Difference between $[a,b]in mathbb R$ and $[a,b]subset mathbb R$?












1












$begingroup$


What is the difference between the following?



Are they both mathematically correct?



begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}



And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.






real-analysis calculus elementary-set-theory notation







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    $in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
    $endgroup$
    – Yanko
    Jan 12 at 16:05








  • 1




    $begingroup$
    As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
    $endgroup$
    – J.G.
    Jan 12 at 16:10










  • $begingroup$
    @J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
    $endgroup$
    – Henning Makholm
    Jan 12 at 16:29










  • $begingroup$
    @HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX owns.
    $endgroup$
    – J.G.
    Jan 12 at 16:34
















1












$begingroup$


What is the difference between the following?



Are they both mathematically correct?



begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}



And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.






real-analysis calculus elementary-set-theory notation







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    $in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
    $endgroup$
    – Yanko
    Jan 12 at 16:05








  • 1




    $begingroup$
    As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
    $endgroup$
    – J.G.
    Jan 12 at 16:10










  • $begingroup$
    @J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
    $endgroup$
    – Henning Makholm
    Jan 12 at 16:29










  • $begingroup$
    @HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX owns.
    $endgroup$
    – J.G.
    Jan 12 at 16:34














1












1








1





$begingroup$


What is the difference between the following?



Are they both mathematically correct?



begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}



And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.






real-analysis calculus elementary-set-theory notation







share|cite|improve this question









$endgroup$




What is the difference between the following?



Are they both mathematically correct?



begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}



And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.






real-analysis calculus elementary-set-theory notation




real-analysis calculus elementary-set-theory notation






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 12 at 16:04









JDoeDoeJDoeDoe

7701614




7701614








  • 1




    $begingroup$
    $in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
    $endgroup$
    – Yanko
    Jan 12 at 16:05








  • 1




    $begingroup$
    As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
    $endgroup$
    – J.G.
    Jan 12 at 16:10










  • $begingroup$
    @J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
    $endgroup$
    – Henning Makholm
    Jan 12 at 16:29










  • $begingroup$
    @HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX owns.
    $endgroup$
    – J.G.
    Jan 12 at 16:34














  • 1




    $begingroup$
    $in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
    $endgroup$
    – Yanko
    Jan 12 at 16:05








  • 1




    $begingroup$
    As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
    $endgroup$
    – J.G.
    Jan 12 at 16:10










  • $begingroup$
    @J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
    $endgroup$
    – Henning Makholm
    Jan 12 at 16:29










  • $begingroup$
    @HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX owns.
    $endgroup$
    – J.G.
    Jan 12 at 16:34








1




1




$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05






$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05






1




1




$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10




$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10












$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29




$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29












$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX owns.
$endgroup$
– J.G.
Jan 12 at 16:34




$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX owns.
$endgroup$
– J.G.
Jan 12 at 16:34










2 Answers
2






active

oldest

votes


















4












$begingroup$

Since $[a,b]$ is a set then only second is validate:



$$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$



If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.






share|cite|improve this answer











$endgroup$





















    5












    $begingroup$

    Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.






    share|cite|improve this answer











    $endgroup$













      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
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3071056%2fdifference-between-a-b-in-mathbb-r-and-a-b-subset-mathbb-r%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      Since $[a,b]$ is a set then only second is validate:



      $$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$



      If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.






      share|cite|improve this answer











      $endgroup$


















        4












        $begingroup$

        Since $[a,b]$ is a set then only second is validate:



        $$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$



        If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.






        share|cite|improve this answer











        $endgroup$
















          4












          4








          4





          $begingroup$

          Since $[a,b]$ is a set then only second is validate:



          $$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$



          If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.






          share|cite|improve this answer











          $endgroup$



          Since $[a,b]$ is a set then only second is validate:



          $$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$



          If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 12 at 16:25









          David C. Ullrich

          61.3k43994




          61.3k43994










          answered Jan 12 at 16:05









          Maria MazurMaria Mazur

          46.9k1260120




          46.9k1260120























              5












              $begingroup$

              Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.






              share|cite|improve this answer











              $endgroup$


















                5












                $begingroup$

                Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.






                share|cite|improve this answer











                $endgroup$
















                  5












                  5








                  5





                  $begingroup$

                  Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.






                  share|cite|improve this answer











                  $endgroup$



                  Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Jan 12 at 16:21

























                  answered Jan 12 at 16:08









                  José Carlos SantosJosé Carlos Santos

                  167k22132235




                  167k22132235






























                      draft saved

                      draft discarded




















































                      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.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3071056%2fdifference-between-a-b-in-mathbb-r-and-a-b-subset-mathbb-r%23new-answer', 'question_page');
                      }
                      );

                      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







                      -

                      Popular posts from this blog

                      Human spaceflight

                      Image
                      Human spaceflight From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Space traveler" redirects here. For other uses, see Space traveler (disambiguation). @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}} Apollo 11 crewmember Buzz Aldrin on the Moon, 1969 International Space Station crewmember Tracy Caldwell Dyson views the Earth, 2010 Space Shuttle Discovery heads into space with a crew aboard, STS-121 in 2006 Inside a space suit on the Canadarm, 1993 Human spaceflight (also referred to as crewed spaceflight or manned spaceflight ) is space travel with a crew or passengers aboard the spacecraft. Spacecraft carrying people may be operated directly, by human crew, or it may be e
                      Read more

                      Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

                      Image
                      0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB]
                      Read more

                      File:DeusFollowingSea.jpg

                      Image
                      File:DeusFollowingSea.jpg From Wikipedia, the free encyclopedia Jump to navigation Jump to search File File history File usage No higher resolution available. DeusFollowingSea.jpg ‎ (238 × 211 pixels, file size: 12 KB, MIME type: image/jpeg ) Summary [ edit ] Media data and Non-free use rationale Description Cover art of the Deus CD Following Sea Author or copyright owner Deus Source (WP:NFCC#4) rocksucker.co.uk Date of publication 2012 Use in article (WP:NFCC#7) Following Sea Purpose of use in article (WP:NFCC#8) to serve as the primary means of visual identification at the top of the article dedicated to the work in question. Not replaceable with free media because (WP:NFCC#1) n.a. Minimal use (WP:NFCC#3) Low res image used in infobox of the article on the album Respect for commercial opportunities (WP:NFCC#2) n.a. Fair use Fair use of co
                      Read more