmathbb{R}tomathbb{R}$ is an element of $P(mathbb{R}timesmathbb{R})$ you have that $mathbb{R}^mathbb{R}$ (all the functions from $mathbb{R}$ to itself) is of cardinality less or equal to the one of $P(mathbb{R}timesmathbb{R})$ which in turn means that $|mathbb{R}^mathbb{R}|le |P(mathbb{R})|$.
Now, since $|P(mathbb{R})| = |2^mathbb{R}|$ which is the set of all functions from $mathbb{R}$ to ${0,1}$, and clearly every function from $mathbb{R}$ into ${0,1}$ is in particular a function from $mathbb{R}$ into itself, we have:
$$|P(mathbb{R})| = |2^mathbb{R}| le |mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})| = |P(mathbb{R})|$$
So all in all we have that $|mathbb{R}^mathbb{R}| = |P(mathbb{R})| = |2^mathbb{R}|$.
I guess that you know that $|mathbb{N}| = |mathbb{N}timesmathbb{N}|$ and thus $|mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$
This means that $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$. Since $fcolonmathbb{R}tomathbb{R}$ is an element of $P(mathbb{R}timesmathbb{R})$ you have that $mathbb{R}^mathbb{R}$ (all the functions from $mathbb{R}$ to itself) is of cardinality less or equal to the one of $P(mathbb{R}timesmathbb{R})$ which in turn means that $|mathbb{R}^mathbb{R}|le |P(mathbb{R})|$.
Now, since $|P(mathbb{R})| = |2^mathbb{R}|$ which is the set of all functions from $mathbb{R}$ to ${0,1}$, and clearly every function from $mathbb{R}$ into ${0,1}$ is in particular a function from $mathbb{R}$ into itself, we have:
$$|P(mathbb{R})| = |2^mathbb{R}| le |mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})| = |P(mathbb{R})|$$
So all in all we have that $|mathbb{R}^mathbb{R}| = |P(mathbb{R})| = |2^mathbb{R}|$.
edited Nov 7 '13 at 5:35
answered Jan 17 '11 at 23:45
Asaf Karagila♦
300k32421751
300k32421751
2
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
2
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
1
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
2
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
|
show 2 more comments
2
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
2
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
1
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
2
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
2
2
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
Did you mean $|P(mathbb{R})|$ at the end of your second paragraph?
– Arturo Magidin
Jan 17 '11 at 23:51
2
2
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
@AsafKaraglia: Could you please detail how and why $|mathbb{R}| = |mathbb{R}timesmathbb{R}|$ and $ mathbb{R} neq mathbb{R}timesmathbb{R} $ $Longrightarrow |P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})|$? I made an incidental edits which I hope will help and referenced math.stackexchange.com/questions/29366/….
– Greek - Area 51 Proposal
Nov 7 '13 at 1:10
1
1
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
@LePressentiment: Don't add color to my posts. Thank you.
– Asaf Karagila♦
Nov 7 '13 at 5:36
2
2
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@AsafKaragila: No problem at all. I'll post my edition separately below. In your previous version (rollback) above, you write that $mathbb{R}^mathbb{R}$ = all the functions from $mathbb{R}$ to itself. Should this be the set of all such functions? Also, will you please to let me know of my previous comment, preceding your comment?
– Greek - Area 51 Proposal
Nov 7 '13 at 9:48
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
@LePressentiment: I don't know what you mean by that. $|A|=|B|$ implies that $|mathcal P(A)|=|mathcal P(B)|$. That's a simple exercise in the definition of cardinalities. And yes $A^B$ is the set of all functions from $B$ into $A$, although it is sometimes denoted by ${}^BA$.
– Asaf Karagila♦
Nov 7 '13 at 10:02
|
show 2 more comments
up vote
3
down vote
This answer is based on, but differs slightly from, user Asaf Karaglia's above.
First, observe that by definition, ${text{all real functions of real variable}}:= {f: ; f: mathbb{R}tomathbb{R}} := mathbb{R}^mathbb{R}$.
The question is about $|{text{all real functions of real variable}}|$, so examine an arbitrary real function of real variable: $f,colon,mathbb{R}tomathbb{R}.$
By inspection, $f,colon,mathbb{R}tomathbb{R} := {(r, f(r)) : r in mathbb{R}} quad subseteq quad P(mathbb{R} times mathbb{R})$.
Thus, $color{green}{|mathbb{R}^{mathbb{R}}| le |P(mathbb{R}timesmathbb{R})|}$.
