Is ${ emptyset }$ is a subset of set ${ emptyset, 1, 2, 3 }$?












0












$begingroup$


Is ${ emptyset }$ a subset of set ${ emptyset, 1, 2, 3 }$?
I know that empty set is subset of every set, but what about ${ emptyset }$? What if 'right set' was just ${1, 2, 3}$? Will it still be true that ${ emptyset }$ is a subset of set ${1, 2, 3}$?






discrete-mathematics elementary-set-theory







share|cite|improve this question











$endgroup$












  • $begingroup$
    The set $A = { emptyset, 1, 2, 3 }$ has four elements. One of them is $emptyset$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:30






  • 1




    $begingroup$
    Correct : $emptyset$ is a subset of every set; thus $emptyset subseteq A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:31










  • $begingroup$
    Now, the question is : is ${ emptyset } subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:32












  • $begingroup$
    ${emptyset}subseteq A$ if and only if $emptysetin A.$ (In general $Bsubseteq A$ if and only if every element of $B$ is an element of $A.$)
    $endgroup$
    – spaceisdarkgreen
    Dec 5 '18 at 15:33


















0












$begingroup$


Is ${ emptyset }$ a subset of set ${ emptyset, 1, 2, 3 }$?
I know that empty set is subset of every set, but what about ${ emptyset }$? What if 'right set' was just ${1, 2, 3}$? Will it still be true that ${ emptyset }$ is a subset of set ${1, 2, 3}$?






discrete-mathematics elementary-set-theory







share|cite|improve this question











$endgroup$












  • $begingroup$
    The set $A = { emptyset, 1, 2, 3 }$ has four elements. One of them is $emptyset$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:30






  • 1




    $begingroup$
    Correct : $emptyset$ is a subset of every set; thus $emptyset subseteq A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:31










  • $begingroup$
    Now, the question is : is ${ emptyset } subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:32












  • $begingroup$
    ${emptyset}subseteq A$ if and only if $emptysetin A.$ (In general $Bsubseteq A$ if and only if every element of $B$ is an element of $A.$)
    $endgroup$
    – spaceisdarkgreen
    Dec 5 '18 at 15:33
















0












0








0





$begingroup$


Is ${ emptyset }$ a subset of set ${ emptyset, 1, 2, 3 }$?
I know that empty set is subset of every set, but what about ${ emptyset }$? What if 'right set' was just ${1, 2, 3}$? Will it still be true that ${ emptyset }$ is a subset of set ${1, 2, 3}$?






discrete-mathematics elementary-set-theory







share|cite|improve this question











$endgroup$




Is ${ emptyset }$ a subset of set ${ emptyset, 1, 2, 3 }$?
I know that empty set is subset of every set, but what about ${ emptyset }$? What if 'right set' was just ${1, 2, 3}$? Will it still be true that ${ emptyset }$ is a subset of set ${1, 2, 3}$?






discrete-mathematics elementary-set-theory




discrete-mathematics elementary-set-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 15:37









Asaf Karagila♦

305k33435766




305k33435766










asked Dec 5 '18 at 15:28









Sergey MalinovSergey Malinov

174




174












  • $begingroup$
    The set $A = { emptyset, 1, 2, 3 }$ has four elements. One of them is $emptyset$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:30






  • 1




    $begingroup$
    Correct : $emptyset$ is a subset of every set; thus $emptyset subseteq A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:31










  • $begingroup$
    Now, the question is : is ${ emptyset } subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:32












  • $begingroup$
    ${emptyset}subseteq A$ if and only if $emptysetin A.$ (In general $Bsubseteq A$ if and only if every element of $B$ is an element of $A.$)
    $endgroup$
    – spaceisdarkgreen
    Dec 5 '18 at 15:33




















  • $begingroup$
    The set $A = { emptyset, 1, 2, 3 }$ has four elements. One of them is $emptyset$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:30






  • 1




    $begingroup$
    Correct : $emptyset$ is a subset of every set; thus $emptyset subseteq A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:31










  • $begingroup$
    Now, the question is : is ${ emptyset } subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$.
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 5 '18 at 15:32












