Showing $a setminus (a setminus b) = a cap b$ using only set notation.












0












$begingroup$


I need to prove that $a setminus (a setminus b) = a cap b$ only through set notations.



I have reached the fact that
$a setminus (a setminus b)$ = {x | x $in$ A $land$ x $notin$ (A $setminus$ B}



Which I then simplify using the De Morgan' formula to
{x | x $in$ A $land$ (x $notin$ A $lor$ x $in$ B)}.



It is evident that this is the same as A $cap$ B (by constructing diagrams and truth tables). But how do I get this expression to {x | x $in$ A $land$ x $in$ B}? Please help!






discrete-mathematics elementary-set-theory boolean-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    Use ${ X}$ for ${ X}$.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:50










  • $begingroup$
    Please edit the question accordingly.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:51
















0












$begingroup$


I need to prove that $a setminus (a setminus b) = a cap b$ only through set notations.



I have reached the fact that
$a setminus (a setminus b)$ = {x | x $in$ A $land$ x $notin$ (A $setminus$ B}



Which I then simplify using the De Morgan' formula to
{x | x $in$ A $land$ (x $notin$ A $lor$ x $in$ B)}.



It is evident that this is the same as A $cap$ B (by constructing diagrams and truth tables). But how do I get this expression to {x | x $in$ A $land$ x $in$ B}? Please help!






discrete-mathematics elementary-set-theory boolean-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    Use ${ X}$ for ${ X}$.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:50










  • $begingroup$
    Please edit the question accordingly.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:51














0












0








0





$begingroup$


I need to prove that $a setminus (a setminus b) = a cap b$ only through set notations.



I have reached the fact that
$a setminus (a setminus b)$ = {x | x $in$ A $land$ x $notin$ (A $setminus$ B}



Which I then simplify using the De Morgan' formula to
{x | x $in$ A $land$ (x $notin$ A $lor$ x $in$ B)}.



It is evident that this is the same as A $cap$ B (by constructing diagrams and truth tables). But how do I get this expression to {x | x $in$ A $land$ x $in$ B}? Please help!






discrete-mathematics elementary-set-theory boolean-algebra







share|cite|improve this question











$endgroup$




I need to prove that $a setminus (a setminus b) = a cap b$ only through set notations.



I have reached the fact that
$a setminus (a setminus b)$ = {x | x $in$ A $land$ x $notin$ (A $setminus$ B}



Which I then simplify using the De Morgan' formula to
{x | x $in$ A $land$ (x $notin$ A $lor$ x $in$ B)}.



It is evident that this is the same as A $cap$ B (by constructing diagrams and truth tables). But how do I get this expression to {x | x $in$ A $land$ x $in$ B}? Please help!






discrete-mathematics elementary-set-theory boolean-algebra




discrete-mathematics elementary-set-theory boolean-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 8 '18 at 10:49









Shaun

9,442113684




9,442113684










asked Dec 8 '18 at 10:25









Leon VladimirovLeon Vladimirov

134




134












  • $begingroup$
    Use ${ X}$ for ${ X}$.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:50










  • $begingroup$
    Please edit the question accordingly.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:51


















  • $begingroup$
    Use ${ X}$ for ${ X}$.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:50










  • $begingroup$
    Please edit the question accordingly.
    $endgroup$
    – Shaun
    Dec 8 '18 at 10:51
















$begingroup$
Use ${ X}$ for ${ X}$.
$endgroup$
– Shaun
Dec 8 '18 at 10:50




$begingroup$
Use ${ X}$ for ${ X}$.
$endgroup$
– Shaun
Dec 8 '18 at 10:50












$begingroup$
Please edit the question accordingly.
$endgroup$
– Shaun
Dec 8 '18 at 10:51




$begingroup$
Please edit the question accordingly.
$endgroup$
– Shaun
Dec 8 '18 at 10:51










2 Answers
2






active

oldest

votes


















1












$begingroup$

Equivalent are:




  • $xin Awedge(xnotin Avee xin B)$

  • $(xin Awedge xnotin A)vee(xin Awedge xin B)$

  • $xin Awedge xin B$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
    $endgroup$
    – Leon Vladimirov
    Dec 8 '18 at 10:30










  • $begingroup$
    By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
    $endgroup$
    – drhab
    Dec 8 '18 at 10:33












  • $begingroup$
    Completely missed that! Understood. Thank you!
    $endgroup$
    – Leon Vladimirov
    Dec 8 '18 at 10:36










  • $begingroup$
    You are welcome
    $endgroup$
    – drhab
    Dec 8 '18 at 10:46



















0












$begingroup$

a - (a - b) = a - (a $cap$ b$^c$) =

a - (a$^c$ $cup$ b)$^c$ = a $cap$ (a$^c$ $cup$ b)

= (a $cap$ a$^c$) $cup$ (a $cap$ b) = a $cap$ b






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%2f3030929%2fshowing-a-setminus-a-setminus-b-a-cap-b-using-only-set-notation%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









    1












    $begingroup$

    Equivalent are:




    • $xin Awedge(xnotin Avee xin B)$

    • $(xin Awedge xnotin A)vee(xin Awedge xin B)$

    • $xin Awedge xin B$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:30










    • $begingroup$
      By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
      $endgroup$
      – drhab
      Dec 8 '18 at 10:33