Before continuing, let's try to simplify $|P(mathbb{R}timesmathbb{R})|$. Observe that $|mathbb{R}| = |mathbb{R}^k| , forall , k in mathbb{N}$. Its proof by mathematical induction requires the induction hypothesis of $|mathbb{R}| = |mathbb{R}^2|$, one proof of which is : $|mathbb{N}| = |mathbb{N}timesmathbb{N}| implies |mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$.
Verily, $mathbb{R} neq mathbb{R}^2$. Howbeit, for infinite sets $A,B$: $|A| = |B| Longrightarrow require{cancel} cancel{Longleftarrow} |P(A)| = |P(B)|$.
(The converse is discussed here.)
Thus, $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})| implies color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$. Now scrutinise $|P(mathbb{R})|$:
● $color{#A9057D}{|P(mathbb{R})| = |2^{mathbb{R}}|}$, where $2^{mathbb{R}} := {f : ; f: mathbb{R} to {0,1}}$,
● Every $f: mathbb{R} to {0,1}$ is a particular case of a function from $mathbb{R}$ to $mathbb{R}$, thus $color{#EC5021}{2^{mathbb{R}} subsetneq mathbb{R}^mathbb{R}}$.
Altogether, $color{#A9057D}{|P(mathbb{R})| =} color{#EC5021}{|2^mathbb{R}| le} color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$
$implies |P(mathbb{R})| qquad qquad quad leq |mathbb{R}^mathbb{R}| leq |P(mathbb{R})| implies color{#A9057D}{underbrace{|P(mathbb{R})|}_{= |2^mathbb{R}|}} = |mathbb{R}^mathbb{R}| $.
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
4
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
add a comment |
up vote
3
down vote
This answer is based on, but differs slightly from, user Asaf Karaglia's above.
First, observe that by definition, ${text{all real functions of real variable}}:= {f: ; f: mathbb{R}tomathbb{R}} := mathbb{R}^mathbb{R}$.
The question is about $|{text{all real functions of real variable}}|$, so examine an arbitrary real function of real variable: $f,colon,mathbb{R}tomathbb{R}.$
By inspection, $f,colon,mathbb{R}tomathbb{R} := {(r, f(r)) : r in mathbb{R}} quad subseteq quad P(mathbb{R} times mathbb{R})$.
Thus, $color{green}{|mathbb{R}^{mathbb{R}}| le |P(mathbb{R}timesmathbb{R})|}$.
Before continuing, let's try to simplify $|P(mathbb{R}timesmathbb{R})|$. Observe that $|mathbb{R}| = |mathbb{R}^k| , forall , k in mathbb{N}$. Its proof by mathematical induction requires the induction hypothesis of $|mathbb{R}| = |mathbb{R}^2|$, one proof of which is : $|mathbb{N}| = |mathbb{N}timesmathbb{N}| implies |mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$.
Verily, $mathbb{R} neq mathbb{R}^2$. Howbeit, for infinite sets $A,B$: $|A| = |B| Longrightarrow require{cancel} cancel{Longleftarrow} |P(A)| = |P(B)|$.
(The converse is discussed here.)
Thus, $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})| implies color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$. Now scrutinise $|P(mathbb{R})|$:
● $color{#A9057D}{|P(mathbb{R})| = |2^{mathbb{R}}|}$, where $2^{mathbb{R}} := {f : ; f: mathbb{R} to {0,1}}$,
● Every $f: mathbb{R} to {0,1}$ is a particular case of a function from $mathbb{R}$ to $mathbb{R}$, thus $color{#EC5021}{2^{mathbb{R}} subsetneq mathbb{R}^mathbb{R}}$.
Altogether, $color{#A9057D}{|P(mathbb{R})| =} color{#EC5021}{|2^mathbb{R}| le} color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$
$implies |P(mathbb{R})| qquad qquad quad leq |mathbb{R}^mathbb{R}| leq |P(mathbb{R})| implies color{#A9057D}{underbrace{|P(mathbb{R})|}_{= |2^mathbb{R}|}} = |mathbb{R}^mathbb{R}| $.
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
4
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
add a comment |
up vote
3
down vote
up vote
3
down vote
This answer is based on, but differs slightly from, user Asaf Karaglia's above.
First, observe that by definition, ${text{all real functions of real variable}}:= {f: ; f: mathbb{R}tomathbb{R}} := mathbb{R}^mathbb{R}$.
