Let $A$ be a finite set. Prove that there exists an $f:n to A$ that is onto $A$ for some $n in omega$ The Next CEO of Stack OverflowProve that f[X] ≼ X iff X is finite.A one-to-one function from a finite set to itself is onto - how to prove by induction?A set is finite, then there exists a bijective map from the set to some natural number?How do I prove that for any set $A$, $|A| < |mathbbN|$ implies that $A$ is finite?Let $n$ and $m$ be two natural numbers. Show that if there is an injection from $n$ into $m$ then $nleq m$.Prove that the set of all finite subsets of $omega$ is denumerableFor given set $A$, show that there exist a set of 'every finite sequence on $A$'.Let f:$A to B$ and $g:B to A$. Prove that…Prove that $p:omega times omega to omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto.Let $n in omega$. Suppose $f:n to A$ is onto $A$. Prove that $A$ is finite.
Can I equip Skullclamp on a creature I am sacrificing?
Written every which way
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
In excess I'm lethal
Rotate a column
Is it my responsibility to learn a new technology in my own time my employer wants to implement?
If/When UK leaves the EU, can a future goverment conduct a referendum to join the EU?
Preparing Indesign booklet with .psd graphics for print
What happens if you roll doubles 3 times then land on "Go to jail?"
sp_blitzCache results Memory grants
Anatomically Correct Strange Women In Ponds Distributing Swords
Why didn't Khan get resurrected in the Genesis Explosion?
Does it take more energy to get to Venus or to Mars?
Why am I allowed to create multiple unique pointers from a single object?
What was the first Unix version to run on a microcomputer?
What is "(CFMCC)" on an ILS approach chart?
How to avoid supervisors with prejudiced views?
Is it possible to search for a directory/file combination?
Solidity! Invalid implicit conversion from string memory to bytes memory requested
What's the best way to handle refactoring a big file?
Interfacing a button to MCU (and PC) with 50m long cable
What is the purpose of the Evocation wizard's Potent Cantrip feature?
Indicator light circuit
How do I go from 300 unfinished/half written blog posts, to published posts?
Let $A$ be a finite set. Prove that there exists an $f:n to A$ that is onto $A$ for some $n in omega$
The Next CEO of Stack OverflowProve that f[X] ≼ X iff X is finite.A one-to-one function from a finite set to itself is onto - how to prove by induction?A set is finite, then there exists a bijective map from the set to some natural number?How do I prove that for any set $A$, $|A| < |mathbbN|$ implies that $A$ is finite?Let $n$ and $m$ be two natural numbers. Show that if there is an injection from $n$ into $m$ then $nleq m$.Prove that the set of all finite subsets of $omega$ is denumerableFor given set $A$, show that there exist a set of 'every finite sequence on $A$'.Let f:$A to B$ and $g:B to A$. Prove that…Prove that $p:omega times omega to omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto.Let $n in omega$. Suppose $f:n to A$ is onto $A$. Prove that $A$ is finite.
$begingroup$
Let $A$ be a finite set. Prove that there exists an $f:n to A$ that is onto $A$ for some $n in omega$.
Here by finite we mean: A set $X$ is finite iff there is a one-to-one function $f:X→n$ for some natural number $n$.
I can see that $A$ being finite implies the existence of some one-to-one function $g:Ato N$ for some $Nin omega$. and thus, its inverse function will be $g^-1:Nto A$, with $g$ onto as needed. But is it this simple? Is it always the case that such an inverse exists, or do we need to prove by induction on $n$?
elementary-set-theory
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
$endgroup$
add a comment |
$begingroup$
Let $A$ be a finite set. Prove that there exists an $f:n to A$ that is onto $A$ for some $n in omega$.
Here by finite we mean: A set $X$ is finite iff there is a one-to-one function $f:X→n$ for some natural number $n$.
I can see that $A$ being finite implies the existence of some one-to-one function $g:Ato N$ for some $Nin omega$. and thus, its inverse function will be $g^-1:Nto A$, with $g$ onto as needed. But is it this simple? Is it always the case that such an inverse exists, or do we need to prove by induction on $n$?
elementary-set-theory
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
$endgroup$
$begingroup$
When you say "$n$" I think you mean "$1, 2, ...., n$." It may also help to define "$omega$" and "natural number" (does it include 0? Is the empty set finite?)
$endgroup$
– Michael
Mar 18 at 22:23
$begingroup$
@Michael it's rather $0,1, dots, n-1$.
$endgroup$
– Berci
Mar 18 at 22:24
$begingroup$
@Berci :?? Is it the case Berci = George W Kush?
$endgroup$
– Michael
Mar 18 at 22:26
1
$begingroup$
With this definition, yes, it's that simple..
$endgroup$
– Berci
Mar 18 at 22:26
$begingroup$
In set theory, natural numbers are inductively encoded as $n=0,1, dots, n-1$, starting with $0:=emptyset, 1:=emptyset$, and so on.
$endgroup$
– Berci
Mar 18 at 22:29
add a comment |
$begingroup$
Let $A$ be a finite set. Prove that there exists an $f:n to A$ that is onto $A$ for some $n in omega$.
Here by finite we mean: A set $X$ is finite iff there is a one-to-one function $f:X→n$ for some natural number $n$.
