Uniqueness of set where subset exactly sums to 1











up vote
1
down vote

favorite












Let $A = {a_1, a_2, ldots, a_I}$ be a set of real numbers, where for all $i in {1,2,ldots,I}$, $0<a_ile1$ and $sum_{j ne i} a_j ge 1$.



I am interested in the set (or sets) $A$ satisfying the following condition:




For every subset $S subset {1,2,ldots,I}$ where $sum_{i in S} a_i ge 1$, there exists a subset $S' subseteq S$ where $sum_{i in S'} a_i = 1$, i.e. that exactly sums to 1.




Is it true that the only set $A$ satisfying this condition is the one where, for all $i in {1,2,ldots,I}$:
$$a_i = a quadlandquad frac{1}{a} text{ is an integer}$$






proof-verification integers







share|cite|improve this question






















  • Could you please give an example of what you're looking for?
    – MJD
    Nov 23 at 14:03










  • @MJD: $A=left{frac{1}{2}, frac{1}{2}, frac{1}{2}right}$ satisfies the condition, while $A=left{frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ doesn't since the sum of $S'=left{frac{1}{3}, frac{1}{2}right}$ does not exactly equal 1.
    – bonna
    Nov 23 at 14:14










  • But for that second example, $A$ doesn't satisfy the $forall iin {1,ldots,I}: sum_{jneq i} a_j geq 1$?
    – Henrik
    Nov 23 at 14:16












  • Sorry. Let the second example be $A=left{frac{1}{3}, frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ and $S'=left{frac{1}{3}, frac{1}{3}, frac{1}{2}right}$
    – bonna
    Nov 23 at 14:20

















up vote
1
down vote

favorite












Let $A = {a_1, a_2, ldots, a_I}$ be a set of real numbers, where for all $i in {1,2,ldots,I}$, $0<a_ile1$ and $sum_{j ne i} a_j ge 1$.



I am interested in the set (or sets) $A$ satisfying the following condition:




For every subset $S subset {1,2,ldots,I}$ where $sum_{i in S} a_i ge 1$, there exists a subset $S' subseteq S$ where $sum_{i in S'} a_i = 1$, i.e. that exactly sums to 1.




Is it true that the only set $A$ satisfying this condition is the one where, for all $i in {1,2,ldots,I}$:
$$a_i = a quadlandquad frac{1}{a} text{ is an integer}$$






proof-verification integers







share|cite|improve this question






















  • Could you please give an example of what you're looking for?
    – MJD
    Nov 23 at 14:03










  • @MJD: $A=left{frac{1}{2}, frac{1}{2}, frac{1}{2}right}$ satisfies the condition, while $A=left{frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ doesn't since the sum of $S'=left{frac{1}{3}, frac{1}{2}right}$ does not exactly equal 1.
    – bonna
    Nov 23 at 14:14










  • But for that second example, $A$ doesn't satisfy the $forall iin {1,ldots,I}: sum_{jneq i} a_j geq 1$?
    – Henrik
    Nov 23 at 14:16












  • Sorry. Let the second example be $A=left{frac{1}{3}, frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ and $S'=left{frac{1}{3}, frac{1}{3}, frac{1}{2}right}$
    – bonna
    Nov 23 at 14:20















up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let $A = {a_1, a_2, ldots, a_I}$ be a set of real numbers, where for all $i in {1,2,ldots,I}$, $0<a_ile1$ and $sum_{j ne i} a_j ge 1$.



I am interested in the set (or sets) $A$ satisfying the following condition:




For every subset $S subset {1,2,ldots,I}$ where $sum_{i in S} a_i ge 1$, there exists a subset $S' subseteq S$ where $sum_{i in S'} a_i = 1$, i.e. that exactly sums to 1.




Is it true that the only set $A$ satisfying this condition is the one where, for all $i in {1,2,ldots,I}$:
$$a_i = a quadlandquad frac{1}{a} text{ is an integer}$$






proof-verification integers







share|cite|improve this question













Let $A = {a_1, a_2, ldots, a_I}$ be a set of real numbers, where for all $i in {1,2,ldots,I}$, $0<a_ile1$ and $sum_{j ne i} a_j ge 1$.



I am interested in the set (or sets) $A$ satisfying the following condition:




For every subset $S subset {1,2,ldots,I}$ where $sum_{i in S} a_i ge 1$, there exists a subset $S' subseteq S$ where $sum_{i in S'} a_i = 1$, i.e. that exactly sums to 1.




Is it true that the only set $A$ satisfying this condition is the one where, for all $i in {1,2,ldots,I}$:
$$a_i = a quadlandquad frac{1}{a} text{ is an integer}$$






proof-verification integers




proof-verification integers






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 23 at 13:58









bonna

858




858












  • Could you please give an example of what you're looking for?
    – MJD
    Nov 23 at 14:03










  • @MJD: $A=left{frac{1}{2}, frac{1}{2}, frac{1}{2}right}$ satisfies the condition, while $A=left{frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ doesn't since the sum of $S'=left{frac{1}{3}, frac{1}{2}right}$ does not exactly equal 1.
    – bonna
    Nov 23 at 14:14










  • But for that second example, $A$ doesn't satisfy the $forall iin {1,ldots,I}: sum_{jneq i} a_j geq 1$?
    – Henrik
    Nov 23 at 14:16












  • Sorry. Let the second example be $A=left{frac{1}{3}, frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ and $S'=left{frac{1}{3}, frac{1}{3}, frac{1}{2}right}$
    – bonna
    Nov 23 at 14:20




















  • Could you please give an example of what you're looking for?
    – MJD
    Nov 23 at 14:03










