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?
notation
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cb7
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up vote
11
down vote
favorite
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?
notation
asked yesterday
cb7
986
add a comment |
up vote
11
down vote
favorite
up vote
11
down vote
favorite
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?
notation
asked yesterday
cb7
986
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?
notation
notation
asked yesterday
cb7
986
asked yesterday
cb7
986
asked yesterday
cb7
986
asked yesterday
cb7
986
asked yesterday
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986
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add a comment |
2 Answers
2
active
oldest
votes
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.
answered yesterday
bof
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
|
show 1 more comment
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 $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.
answered yesterday
mathnoob
93213
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered yesterday
bof
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
|
show 1 more comment
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.
answered yesterday
bof
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
|
show 1 more comment
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.
answered yesterday
bof
48.7k451116
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.
answered yesterday
bof
48.7k451116
answered yesterday
bof
48.7k451116
answered yesterday
bof
48.7k451116
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
|
show 1 more comment
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
|
show 1 more comment
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 $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.
answered yesterday
mathnoob
93213
add a comment |
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 $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.
answered yesterday
mathnoob
93213
add a comment |
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 $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.
answered yesterday
mathnoob
93213
$S$ 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 $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.
answered yesterday
mathnoob
93213
answered yesterday
mathnoob
93213
answered yesterday
mathnoob
93213
answered yesterday
mathnoob
93213
93213
add a comment |
add a comment |
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