I can see that $A$ being finite implies the existence of some one-to-one function $g:Ato N$ for some $Nin omega$. and thus, its inverse function will be $g^-1:Nto A$, with $g$ onto as needed. But is it this simple? Is it always the case that such an inverse exists, or do we need to prove by induction on $n$?
elementary-set-theory
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
$endgroup$
Let $A$ be a finite set. Prove that there exists an $f:n to A$ that is onto $A$ for some $n in omega$.
Here by finite we mean: A set $X$ is finite iff there is a one-to-one function $f:X→n$ for some natural number $n$.
I can see that $A$ being finite implies the existence of some one-to-one function $g:Ato N$ for some $Nin omega$. and thus, its inverse function will be $g^-1:Nto A$, with $g$ onto as needed. But is it this simple? Is it always the case that such an inverse exists, or do we need to prove by induction on $n$?
elementary-set-theory
elementary-set-theory
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
asked Mar 18 at 22:13
George W KushGeorge W Kush
256
256
$begingroup$
When you say "$n$" I think you mean "$1, 2, ...., n$." It may also help to define "$omega$" and "natural number" (does it include 0? Is the empty set finite?)
$endgroup$
– Michael
Mar 18 at 22:23
$begingroup$
@Michael it's rather $0,1, dots, n-1$.
$endgroup$
– Berci
Mar 18 at 22:24
$begingroup$
@Berci :?? Is it the case Berci = George W Kush?
$endgroup$
– Michael
Mar 18 at 22:26
1
$begingroup$
With this definition, yes, it's that simple..
$endgroup$
– Berci
Mar 18 at 22:26
$begingroup$
In set theory, natural numbers are inductively encoded as $n=0,1, dots, n-1$, starting with $0:=emptyset, 1:=emptyset$, and so on.
$endgroup$
– Berci
Mar 18 at 22:29
add a comment |
$begingroup$
When you say "$n$" I think you mean "$1, 2, ...., n$." It may also help to define "$omega$" and "natural number" (does it include 0? Is the empty set finite?)
$endgroup$
– Michael
Mar 18 at 22:23
$begingroup$
@Michael it's rather $0,1, dots, n-1$.
$endgroup$
– Berci
Mar 18 at 22:24
$begingroup$
@Berci :?? Is it the case Berci = George W Kush?
$endgroup$
– Michael
Mar 18 at 22:26
1
$begingroup$
With this definition, yes, it's that simple..
$endgroup$
– Berci
Mar 18 at 22:26
$begingroup$
In set theory, natural numbers are inductively encoded as $n=0,1, dots, n-1$, starting with $0:=emptyset, 1:=emptyset$, and so on.
$endgroup$
– Berci
Mar 18 at 22:29
$begingroup$
When you say "$n$" I think you mean "$1, 2, ...., n$." It may also help to define "$omega$" and "natural number" (does it include 0? Is the empty set finite?)
$endgroup$
– Michael
Mar 18 at 22:23
$begingroup$
When you say "$n$" I think you mean "$1, 2, ...., n$." It may also help to define "$omega$" and "natural number" (does it include 0? Is the empty set finite?)
$endgroup$
– Michael
Mar 18 at 22:23
$begingroup$
@Michael it's rather $0,1, dots, n-1$.
$endgroup$
– Berci
Mar 18 at 22:24
$begingroup$
@Michael it's rather $0,1, dots, n-1$.
$endgroup$
– Berci
Mar 18 at 22:24
$begingroup$
@Berci :?? Is it the case Berci = George W Kush?
$endgroup$
– Michael
Mar 18 at 22:26
$begingroup$
@Berci :?? Is it the case Berci = George W Kush?
$endgroup$
– Michael
Mar 18 at 22:26
1
1
$begingroup$
With this definition, yes, it's that simple..
$endgroup$
– Berci
Mar 18 at 22:26
$begingroup$
With this definition, yes, it's that simple..
$endgroup$
– Berci
Mar 18 at 22:26
$begingroup$
In set theory, natural numbers are inductively encoded as $n=0,1, dots, n-1$, starting with $0:=emptyset, 1:=emptyset$, and so on.
$endgroup$
– Berci
Mar 18 at 22:29
$begingroup$
In set theory, natural numbers are inductively encoded as $n=0,1, dots, n-1$, starting with $0:=emptyset, 1:=emptyset$, and so on.
$endgroup$
– Berci
Mar 18 at 22:29
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3153387%2flet-a-be-a-finite-set-prove-that-there-exists-an-fn-to-a-that-is-onto-a%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3153387%2flet-a-be-a-finite-set-prove-that-there-exists-an-fn-to-a-that-is-onto-a%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
When you say "$n$" I think you mean "$1, 2, ...., n$." It may also help to define "$omega$" and "natural number" (does it include 0? Is the empty set finite?)
$endgroup$
– Michael
Mar 18 at 22:23
$begingroup$
@Michael it's rather $0,1, dots, n-1$.
$endgroup$
– Berci
Mar 18 at 22:24
$begingroup$
@Berci :?? Is it the case Berci = George W Kush?
$endgroup$
– Michael
Mar 18 at 22:26
1
$begingroup$
With this definition, yes, it's that simple..
$endgroup$
– Berci
Mar 18 at 22:26
$begingroup$
In set theory, natural numbers are inductively encoded as $n=0,1, dots, n-1$, starting with $0:=emptyset, 1:=emptyset$, and so on.
$endgroup$
– Berci
Mar 18 at 22:29