What do maximal subsets of $mathbbR$ that are closed under addition and multiplication and don't contain additive inverses look like?Linear Span propertiesWhen does solution of $Ax = b$ exists, with $x_i in [0, 1] ~forall i in 1, ldots, N$?A subset that is closed under multiplication but not addition?How to prove a type of functions is a subspace of the vector space of all functions.Closed under vector addition and scalar multiplicationNot subspace, but closed under addition and under taking additive inverses?How do i show that functions are closed under additionA subset that is closed under addition and scalar multiplication but is not a subspaceShowing that set $mathbbS = a + bimid a,binmathbbQ$ is closed under addition and multiplicationWhy we only need to verify additive identity, and closed under addition and scalar multiplication for subspace?
How old can references or sources in a thesis be?
Watching something be written to a file live with tail
Replacing matching entries in one column of a file by another column from a different file
What's that red-plus icon near a text?
What typically incentivizes a professor to change jobs to a lower ranking university?
I'm flying to France today and my passport expires in less than 2 months
Theorems that impeded progress
How is the claim "I am in New York only if I am in America" the same as "If I am in New York, then I am in America?
Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)
Languages that we cannot (dis)prove to be Context-Free
A case of the sniffles
how to check a propriety using r studio
Why do I get two different answers for this counting problem?
Rock identification in KY
Was any UN Security Council vote triple-vetoed?
How to format long polynomial?
If human space travel is limited by the G force vulnerability, is there a way to counter G forces?
Does detail obscure or enhance action?
How much of data wrangling is a data scientist's job?
NMaximize is not converging to a solution
Filter any system log file by date or date range
Is it legal for company to use my work email to pretend I still work there?
What does it mean to describe someone as a butt steak?
Can you really stack all of this on an Opportunity Attack?
What do maximal subsets of $mathbbR$ that are closed under addition and multiplication and don't contain additive inverses look like?
Linear Span propertiesWhen does solution of $Ax = b$ exists, with $x_i in [0, 1] ~forall i in 1, ldots, N$?A subset that is closed under multiplication but not addition?How to prove a type of functions is a subspace of the vector space of all functions.Closed under vector addition and scalar multiplicationNot subspace, but closed under addition and under taking additive inverses?How do i show that functions are closed under additionA subset that is closed under addition and scalar multiplication but is not a subspaceShowing that set $mathbbS = a + bimid a,binmathbbQ$ is closed under addition and multiplicationWhy we only need to verify additive identity, and closed under addition and scalar multiplication for subspace?
$begingroup$
Let's say we have a subset $A subsetmathbbR$ that satisfies the following conditions:
$forall x,y in A: x+y,xy subset A$
$A cup (-A) = mathbbR$
$A cap (-A) = 0$
We know that $ A = [0,infty)$ satisfies these conditions, but is this the only set with these properties?
linear-algebra
asked Mar 29 at 9:48
NicolasNicolas
522
$endgroup$
add a comment |
$begingroup$
Let's say we have a subset $A subsetmathbbR$ that satisfies the following conditions:
$forall x,y in A: x+y,xy subset A$
$A cup (-A) = mathbbR$
$A cap (-A) = 0$
We know that $ A = [0,infty)$ satisfies these conditions, but is this the only set with these properties?
linear-algebra
asked Mar 29 at 9:48
NicolasNicolas
522
$endgroup$
add a comment |
$begingroup$
Let's say we have a subset $A subsetmathbbR$ that satisfies the following conditions:
$forall x,y in A: x+y,xy subset A$
$A cup (-A) = mathbbR$
$A cap (-A) = 0$
We know that $ A = [0,infty)$ satisfies these conditions, but is this the only set with these properties?
linear-algebra
asked Mar 29 at 9:48
NicolasNicolas
522
$endgroup$
Let's say we have a subset $A subsetmathbbR$ that satisfies the following conditions:
$forall x,y in A: x+y,xy subset A$
$A cup (-A) = mathbbR$
$A cap (-A) = 0$
We know that $ A = [0,infty)$ satisfies these conditions, but is this the only set with these properties?
linear-algebra
linear-algebra
asked Mar 29 at 9:48
NicolasNicolas
522
asked Mar 29 at 9:48
NicolasNicolas
522
asked Mar 29 at 9:48
NicolasNicolas
522
asked Mar 29 at 9:48
NicolasNicolas
522
asked Mar 29 at 9:48
NicolasNicolas
522
522
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes, if $xgeq0$ then either $sqrt x$ or $-sqrt x$ is in $A$, and so is its square.
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
$endgroup$
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
add a comment |
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%2f3166953%2fwhat-do-maximal-subsets-of-mathbbr-that-are-closed-under-addition-and-multi%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes, if $xgeq0$ then either $sqrt x$ or $-sqrt x$ is in $A$, and so is its square.
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
$endgroup$
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
add a comment |
$begingroup$
Yes, if $xgeq0$ then either $sqrt x$ or $-sqrt x$ is in $A$, and so is its square.
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
$endgroup$
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
add a comment |
$begingroup$
Yes, if $xgeq0$ then either $sqrt x$ or $-sqrt x$ is in $A$, and so is its square.
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
$endgroup$
Yes, if $xgeq0$ then either $sqrt x$ or $-sqrt x$ is in $A$, and so is its square.
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
answered Mar 29 at 10:23
Chris CulterChris Culter
21.5k43888
21.5k43888
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
add a comment |
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
Why does that matter though? How can I explicitly disprove that there is no subset $A$ that contains numbers like $e$ or $pi$?
$endgroup$
– Nicolas
Mar 29 at 10:28
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
$begingroup$
@Nicholas, let $rinmathbbR_>0$ and suppose that $-rin A$. As $Acup(-A)=mathbbR$ we have that $sqrtrin A$ or $-sqrtrin A$, then $(pmsqrtr)^2=rin A$ which is a contradiction.
$endgroup$
– Floris Claassens
Mar 29 at 10:36
add a comment |
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%2f3166953%2fwhat-do-maximal-subsets-of-mathbbr-that-are-closed-under-addition-and-multi%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