The question is about $|{text{all real functions of real variable}}|$, so examine an arbitrary real function of real variable: $f,colon,mathbb{R}tomathbb{R}.$
By inspection, $f,colon,mathbb{R}tomathbb{R} := {(r, f(r)) : r in mathbb{R}} quad subseteq quad P(mathbb{R} times mathbb{R})$.
Thus, $color{green}{|mathbb{R}^{mathbb{R}}| le |P(mathbb{R}timesmathbb{R})|}$.
Before continuing, let's try to simplify $|P(mathbb{R}timesmathbb{R})|$. Observe that $|mathbb{R}| = |mathbb{R}^k| , forall , k in mathbb{N}$. Its proof by mathematical induction requires the induction hypothesis of $|mathbb{R}| = |mathbb{R}^2|$, one proof of which is : $|mathbb{N}| = |mathbb{N}timesmathbb{N}| implies |mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$.
Verily, $mathbb{R} neq mathbb{R}^2$. Howbeit, for infinite sets $A,B$: $|A| = |B| Longrightarrow require{cancel} cancel{Longleftarrow} |P(A)| = |P(B)|$.
(The converse is discussed here.)
Thus, $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})| implies color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$. Now scrutinise $|P(mathbb{R})|$:
● $color{#A9057D}{|P(mathbb{R})| = |2^{mathbb{R}}|}$, where $2^{mathbb{R}} := {f : ; f: mathbb{R} to {0,1}}$,
● Every $f: mathbb{R} to {0,1}$ is a particular case of a function from $mathbb{R}$ to $mathbb{R}$, thus $color{#EC5021}{2^{mathbb{R}} subsetneq mathbb{R}^mathbb{R}}$.
Altogether, $color{#A9057D}{|P(mathbb{R})| =} color{#EC5021}{|2^mathbb{R}| le} color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$
$implies |P(mathbb{R})| qquad qquad quad leq |mathbb{R}^mathbb{R}| leq |P(mathbb{R})| implies color{#A9057D}{underbrace{|P(mathbb{R})|}_{= |2^mathbb{R}|}} = |mathbb{R}^mathbb{R}| $.
This answer is based on, but differs slightly from, user Asaf Karaglia's above.
First, observe that by definition, ${text{all real functions of real variable}}:= {f: ; f: mathbb{R}tomathbb{R}} := mathbb{R}^mathbb{R}$.
The question is about $|{text{all real functions of real variable}}|$, so examine an arbitrary real function of real variable: $f,colon,mathbb{R}tomathbb{R}.$
By inspection, $f,colon,mathbb{R}tomathbb{R} := {(r, f(r)) : r in mathbb{R}} quad subseteq quad P(mathbb{R} times mathbb{R})$.
Thus, $color{green}{|mathbb{R}^{mathbb{R}}| le |P(mathbb{R}timesmathbb{R})|}$.
Before continuing, let's try to simplify $|P(mathbb{R}timesmathbb{R})|$. Observe that $|mathbb{R}| = |mathbb{R}^k| , forall , k in mathbb{N}$. Its proof by mathematical induction requires the induction hypothesis of $|mathbb{R}| = |mathbb{R}^2|$, one proof of which is : $|mathbb{N}| = |mathbb{N}timesmathbb{N}| implies |mathbb{R}| = |2^{mathbb{N}}| = |2^{mathbb{N}timesmathbb{N}}| = |2^mathbb{N}times 2^mathbb{N}| = |mathbb{R}timesmathbb{R}|$.
Verily, $mathbb{R} neq mathbb{R}^2$. Howbeit, for infinite sets $A,B$: $|A| = |B| Longrightarrow require{cancel} cancel{Longleftarrow} |P(A)| = |P(B)|$.
(The converse is discussed here.)
Thus, $|P(mathbb{R})| = |P(mathbb{R}timesmathbb{R})| implies color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$. Now scrutinise $|P(mathbb{R})|$:
● $color{#A9057D}{|P(mathbb{R})| = |2^{mathbb{R}}|}$, where $2^{mathbb{R}} := {f : ; f: mathbb{R} to {0,1}}$,
● Every $f: mathbb{R} to {0,1}$ is a particular case of a function from $mathbb{R}$ to $mathbb{R}$, thus $color{#EC5021}{2^{mathbb{R}} subsetneq mathbb{R}^mathbb{R}}$.