  • @MJD: $A=left{frac{1}{2}, frac{1}{2}, frac{1}{2}right}$ satisfies the condition, while $A=left{frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ doesn't since the sum of $S'=left{frac{1}{3}, frac{1}{2}right}$ does not exactly equal 1.
    – bonna
    Nov 23 at 14:14










  • But for that second example, $A$ doesn't satisfy the $forall iin {1,ldots,I}: sum_{jneq i} a_j geq 1$?
    – Henrik
    Nov 23 at 14:16












  • Sorry. Let the second example be $A=left{frac{1}{3}, frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ and $S'=left{frac{1}{3}, frac{1}{3}, frac{1}{2}right}$
    – bonna
    Nov 23 at 14:20


















Could you please give an example of what you're looking for?
– MJD
Nov 23 at 14:03




Could you please give an example of what you're looking for?
– MJD
Nov 23 at 14:03












@MJD: $A=left{frac{1}{2}, frac{1}{2}, frac{1}{2}right}$ satisfies the condition, while $A=left{frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ doesn't since the sum of $S'=left{frac{1}{3}, frac{1}{2}right}$ does not exactly equal 1.
– bonna
Nov 23 at 14:14




@MJD: $A=left{frac{1}{2}, frac{1}{2}, frac{1}{2}right}$ satisfies the condition, while $A=left{frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ doesn't since the sum of $S'=left{frac{1}{3}, frac{1}{2}right}$ does not exactly equal 1.
– bonna
Nov 23 at 14:14












But for that second example, $A$ doesn't satisfy the $forall iin {1,ldots,I}: sum_{jneq i} a_j geq 1$?
– Henrik
Nov 23 at 14:16






But for that second example, $A$ doesn't satisfy the $forall iin {1,ldots,I}: sum_{jneq i} a_j geq 1$?
– Henrik
Nov 23 at 14:16














Sorry. Let the second example be $A=left{frac{1}{3}, frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ and $S'=left{frac{1}{3}, frac{1}{3}, frac{1}{2}right}$
– bonna
Nov 23 at 14:20






Sorry. Let the second example be $A=left{frac{1}{3}, frac{1}{3}, frac{1}{2}, frac{1}{2}right}$ and $S'=left{frac{1}{3}, frac{1}{3}, frac{1}{2}right}$
– bonna
Nov 23 at 14:20












1 Answer
1






active

oldest

votes

















up vote
0
down vote



accepted










No, it is not true. Consider the following counter-example where $a_i ne a, forall i$:



$$A = {a_1, a_2, a_3, a_4}, text{ where } a_1 = a_2 = frac{1}{4} text{ and } a_3 = a_4 = frac{1}{2}$$



This satisfies the condition that all the subsets $$S = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}, {a_1, a_3, a_4}, {a_2, a_3, a_4}$$ where $sum_{i in S} a_i ge 1$, have subsets $$S' = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}$$ where $sum_{i in S'} a_i = 1$.






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: 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%2f3010394%2funiqueness-of-set-where-subset-exactly-sums-to-1%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








    up vote
    0
    down vote



    accepted










    No, it is not true. Consider the following counter-example where $a_i ne a, forall i$:



    $$A = {a_1, a_2, a_3, a_4}, text{ where } a_1 = a_2 = frac{1}{4} text{ and } a_3 = a_4 = frac{1}{2}$$



    This satisfies the condition that all the subsets $$S = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}, {a_1, a_3, a_4}, {a_2, a_3, a_4}$$ where $sum_{i in S} a_i ge 1$, have subsets $$S' = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}$$ where $sum_{i in S'} a_i = 1$.






    share|cite|improve this answer

























      up vote
      0
      down vote



      accepted










      No, it is not true. Consider the following counter-example where $a_i ne a, forall i$:



      $$A = {a_1, a_2, a_3, a_4}, text{ where } a_1 = a_2 = frac{1}{4} text{ and } a_3 = a_4 = frac{1}{2}$$



      This satisfies the condition that all the subsets $$S = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}, {a_1, a_3, a_4}, {a_2, a_3, a_4}$$ where $sum_{i in S} a_i ge 1$, have subsets $$S' = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}$$ where $sum_{i in S'} a_i = 1$.






      share|cite|improve this answer























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        No, it is not true. Consider the following counter-example where $a_i ne a, forall i$:



        $$A = {a_1, a_2, a_3, a_4}, text{ where } a_1 = a_2 = frac{1}{4} text{ and } a_3 = a_4 = frac{1}{2}$$



        This satisfies the condition that all the subsets $$S = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}, {a_1, a_3, a_4}, {a_2, a_3, a_4}$$ where $sum_{i in S} a_i ge 1$, have subsets $$S' = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}$$ where $sum_{i in S'} a_i = 1$.






        share|cite|improve this answer












        No, it is not true. Consider the following counter-example where $a_i ne a, forall i$:



        $$A = {a_1, a_2, a_3, a_4}, text{ where } a_1 = a_2 = frac{1}{4} text{ and } a_3 = a_4 = frac{1}{2}$$



        This satisfies the condition that all the subsets $$S = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}, {a_1, a_3, a_4}, {a_2, a_3, a_4}$$ where $sum_{i in S} a_i ge 1$, have subsets $$S' = {a_1, a_2, a_3}, {a_1, a_2, a_4}, {a_3, a_4}$$ where $sum_{i in S'} a_i = 1$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 23 at 15:12









        bonna

        858




        858






























            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.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • 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.


            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%2f3010394%2funiqueness-of-set-where-subset-exactly-sums-to-1%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

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Are they similar matrix

            Image
            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
            Read more