Does “V contains S” have two different meanings?











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Talking in terms of sets, I would take the above to mean $S in V$. But my course's notes says




Let $S$ be a subset of a vector space $V$, the span of $S$, denoted $Span(S)$ is the smallest subspace of $V$ that contains $S$.




Which confuses me, because $V$ contains vectors, whereas $S$ is a set, not a vector, so by my definition, $V$ cannot contains $S$.



So by "$V$ contains $S$" I assume it means $S subseteq V$, right? Is this considered correct also?










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    up vote
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    Talking in terms of sets, I would take the above to mean $S in V$. But my course's notes says




    Let $S$ be a subset of a vector space $V$, the span of $S$, denoted $Span(S)$ is the smallest subspace of $V$ that contains $S$.




    Which confuses me, because $V$ contains vectors, whereas $S$ is a set, not a vector, so by my definition, $V$ cannot contains $S$.



    So by "$V$ contains $S$" I assume it means $S subseteq V$, right? Is this considered correct also?










    share|cite|improve this question
























      up vote
      11
      down vote

      favorite
      1









      up vote
      11
      down vote

      favorite
      1






      1





      Talking in terms of sets, I would take the above to mean $S in V$. But my course's notes says




      Let $S$ be a subset of a vector space $V$, the span of $S$, denoted $Span(S)$ is the smallest subspace of $V$ that contains $S$.




      Which confuses me, because $V$ contains vectors, whereas $S$ is a set, not a vector, so by my definition, $V$ cannot contains $S$.



      So by "$V$ contains $S$" I assume it means $S subseteq V$, right? Is this considered correct also?










      share|cite|improve this question













      Talking in terms of sets, I would take the above to mean $S in V$. But my course's notes says




      Let $S$ be a subset of a vector space $V$, the span of $S$, denoted $Span(S)$ is the smallest subspace of $V$ that contains $S$.




      Which confuses me, because $V$ contains vectors, whereas $S$ is a set, not a vector, so by my definition, $V$ cannot contains $S$.



      So by "$V$ contains $S$" I assume it means $S subseteq V$, right? Is this considered correct also?







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      asked yesterday









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          Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some writers make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some do just the opposite, and say that a set includes its elements and contains its subsets. Sorry about that.






          share|cite|improve this answer





















          • In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
            – Barry Cipra
            yesterday












          • @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
            – bof
            yesterday






          • 4




            I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
            – Mong H. Ng
            yesterday










          • @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
            – bof
            21 hours ago






          • 3




            @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
            – Ilmari Karonen
            14 hours ago




















          up vote
          1
          down vote













          $S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= {v_1,v_2,...}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)={a_1v_1+a_2v_2+...|a_i in F,v_i in S}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.



          I mean you could have $S subset V$. Take for example $V=mathbb{R}^3$ and $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.






          share|cite|improve this answer





















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            up vote
            16
            down vote



            accepted










            Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some writers make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some do just the opposite, and say that a set includes its elements and contains its subsets. Sorry about that.






            share|cite|improve this answer





















            • In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
              – Barry Cipra
              yesterday












            • @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
              – bof
              yesterday






            • 4




              I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
              – Mong H. Ng
              yesterday










            • @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
              – bof
              21 hours ago






            • 3




              @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
              – Ilmari Karonen
              14 hours ago

















            up vote
            16
            down vote



            accepted










            Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some writers make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some do just the opposite, and say that a set includes its elements and contains its subsets. Sorry about that.






            share|cite|improve this answer





















            • In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
              – Barry Cipra
              yesterday












            • @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
              – bof
              yesterday






            • 4




              I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
              – Mong H. Ng
              yesterday










            • @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
              – bof
              21 hours ago






            • 3




              @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
              – Ilmari Karonen
              14 hours ago















            up vote
            16
            down vote



            accepted







            up vote
            16
            down vote



            accepted






            Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some writers make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some do just the opposite, and say that a set includes its elements and contains its subsets. Sorry about that.






            share|cite|improve this answer












            Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some writers make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some do just the opposite, and say that a set includes its elements and contains its subsets. Sorry about that.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered yesterday









            bof

            48.7k451116




            48.7k451116












            • In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
              – Barry Cipra
              yesterday












            • @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
              – bof
              yesterday






            • 4




              I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
              – Mong H. Ng
              yesterday










            • @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
              – bof
              21 hours ago






            • 3




              @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
              – Ilmari Karonen
              14 hours ago




















            • In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
              – Barry Cipra
              yesterday












            • @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
              – bof
              yesterday






            • 4




              I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
              – Mong H. Ng
              yesterday










            • @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
              – bof
              21 hours ago






            • 3




              @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
              – Ilmari Karonen
              14 hours ago


















            In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
            – Barry Cipra
            yesterday






            In Naive Set Theory (1960), Paul Halmos has sets containing their elements and including their subsets. Do you know of any books that do the opposite? (I suspect some authors may use one word, or both, in both senses, but it'd be good to have an example or two of that as well.)
            – Barry Cipra
            yesterday














            @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
            – bof
            yesterday




            @BarryCipra I recall that C. St. J. A. Nash-Williams, in (at least) one of his series of papers on well-quasi-ordering, stated explicitly that sets include their elements and contain their subsets. I recall being surprised by this, as I was used to seeing it done the other way. Unfortunately I don't have an exact reference.
            – bof
            yesterday




            4




            4




            I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
            – Mong H. Ng
            yesterday




            I thought there isnt really an issue, as the context usually gives you a hint about whether it is referring to a set or a single element.
            – Mong H. Ng
            yesterday












            @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
            – bof
            21 hours ago




            @MongH.Ng Yes, it is usually possible to guess the author's intent from the context. However, mathematics is supposed to be an exact science. Of course we have to guess what statements we should try to prove, and we have to guess what methods to use; but some of us would prefer not to guess at the meaning of the words.
            – bof
            21 hours ago




            3




            3




            @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
            – Ilmari Karonen
            14 hours ago






            @AlexVong: I remember one lecturer I listened to using just that convention, except that he used a blackletter $mathfrak A$ for the set of sets. I didn't feel like trying to repeatedly draw that in my notes, so I substituted a curly $mathscr A$ for it, only to find him a few minutes later introducing an actual $mathscr A$ for, I think, a set of functions over $A$.
            – Ilmari Karonen
            14 hours ago












            up vote
            1
            down vote













            $S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= {v_1,v_2,...}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)={a_1v_1+a_2v_2+...|a_i in F,v_i in S}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.



            I mean you could have $S subset V$. Take for example $V=mathbb{R}^3$ and $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.






            share|cite|improve this answer

























              up vote
              1
              down vote













              $S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= {v_1,v_2,...}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)={a_1v_1+a_2v_2+...|a_i in F,v_i in S}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.



              I mean you could have $S subset V$. Take for example $V=mathbb{R}^3$ and $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                $S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= {v_1,v_2,...}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)={a_1v_1+a_2v_2+...|a_i in F,v_i in S}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.



                I mean you could have $S subset V$. Take for example $V=mathbb{R}^3$ and $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.






                share|cite|improve this answer












                $S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= {v_1,v_2,...}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)={a_1v_1+a_2v_2+...|a_i in F,v_i in S}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.



                I mean you could have $S subset V$. Take for example $V=mathbb{R}^3$ and $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered yesterday









                mathnoob

                93213




                93213






























                     

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