Let $a,b,c$ be real numbers and denote by $textmid(a,b,c)$ the 'value in the middle'.

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite













Let $a,b,c$ be real numbers and denote by mid$(a,b,c)$ the 'value in the middle'.



Show that $textmid(a,b,c) = infsupa,b, supa,c, supb,c$.




Suppose that $a<b<c$. Then, it follows that $infb,c,c= b$, so it works, but I don't know how to formally prove this.




real-analysis




share|cite|improve this question

















  • 1




    The value in the middle? The average value? Or the median?
    – Calvin Khor
    Aug 2 at 7:36






  • 1




    @CalvinKhor The one of the three values which lies between the other two. So, I guess the median would be appropriate.
    – Arthur
    Aug 2 at 7:40











  • That's called the median
    – Calvin Khor
    Aug 2 at 7:40














up vote
2
down vote

favorite













Let $a,b,c$ be real numbers and denote by mid$(a,b,c)$ the 'value in the middle'.



Show that $textmid(a,b,c) = infsupa,b, supa,c, supb,c$.




Suppose that $a<b<c$. Then, it follows that $infb,c,c= b$, so it works, but I don't know how to formally prove this.




real-analysis




share|cite|improve this question

















  • 1




    The value in the middle? The average value? Or the median?
    – Calvin Khor
    Aug 2 at 7:36






  • 1




    @CalvinKhor The one of the three values which lies between the other two. So, I guess the median would be appropriate.
    – Arthur
    Aug 2 at 7:40











  • That's called the median
    – Calvin Khor
    Aug 2 at 7:40












up vote
2
down vote

favorite









up vote
2
down vote

favorite












Let $a,b,c$ be real numbers and denote by mid$(a,b,c)$ the 'value in the middle'.



Show that $textmid(a,b,c) = infsupa,b, supa,c, supb,c$.




Suppose that $a<b<c$. Then, it follows that $infb,c,c= b$, so it works, but I don't know how to formally prove this.




real-analysis




share|cite|improve this question














Let $a,b,c$ be real numbers and denote by mid$(a,b,c)$ the 'value in the middle'.



Show that $textmid(a,b,c) = infsupa,b, supa,c, supb,c$.




Suppose that $a<b<c$. Then, it follows that $infb,c,c= b$, so it works, but I don't know how to formally prove this.





real-analysis





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 2 at 12:18









David G. Stork

7,3302728




7,3302728









asked Aug 2 at 7:26









Sihyun Kim

701210




701210







  • 1




    The value in the middle? The average value? Or the median?
    – Calvin Khor
    Aug 2 at 7:36






  • 1




    @CalvinKhor The one of the three values which lies between the other two. So, I guess the median would be appropriate.
    – Arthur
    Aug 2 at 7:40











  • That's called the median
    – Calvin Khor
    Aug 2 at 7:40












  • 1




    The value in the middle? The average value? Or the median?
    – Calvin Khor
    Aug 2 at 7:36






  • 1




    @CalvinKhor The one of the three values which lies between the other two. So, I guess the median would be appropriate.
    – Arthur
    Aug 2 at 7:40











  • That's called the median
    – Calvin Khor
    Aug 2 at 7:40







1




1




The value in the middle? The average value? Or the median?
– Calvin Khor
Aug 2 at 7:36




The value in the middle? The average value? Or the median?
– Calvin Khor
Aug 2 at 7:36




1




1




@CalvinKhor The one of the three values which lies between the other two. So, I guess the median would be appropriate.
– Arthur
Aug 2 at 7:40





@CalvinKhor The one of the three values which lies between the other two. So, I guess the median would be appropriate.
– Arthur
Aug 2 at 7:40













That's called the median
– Calvin Khor
Aug 2 at 7:40




That's called the median
– Calvin Khor
Aug 2 at 7:40










2 Answers
2






active

oldest

votes

















up vote
4
down vote













Without loss of generality $ale ble c$ so the equation reduces to the trivial $b=infb,,c,,c$.






share|cite|improve this answer




























    up vote
    0
    down vote













    There are (at least) two options that you can use to prove it:



    The first one is likely to be quite tedious. You consider all six possible orders that $a, b, c$ can be in, and prove the equation in each case.