Altogether, $color{#A9057D}{|P(mathbb{R})| =} color{#EC5021}{|2^mathbb{R}| le} color{green}{|mathbb{R}^mathbb{R}| le |P(mathbb{R}timesmathbb{R})|} = |P(mathbb{R})|$
$implies |P(mathbb{R})| qquad qquad quad leq |mathbb{R}^mathbb{R}| leq |P(mathbb{R})| implies color{#A9057D}{underbrace{|P(mathbb{R})|}_{= |2^mathbb{R}|}} = |mathbb{R}^mathbb{R}| $.
edited Apr 13 '17 at 12:19
Community♦
1
1
answered Nov 7 '13 at 9:49
Greek - Area 51 Proposal
3,100668103
3,100668103
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
4
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
add a comment |
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
4
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
Downvoters, pursuant to my edit, please let me know of further sugggestions which would be more instructive than a mere downvote.
– Greek - Area 51 Proposal
Nov 8 '13 at 7:35
4
4
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
Use less colors, so people with disabilities could read this without getting a headache.
– Asaf Karagila♦
Nov 8 '13 at 8:31
add a comment |
up vote
2
down vote
This is irrelevent here, still it is 'relevent'. The cardinality of set of all continuous function from $mathbb{R}$ to $mathbb{R}$ $(C(mathbb{R},mathbb{R}))$ is $2 ^ mathbb{N_0} = mathfrak{c}$ because any such function is determined by its value on rationals. hence #$(C(mathbb{R},mathbb{R}))$ = # $mathbb{R}^mathbb{Q}$ which has cardinality $2^mathbb{N_0}$.
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
add a comment |
up vote
2
down vote
This is irrelevent here, still it is 'relevent'. The cardinality of set of all continuous function from $mathbb{R}$ to $mathbb{R}$ $(C(mathbb{R},mathbb{R}))$ is $2 ^ mathbb{N_0} = mathfrak{c}$ because any such function is determined by its value on rationals. hence #$(C(mathbb{R},mathbb{R}))$ = # $mathbb{R}^mathbb{Q}$ which has cardinality $2^mathbb{N_0}$.
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
add a comment |
up vote
2
down vote
up vote
2
down vote
This is irrelevent here, still it is 'relevent'. The cardinality of set of all continuous function from $mathbb{R}$ to $mathbb{R}$ $(C(mathbb{R},mathbb{R}))$ is $2 ^ mathbb{N_0} = mathfrak{c}$ because any such function is determined by its value on rationals. hence #$(C(mathbb{R},mathbb{R}))$ = # $mathbb{R}^mathbb{Q}$ which has cardinality $2^mathbb{N_0}$.
This is irrelevent here, still it is 'relevent'. The cardinality of set of all continuous function from $mathbb{R}$ to $mathbb{R}$ $(C(mathbb{R},mathbb{R}))$ is $2 ^ mathbb{N_0} = mathfrak{c}$ because any such function is determined by its value on rationals. hence #$(C(mathbb{R},mathbb{R}))$ = # $mathbb{R}^mathbb{Q}$ which has cardinality $2^mathbb{N_0}$.
edited Apr 29 '17 at 22:03
loved.by.Jesus
1319
1319
answered Nov 7 '13 at 6:14
GA316
2,6401132
2,6401132
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
add a comment |
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
Yes, it's relevant, because it sets a lower bound on the cardinality of the question, i.e., $|mathbb{R}^{mathbb{R}}| geq |mathbb{R}| = mathfrak{c}$. Moreover, it is a nice, easy-to-understand, result :)
– loved.by.Jesus
Apr 29 '17 at 21:48
add a comment |
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You have used the assumption that the Cardinality of power set of R is equal to the Cardinality of functions from real to {0,1}. How to we prove that as well?
– user39246
Sep 2 '12 at 10:59
@krishnanem: Please ask new questions in a new thread, rather than as an answer to a previous question. Furthermore, this question was asked and answered several times before.
– Asaf Karagila♦
Sep 2 '12 at 11:17
@AsafKaragila: I have moved this non-answer to a comment so that krishnanem can pose it as a new question and can read your comment.
– robjohn♦
Sep 2 '12 at 20:39
@robjohn: I think Michael's comment would have been a more suitable choice over mine.
– Asaf Karagila♦
Sep 2 '12 at 21:44
@AsafKaragila: The point is to have krishnanem post the comment as a new question. Your comment solely addresses that point. Michael's also addresses the misplaced question and might encourage a reply.
– robjohn♦
Sep 3 '12 at 3:18