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







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$endgroup$












  • $begingroup$
    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











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    2 Answers
    2






    active

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    2 Answers
    2






    active

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    active

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    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


















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