    The second is similar: You still consider a specific ordering of $a, b, c$ and prove the equation under that assumption. However, you don't consider every possible order. Instead, you notice that swapping any two variables in the equation doesn't change the value of either side of the equation. e.g. If you swap the values of $a$ and $b$, then the left hand side becomes
    $$ operatornamemid(b, a, c) = operatornamemid(a, b, c) $$ and the right hand side similarly remains the same. Thus you can reorder $a, b, c$ as you please, and so it is enough to prove the equation for the specific case $a leq b leq c$, which you have already done.






    share|cite|improve this answer





















      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',
      convertImagesToLinks: true,
      noModals: false,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      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%2f2869786%2flet-a-b-c-be-real-numbers-and-denote-by-textmida-b-c-the-value-in-the%23new-answer', 'question_page');

      );

      Post as a guest

















      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote













      Without loss of generality $ale ble c$ so the equation reduces to the trivial $b=infb,,c,,c$.






      share|cite|improve this answer

























        up vote
        4
        down vote













        Without loss of generality $ale ble c$ so the equation reduces to the trivial $b=infb,,c,,c$.






        share|cite|improve this answer























          up vote
          4
          down vote










          up vote
          4
          down vote









          Without loss of generality $ale ble c$ so the equation reduces to the trivial $b=infb,,c,,c$.






          share|cite|improve this answer













          Without loss of generality $ale ble c$ so the equation reduces to the trivial $b=infb,,c,,c$.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Aug 2 at 7:39









          J.G.

          12.9k11423




          12.9k11423




















              up vote
              0
              down vote













              There are (at least) two options that you can use to prove it:



              The first one is likely to be quite tedious. You consider all six possible orders that $a, b, c$ can be in, and prove the equation in each case.



              The second is similar: You still consider a specific ordering of $a, b, c$ and prove the equation under that assumption. However, you don't consider every possible order. Instead, you notice that swapping any two variables in the equation doesn't change the value of either side of the equation. e.g. If you swap the values of $a$ and $b$, then the left hand side becomes
              $$ operatornamemid(b, a, c) = operatornamemid(a, b, c) $$ and the right hand side similarly remains the same. Thus you can reorder $a, b, c$ as you please, and so it is enough to prove the equation for the specific case $a leq b leq c$, which you have already done.






              share|cite|improve this answer

























                up vote
                0
                down vote













                There are (at least) two options that you can use to prove it:



                The first one is likely to be quite tedious. You consider all six possible orders that $a, b, c$ can be in, and prove the equation in each case.



                The second is similar: You still consider a specific ordering of $a, b, c$ and prove the equation under that assumption. However, you don't consider every possible order. Instead, you notice that swapping any two variables in the equation doesn't change the value of either side of the equation. e.g. If you swap the values of $a$ and $b$, then the left hand side becomes
                $$ operatornamemid(b, a, c) = operatornamemid(a, b, c) $$ and the right hand side similarly remains the same. Thus you can reorder $a, b, c$ as you please, and so it is enough to prove the equation for the specific case $a leq b leq c$, which you have already done.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  There are (at least) two options that you can use to prove it:



                  The first one is likely to be quite tedious. You consider all six possible orders that $a, b, c$ can be in, and prove the equation in each case.



                  The second is similar: You still consider a specific ordering of $a, b, c$ and prove the equation under that assumption. However, you don't consider every possible order. Instead, you notice that swapping any two variables in the equation doesn't change the value of either side of the equation. e.g. If you swap the values of $a$ and $b$, then the left hand side becomes
                  $$ operatornamemid(b, a, c) = operatornamemid(a, b, c) $$ and the right hand side similarly remains the same. Thus you can reorder $a, b, c$ as you please, and so it is enough to prove the equation for the specific case $a leq b leq c$, which you have already done.






                  share|cite|improve this answer













                  There are (at least) two options that you can use to prove it:



                  The first one is likely to be quite tedious. You consider all six possible orders that $a, b, c$ can be in, and prove the equation in each case.



                  The second is similar: You still consider a specific ordering of $a, b, c$ and prove the equation under that assumption. However, you don't consider every possible order. Instead, you notice that swapping any two variables in the equation doesn't change the value of either side of the equation. e.g. If you swap the values of $a$ and $b$, then the left hand side becomes
                  $$ operatornamemid(b, a, c) = operatornamemid(a, b, c) $$ and the right hand side similarly remains the same. Thus you can reorder $a, b, c$ as you please, and so it is enough to prove the equation for the specific case $a leq b leq c$, which you have already done.







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Aug 2 at 7:50









                  Dylan

                  5,67231134




                  5,67231134






















                       

                      draft saved


                      draft discarded


























                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function ()
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2869786%2flet-a-b-c-be-real-numbers-and-denote-by-textmida-b-c-the-value-in-the%23new-answer', 'question_page');

                      );

                      Post as a guest
































































                      Comments

                      Post a Comment

                      Popular posts from this blog

                      What is the equation of a 3D cone with generalised tilt?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
                      Read more

                      Relationship between determinant of matrix and determinant of adjoint?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
                      Read more

                      Color the edges and diagonals of a regular polygon

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
                      Read more