  • $begingroup$
    ${emptyset}subseteq A$ if and only if $emptysetin A.$ (In general $Bsubseteq A$ if and only if every element of $B$ is an element of $A.$)
    $endgroup$
    – spaceisdarkgreen
    Dec 5 '18 at 15:33


















$begingroup$
The set $A = { emptyset, 1, 2, 3 }$ has four elements. One of them is $emptyset$.
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:30




$begingroup$
The set $A = { emptyset, 1, 2, 3 }$ has four elements. One of them is $emptyset$.
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:30




1




1




$begingroup$
Correct : $emptyset$ is a subset of every set; thus $emptyset subseteq A$.
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:31




$begingroup$
Correct : $emptyset$ is a subset of every set; thus $emptyset subseteq A$.
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:31












$begingroup$
Now, the question is : is ${ emptyset } subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$.
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:32






$begingroup$
Now, the question is : is ${ emptyset } subseteq A$ ? We have to apply the def of subset ... or lists all the subsetts of $A$.
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:32














$begingroup$
${emptyset}subseteq A$ if and only if $emptysetin A.$ (In general $Bsubseteq A$ if and only if every element of $B$ is an element of $A.$)
$endgroup$
– spaceisdarkgreen
Dec 5 '18 at 15:33






$begingroup$
${emptyset}subseteq A$ if and only if $emptysetin A.$ (In general $Bsubseteq A$ if and only if every element of $B$ is an element of $A.$)
$endgroup$
– spaceisdarkgreen
Dec 5 '18 at 15:33












2 Answers
2






active

oldest

votes


















1












$begingroup$

The set ${x}$ is a subset of ${a,b,c}$ if and only if $x$ is equal to $a$, $b$ or $c$. So, unless you have a weird definition of $1$, $2$ or $3$, ${emptyset}nsubseteq{1,2,3}$. But ${emptyset}subseteq{emptyset, 1,2,3}$



Remark: The most usual construction of $Bbb N$ from ZFC defines $0=emptyset$ and $1={emptyset}$.






share|cite|improve this answer











$endgroup$





















    0












    $begingroup$

    If you are asking about $emptyset$, then it is indeed the subset of any set.



    If you are asking about $E = {emptyset, 1,2,3}$, then $emptyset in E$ and $emptyset subset E$.



    Moreover, if $S = {emptyset}$ then $S subset E$.






    share|cite|improve this answer











    $endgroup$









    • 2




      $begingroup$
      It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
      $endgroup$
      – Henning Makholm
      Dec 5 '18 at 15:33










    • $begingroup$
      @HenningMakholm OP changed the question twice, will redo
      $endgroup$
      – gt6989b
      Dec 5 '18 at 15:34











    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%2f3027195%2fis-emptyset-is-a-subset-of-set-emptyset-1-2-3%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












    $begingroup$

    The set ${x}$ is a subset of ${a,b,c}$ if and only if $x$ is equal to $a$, $b$ or $c$. So, unless you have a weird definition of $1$, $2$ or $3$, ${emptyset}nsubseteq{1,2,3}$. But ${emptyset}subseteq{emptyset, 1,2,3}$



    Remark: The most usual construction of $Bbb N$ from ZFC defines $0=emptyset$ and $1={emptyset}$.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      The set ${x}$ is a subset of ${a,b,c}$ if and only if $x$ is equal to $a$, $b$ or $c$. So, unless you have a weird definition of $1$, $2$ or $3$, ${emptyset}nsubseteq{1,2,3}$. But ${emptyset}subseteq{emptyset, 1,2,3}$



      Remark: The most usual construction of $Bbb N$ from ZFC defines $0=emptyset$ and $1={emptyset}$.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        The set ${x}$ is a subset of ${a,b,c}$ if and only if $x$ is equal to $a$, $b$ or $c$. So, unless you have a weird definition of $1$, $2$ or $3$, ${emptyset}nsubseteq{1,2,3}$. But ${emptyset}subseteq{emptyset, 1,2,3}$



        Remark: The most usual construction of $Bbb N$ from ZFC defines $0=emptyset$ and $1={emptyset}$.