    • $begingroup$
      Completely missed that! Understood. Thank you!
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:36










    • $begingroup$
      You are welcome
      $endgroup$
      – drhab
      Dec 8 '18 at 10:46
















    1












    $begingroup$

    Equivalent are:




    • $xin Awedge(xnotin Avee xin B)$

    • $(xin Awedge xnotin A)vee(xin Awedge xin B)$

    • $xin Awedge xin B$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:30










    • $begingroup$
      By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
      $endgroup$
      – drhab
      Dec 8 '18 at 10:33












    • $begingroup$
      Completely missed that! Understood. Thank you!
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:36










    • $begingroup$
      You are welcome
      $endgroup$
      – drhab
      Dec 8 '18 at 10:46














    1












    1








    1





    $begingroup$

    Equivalent are:




    • $xin Awedge(xnotin Avee xin B)$

    • $(xin Awedge xnotin A)vee(xin Awedge xin B)$

    • $xin Awedge xin B$






    share|cite|improve this answer









    $endgroup$



    Equivalent are:




    • $xin Awedge(xnotin Avee xin B)$

    • $(xin Awedge xnotin A)vee(xin Awedge xin B)$

    • $xin Awedge xin B$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 8 '18 at 10:29









    drhabdrhab

    103k545136




    103k545136












    • $begingroup$
      Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:30










    • $begingroup$
      By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
      $endgroup$
      – drhab
      Dec 8 '18 at 10:33












    • $begingroup$
      Completely missed that! Understood. Thank you!
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:36










    • $begingroup$
      You are welcome
      $endgroup$
      – drhab
      Dec 8 '18 at 10:46


















    • $begingroup$
      Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:30










    • $begingroup$
      By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
      $endgroup$
      – drhab
      Dec 8 '18 at 10:33












    • $begingroup$
      Completely missed that! Understood. Thank you!
      $endgroup$
      – Leon Vladimirov
      Dec 8 '18 at 10:36










    • $begingroup$
      You are welcome
      $endgroup$
      – drhab
      Dec 8 '18 at 10:46
















    $begingroup$
    Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
    $endgroup$
    – Leon Vladimirov
    Dec 8 '18 at 10:30




    $begingroup$
    Thank you for your answer! Could you please explain how do I get from line 1 to line 2?
    $endgroup$
    – Leon Vladimirov
    Dec 8 '18 at 10:30












    $begingroup$
    By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
    $endgroup$
    – drhab
    Dec 8 '18 at 10:33






    $begingroup$
    By applying distributivity of $wedge $ wrt $vee $. Not familiar with that yet? A Boolean algebra is distributive lattice.
    $endgroup$
    – drhab
    Dec 8 '18 at 10:33














    $begingroup$
    Completely missed that! Understood. Thank you!
    $endgroup$
    – Leon Vladimirov
    Dec 8 '18 at 10:36




    $begingroup$
    Completely missed that! Understood. Thank you!
    $endgroup$
    – Leon Vladimirov
    Dec 8 '18 at 10:36












    $begingroup$
    You are welcome
    $endgroup$
    – drhab
    Dec 8 '18 at 10:46




    $begingroup$
    You are welcome
    $endgroup$
    – drhab
    Dec 8 '18 at 10:46











    0












    $begingroup$

    a - (a - b) = a - (a $cap$ b$^c$) =

    a - (a$^c$ $cup$ b)$^c$ = a $cap$ (a$^c$ $cup$ b)

    = (a $cap$ a$^c$) $cup$ (a $cap$ b) = a $cap$ b






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      a - (a - b) = a - (a $cap$ b$^c$) =

      a - (a$^c$ $cup$ b)$^c$ = a $cap$ (a$^c$ $cup$ b)

      = (a $cap$ a$^c$) $cup$ (a $cap$ b) = a $cap$ b






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        a - (a - b) = a - (a $cap$ b$^c$) =

        a - (a$^c$ $cup$ b)$^c$ = a $cap$ (a$^c$ $cup$ b)

        = (a $cap$ a$^c$) $cup$ (a $cap$ b) = a $cap$ b






        share|cite|improve this answer









        $endgroup$



        a - (a - b) = a - (a $cap$ b$^c$) =

        a - (a$^c$ $cup$ b)$^c$ = a $cap$ (a$^c$ $cup$ b)

        = (a $cap$ a$^c$) $cup$ (a $cap$ b) = a $cap$ b







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 8 '18 at 10:41









        William ElliotWilliam Elliot

        8,6922720




        8,6922720






























            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%2f3030929%2fshowing-a-setminus-a-setminus-b-a-cap-b-using-only-set-notation%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

            mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924

            Image
            0 I just uploaded a new plugin to my wordpress site and I got this warning: mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924 Usually if I have a problem, I can find answers on google but this one seems pretty particular and I'm not finding any info on it except for this forum. I'm not terribly techie but if someone can tell me how to read this and how to change it I can follow directions pretty well. Thanks All! PS If I don't have the proper tags here could someone add the appropriate tags? php mysql wordpress share | improve this question asked Nov 16 '18 at 20:41 ...
            Read more

            How to change which sound is reproduced for terminal bell?

            Image
            3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
            Read more

            Can I use Tabulator js library in my java Spring + Thymeleaf project?

            Image
            0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
            Read more