Are the elements of this set distinct? Can it be considered a set if they are not?












0












$begingroup$


I'm currently reading Book of Proof by Richard Hammack. In the chapter on sets, he gives this as an example:




X = { n2 : n ∈ Z }




If n can be any integer and (-n)2 = (n)2, are the elements of this set distinct? If they are not, is this a set?



To my understanding, sets cannot have repeated elements. However, on the wikipedia page for sets it defines a set as




a collection of distinct objects, considered as an object in its own right




but later says




In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}.




So can sets have repeated elements?






elementary-set-theory







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    I'm currently reading Book of Proof by Richard Hammack. In the chapter on sets, he gives this as an example:




    X = { n2 : n ∈ Z }




    If n can be any integer and (-n)2 = (n)2, are the elements of this set distinct? If they are not, is this a set?



    To my understanding, sets cannot have repeated elements. However, on the wikipedia page for sets it defines a set as




    a collection of distinct objects, considered as an object in its own right




    but later says




    In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}.




    So can sets have repeated elements?






    elementary-set-theory







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I'm currently reading Book of Proof by Richard Hammack. In the chapter on sets, he gives this as an example:




      X = { n2 : n ∈ Z }




      If n can be any integer and (-n)2 = (n)2, are the elements of this set distinct? If they are not, is this a set?



      To my understanding, sets cannot have repeated elements. However, on the wikipedia page for sets it defines a set as




      a collection of distinct objects, considered as an object in its own right




      but later says




      In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}.




      So can sets have repeated elements?






      elementary-set-theory







      share|cite|improve this question









      $endgroup$




      I'm currently reading Book of Proof by Richard Hammack. In the chapter on sets, he gives this as an example:




      X = { n2 : n ∈ Z }




      If n can be any integer and (-n)2 = (n)2, are the elements of this set distinct? If they are not, is this a set?



      To my understanding, sets cannot have repeated elements. However, on the wikipedia page for sets it defines a set as




      a collection of distinct objects, considered as an object in its own right




      but later says




      In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}.




      So can sets have repeated elements?






      elementary-set-theory




      elementary-set-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 4 '18 at 14:16









      Copeland CorleyCopeland Corley

      82




      82






















          3 Answers
          3






          active

          oldest

          votes


















          0












          $begingroup$

          Think of it like this: an element $x$ of a set $A$ can be mentioned in the notation of the set more than once.



          Here ${11,6,6}$ is a notation of the unique set that only has the elements $6$ and $11$ (and is completely determined by that fact) and in this notation element $6$ is mentioned twice. That is allowed.



          That's all.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:36










          • $begingroup$
            By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
            $endgroup$
            – drhab
            Dec 4 '18 at 14:58












          • $begingroup$
            One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 15:33










          • $begingroup$
            Yes, they do. The cardinality is $2$ as you will understand.
            $endgroup$
            – drhab
            Dec 4 '18 at 16:40



















          0












          $begingroup$

          Yes, it is a set. Note that ${1,1}$ and ${1}$ are the same set. So$$X={1,1,4,4,9,9,16,16,ldots}={1,4,9,16,ldots}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:39










          • $begingroup$
            I have no experience whatsoever with multisets.
            $endgroup$
            – José Carlos Santos
            Dec 4 '18 at 14:40



















          0












          $begingroup$

          The notation allows to repeat the elements, but as they are identical, they "count once" only. So, strictly,



          $${11,6,6}={11,6}.$$



          True replication requires a multiset.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:37










          • $begingroup$
            @CopelandCorley: of course. Multisets aren't so often met.
            $endgroup$
            – Yves Daoust
            Dec 4 '18 at 14:55











          Your Answer





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          3 Answers
          3






          active

          oldest

          votes








          3 Answers
          3






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          0












          $begingroup$

          Think of it like this: an element $x$ of a set $A$ can be mentioned in the notation of the set more than once.



          Here ${11,6,6}$ is a notation of the unique set that only has the elements $6$ and $11$ (and is completely determined by that fact) and in this notation element $6$ is mentioned twice. That is allowed.



          That's all.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:36










          • $begingroup$
            By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
            $endgroup$
            – drhab
            Dec 4 '18 at 14:58












          • $begingroup$
            One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 15:33










          • $begingroup$
            Yes, they do. The cardinality is $2$ as you will understand.
            $endgroup$
            – drhab
            Dec 4 '18 at 16:40
















          0












          $begingroup$

          Think of it like this: an element $x$ of a set $A$ can be mentioned in the notation of the set more than once.



          Here ${11,6,6}$ is a notation of the unique set that only has the elements $6$ and $11$ (and is completely determined by that fact) and in this notation element $6$ is mentioned twice. That is allowed.



          That's all.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:36










          • $begingroup$
            By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
            $endgroup$
            – drhab
            Dec 4 '18 at 14:58












          • $begingroup$
            One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 15:33










          • $begingroup$
            Yes, they do. The cardinality is $2$ as you will understand.
            $endgroup$
            – drhab
            Dec 4 '18 at 16:40














          0












          0








          0





          $begingroup$

          Think of it like this: an element $x$ of a set $A$ can be mentioned in the notation of the set more than once.



          Here ${11,6,6}$ is a notation of the unique set that only has the elements $6$ and $11$ (and is completely determined by that fact) and in this notation element $6$ is mentioned twice. That is allowed.



          That's all.






          share|cite|improve this answer









          $endgroup$



          Think of it like this: an element $x$ of a set $A$ can be mentioned in the notation of the set more than once.



          Here ${11,6,6}$ is a notation of the unique set that only has the elements $6$ and $11$ (and is completely determined by that fact) and in this notation element $6$ is mentioned twice. That is allowed.