        share|cite|improve this answer











        $endgroup$



        The set ${x}$ is a subset of ${a,b,c}$ if and only if $x$ is equal to $a$, $b$ or $c$. So, unless you have a weird definition of $1$, $2$ or $3$, ${emptyset}nsubseteq{1,2,3}$. But ${emptyset}subseteq{emptyset, 1,2,3}$



        Remark: The most usual construction of $Bbb N$ from ZFC defines $0=emptyset$ and $1={emptyset}$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 5 '18 at 15:36

























        answered Dec 5 '18 at 15:33









        ajotatxeajotatxe

        53.9k24090




        53.9k24090























            0












            $begingroup$

            If you are asking about $emptyset$, then it is indeed the subset of any set.



            If you are asking about $E = {emptyset, 1,2,3}$, then $emptyset in E$ and $emptyset subset E$.



            Moreover, if $S = {emptyset}$ then $S subset E$.






            share|cite|improve this answer











            $endgroup$









            • 2




              $begingroup$
              It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
              $endgroup$
              – Henning Makholm
              Dec 5 '18 at 15:33










            • $begingroup$
              @HenningMakholm OP changed the question twice, will redo
              $endgroup$
              – gt6989b
              Dec 5 '18 at 15:34
















            0












            $begingroup$

            If you are asking about $emptyset$, then it is indeed the subset of any set.



            If you are asking about $E = {emptyset, 1,2,3}$, then $emptyset in E$ and $emptyset subset E$.



            Moreover, if $S = {emptyset}$ then $S subset E$.






            share|cite|improve this answer











            $endgroup$









            • 2




              $begingroup$
              It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
              $endgroup$
              – Henning Makholm
              Dec 5 '18 at 15:33










            • $begingroup$
              @HenningMakholm OP changed the question twice, will redo
              $endgroup$
              – gt6989b
              Dec 5 '18 at 15:34














            0












            0








            0





            $begingroup$

            If you are asking about $emptyset$, then it is indeed the subset of any set.



            If you are asking about $E = {emptyset, 1,2,3}$, then $emptyset in E$ and $emptyset subset E$.



            Moreover, if $S = {emptyset}$ then $S subset E$.






            share|cite|improve this answer











            $endgroup$



            If you are asking about $emptyset$, then it is indeed the subset of any set.



            If you are asking about $E = {emptyset, 1,2,3}$, then $emptyset in E$ and $emptyset subset E$.



            Moreover, if $S = {emptyset}$ then $S subset E$.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 5 '18 at 15:35

























            answered Dec 5 '18 at 15:31









            gt6989bgt6989b

            34.6k22456




            34.6k22456








            • 2




              $begingroup$
              It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
              $endgroup$
              – Henning Makholm
              Dec 5 '18 at 15:33










            • $begingroup$
              @HenningMakholm OP changed the question twice, will redo
              $endgroup$
              – gt6989b
              Dec 5 '18 at 15:34














            • 2




              $begingroup$
              It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
              $endgroup$
              – Henning Makholm
              Dec 5 '18 at 15:33










            • $begingroup$
              @HenningMakholm OP changed the question twice, will redo
              $endgroup$
              – gt6989b
              Dec 5 '18 at 15:34








            2




            2




            $begingroup$
            It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
            $endgroup$
            – Henning Makholm
            Dec 5 '18 at 15:33




            $begingroup$
            It looks to me like the question is asking whether ${varnothing}subseteq{varnothing,1,2,3}$, which is not what you're answering here.
            $endgroup$
            – Henning Makholm
            Dec 5 '18 at 15:33












            $begingroup$
            @HenningMakholm OP changed the question twice, will redo
            $endgroup$
            – gt6989b
            Dec 5 '18 at 15:34




            $begingroup$
            @HenningMakholm OP changed the question twice, will redo
            $endgroup$
            – gt6989b
            Dec 5 '18 at 15:34


















            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%2f3027195%2fis-emptyset-is-a-subset-of-set-emptyset-1-2-3%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

            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

            Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

            Image
            up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
            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