Let $a in A and U in C such that a in U $. Prove that $[a] = U $.












0














If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition







share|cite|improve this question




















  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53
















0














If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition







share|cite|improve this question




















  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53














0












0








0







If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition







share|cite|improve this question















If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition




equivalence-relations set-partition






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 30 '18 at 3:05









Graham Kemp

84.7k43378




84.7k43378










asked Nov 30 '18 at 1:43









ClaireClaire

556




556








  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53














  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53








1




1




First, what is $[a]$?
– Graham Kemp
Nov 30 '18 at 1:49




First, what is $[a]$?
– Graham Kemp
Nov 30 '18 at 1:49












$[a]$ is an equivalence class @GrahamKemp
– Claire
Nov 30 '18 at 1:50




$[a]$ is an equivalence class @GrahamKemp
– Claire
Nov 30 '18 at 1:50












For what equivalence relation?
– Graham Kemp
Nov 30 '18 at 1:51






For what equivalence relation?
– Graham Kemp
Nov 30 '18 at 1:51














I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
– JMoravitz
Nov 30 '18 at 1:51






I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
– JMoravitz
Nov 30 '18 at 1:51






1




1




Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
– Claire
Nov 30 '18 at 1:53




Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
– Claire
Nov 30 '18 at 1:53










1 Answer
1






active

oldest

votes


















2














This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



Here you are asked to prove if $ain$ the partitioning set $U$ that:



$[a] ={xin A| asim x} = U$.



As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



$[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



That's all.






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',
    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%2f3019507%2flet-a-in-a-and-u-in-c-such-that-a-in-u-prove-that-a-u%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









    2














    This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



    In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



    Here you are asked to prove if $ain$ the partitioning set $U$ that:



    $[a] ={xin A| asim x} = U$.



    As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



    $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



    That's all.






    share|cite|improve this answer


























      2














      This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



      In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



      Here you are asked to prove if $ain$ the partitioning set $U$ that:



      $[a] ={xin A| asim x} = U$.



      As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



      $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



      That's all.






      share|cite|improve this answer
























        2












        2








        2






        This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



        In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



        Here you are asked to prove if $ain$ the partitioning set $U$ that:



        $[a] ={xin A| asim x} = U$.



        As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



        $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



        That's all.






        share|cite|improve this answer












        This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



        In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



        Here you are asked to prove if $ain$ the partitioning set $U$ that:



        $[a] ={xin A| asim x} = U$.



        As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



        $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



        That's all.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 30 '18 at 2:37









        fleabloodfleablood

        68.5k22685




        68.5k22685






























            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%2f3019507%2flet-a-in-a-and-u-in-c-such-that-a-in-u-prove-that-a-u%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