          That's all.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 '18 at 14:27









          drhabdrhab

          102k545136




          102k545136












          • $begingroup$
            So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:36










          • $begingroup$
            By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
            $endgroup$
            – drhab
            Dec 4 '18 at 14:58












          • $begingroup$
            One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 15:33










          • $begingroup$
            Yes, they do. The cardinality is $2$ as you will understand.
            $endgroup$
            – drhab
            Dec 4 '18 at 16:40


















          • $begingroup$
            So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:36










          • $begingroup$
            By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
            $endgroup$
            – drhab
            Dec 4 '18 at 14:58












          • $begingroup$
            One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 15:33










          • $begingroup$
            Yes, they do. The cardinality is $2$ as you will understand.
            $endgroup$
            – drhab
            Dec 4 '18 at 16:40
















          $begingroup$
          So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 14:36




          $begingroup$
          So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 14:36












          $begingroup$
          By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
          $endgroup$
          – drhab
          Dec 4 '18 at 14:58






          $begingroup$
          By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a,a,b}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets.
          $endgroup$
          – drhab
          Dec 4 '18 at 14:58














          $begingroup$
          One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 15:33




          $begingroup$
          One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 15:33












          $begingroup$
          Yes, they do. The cardinality is $2$ as you will understand.
          $endgroup$
          – drhab
          Dec 4 '18 at 16:40




          $begingroup$
          Yes, they do. The cardinality is $2$ as you will understand.
          $endgroup$
          – drhab
          Dec 4 '18 at 16:40











          0












          $begingroup$

          Yes, it is a set. Note that ${1,1}$ and ${1}$ are the same set. So$$X={1,1,4,4,9,9,16,16,ldots}={1,4,9,16,ldots}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:39










          • $begingroup$
            I have no experience whatsoever with multisets.
            $endgroup$
            – José Carlos Santos
            Dec 4 '18 at 14:40
















          0












          $begingroup$

          Yes, it is a set. Note that ${1,1}$ and ${1}$ are the same set. So$$X={1,1,4,4,9,9,16,16,ldots}={1,4,9,16,ldots}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:39










          • $begingroup$
            I have no experience whatsoever with multisets.
            $endgroup$
            – José Carlos Santos
            Dec 4 '18 at 14:40














          0












          0








          0





          $begingroup$

          Yes, it is a set. Note that ${1,1}$ and ${1}$ are the same set. So$$X={1,1,4,4,9,9,16,16,ldots}={1,4,9,16,ldots}$$






          share|cite|improve this answer









          $endgroup$



          Yes, it is a set. Note that ${1,1}$ and ${1}$ are the same set. So$$X={1,1,4,4,9,9,16,16,ldots}={1,4,9,16,ldots}$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 '18 at 14:19









          José Carlos SantosJosé Carlos Santos

          164k22131234




          164k22131234












          • $begingroup$
            Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:39










          • $begingroup$
            I have no experience whatsoever with multisets.
            $endgroup$
            – José Carlos Santos
            Dec 4 '18 at 14:40


















          • $begingroup$
            Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:39










          • $begingroup$
            I have no experience whatsoever with multisets.
            $endgroup$
            – José Carlos Santos
            Dec 4 '18 at 14:40
















          $begingroup$
          Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 14:39




          $begingroup$
          Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 14:39












          $begingroup$
          I have no experience whatsoever with multisets.
          $endgroup$
          – José Carlos Santos
          Dec 4 '18 at 14:40




          $begingroup$
          I have no experience whatsoever with multisets.
          $endgroup$
          – José Carlos Santos
          Dec 4 '18 at 14:40











          0












          $begingroup$

          The notation allows to repeat the elements, but as they are identical, they "count once" only. So, strictly,



          $${11,6,6}={11,6}.$$



          True replication requires a multiset.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:37










          • $begingroup$
            @CopelandCorley: of course. Multisets aren't so often met.
            $endgroup$
            – Yves Daoust
            Dec 4 '18 at 14:55
















          0












          $begingroup$

          The notation allows to repeat the elements, but as they are identical, they "count once" only. So, strictly,



          $${11,6,6}={11,6}.$$



          True replication requires a multiset.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:37










          • $begingroup$
            @CopelandCorley: of course. Multisets aren't so often met.
            $endgroup$
            – Yves Daoust
            Dec 4 '18 at 14:55














          0












          0








          0





          $begingroup$

          The notation allows to repeat the elements, but as they are identical, they "count once" only. So, strictly,



          $${11,6,6}={11,6}.$$



          True replication requires a multiset.






          share|cite|improve this answer









          $endgroup$



          The notation allows to repeat the elements, but as they are identical, they "count once" only. So, strictly,



          $${11,6,6}={11,6}.$$



          True replication requires a multiset.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 '18 at 14:28









          Yves DaoustYves Daoust

          129k675227




          129k675227












          • $begingroup$
            How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:37










          • $begingroup$
            @CopelandCorley: of course. Multisets aren't so often met.
            $endgroup$
            – Yves Daoust
            Dec 4 '18 at 14:55


















          • $begingroup$
            How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
            $endgroup$
            – Copeland Corley
            Dec 4 '18 at 14:37










          • $begingroup$
            @CopelandCorley: of course. Multisets aren't so often met.
            $endgroup$
            – Yves Daoust
            Dec 4 '18 at 14:55
















          $begingroup$
          How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 14:37




          $begingroup$
          How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context?
          $endgroup$
          – Copeland Corley
          Dec 4 '18 at 14:37












          $begingroup$
          @CopelandCorley: of course. Multisets aren't so often met.
          $endgroup$
          – Yves Daoust
          Dec 4 '18 at 14:55




          $begingroup$
          @CopelandCorley: of course. Multisets aren't so often met.
          $endgroup$
          – Yves Daoust
          Dec 4 '18 at 14:55


















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

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          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:"
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