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
asked Nov 25 at 2:08
cb7
1036
add a comment |
up vote
12
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 Nov 25 at 2:08
cb7
1036
add a comment |
up vote
12
down vote
favorite
up vote
12
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 Nov 25 at 2:08
cb7
1036
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 Nov 25 at 2:08
cb7
1036
asked Nov 25 at 2:08
cb7
1036
asked Nov 25 at 2:08
cb7
1036
asked Nov 25 at 2:08
cb7
1036
asked Nov 25 at 2:08
cb7
1036
1036
add a comment |
add a comment |
3 Answers
3
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 authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:
To avoid confusion, we shall say that a set includes its elements and contains its subsets.
answered Nov 25 at 2:19
bof
49k452116
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
Nov 25 at 2:29
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
Nov 25 at 4:26
1
@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
Nov 25 at 7:01
1
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
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
Nov 25 at 14:03
|
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 Nov 25 at 2:13
mathnoob
1,133116
add a comment |
up vote
0
down vote
I think this is an example of a function being implicitly applied to a set of its possible inputs in a point-wise fashion. Namely, since it is clear what it means for an element s of span(S) to be contained in V, we can (by point-wise extension) also give meaning to the statement that span(S) is contained in V.
The function this is applied to in this case is simply:
$$in_V : S to mathrm{Bool} : s mapsto [s in V].$$
answered Nov 26 at 9:31
hkBst
33817
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
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 authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:
To avoid confusion, we shall say that a set includes its elements and contains its subsets.
answered Nov 25 at 2:19
bof
49k452116
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
Nov 25 at 2:29
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
Nov 25 at 4:26
1
@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
Nov 25 at 7:01
1
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
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
Nov 25 at 14:03
|
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 authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:
To avoid confusion, we shall say that a set includes its elements and contains its subsets.
answered Nov 25 at 2:19
bof
49k452116
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
Nov 25 at 2:29
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
Nov 25 at 4:26
1
@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
Nov 25 at 7:01
1
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
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
Nov 25 at 14:03
|
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 authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:
To avoid confusion, we shall say that a set includes its elements and contains its subsets.
answered Nov 25 at 2:19
bof
49k452116
Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:
To avoid confusion, we shall say that a set includes its elements and contains its subsets.
answered Nov 25 at 2:19
bof
49k452116
edited Nov 26 at 5:46
answered Nov 25 at 2:19
bof
49k452116
answered Nov 25 at 2:19
bof
49k452116
answered Nov 25 at 2:19
bof
49k452116
49k452116
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
Nov 25 at 2:29
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
Nov 25 at 4:26
1
@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
Nov 25 at 7:01
1
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
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
Nov 25 at 14:03
|
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
Nov 25 at 2:29
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
Nov 25 at 4:26
1
@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
Nov 25 at 7:01
1
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
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
Nov 25 at 14:03
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
Nov 25 at 2:29
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
Nov 25 at 2:29
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
Nov 25 at 4:26
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
Nov 25 at 4:26
1
1
@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
Nov 25 at 7:01
@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
Nov 25 at 7:01
1
1
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
I have seen professor uses lowercase for element ($a$), uppercase for set of elements ($A$) and "curly"-case for set of sets of elements ($mathscr{A}$), but this convention breaks down when you have more than three levels.
– Alex Vong
Nov 25 at 10:31
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
Nov 25 at 14:03
@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
Nov 25 at 14:03
|
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 Nov 25 at 2:13
mathnoob
1,133116
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 Nov 25 at 2:13
mathnoob
1,133116
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 Nov 25 at 2:13
mathnoob
1,133116
$S$ 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 Nov 25 at 2:13
mathnoob
1,133116
answered Nov 25 at 2:13
mathnoob
1,133116
answered Nov 25 at 2:13
mathnoob
1,133116
answered Nov 25 at 2:13
mathnoob
1,133116
1,133116
add a comment |
add a comment |
up vote
0
down vote
I think this is an example of a function being implicitly applied to a set of its possible inputs in a point-wise fashion. Namely, since it is clear what it means for an element s of span(S) to be contained in V, we can (by point-wise extension) also give meaning to the statement that span(S) is contained in V.
The function this is applied to in this case is simply:
$$in_V : S to mathrm{Bool} : s mapsto [s in V].$$
answered Nov 26 at 9:31
hkBst
33817
add a comment |
up vote
0
down vote
I think this is an example of a function being implicitly applied to a set of its possible inputs in a point-wise fashion. Namely, since it is clear what it means for an element s of span(S) to be contained in V, we can (by point-wise extension) also give meaning to the statement that span(S) is contained in V.
The function this is applied to in this case is simply:
$$in_V : S to mathrm{Bool} : s mapsto [s in V].$$
answered Nov 26 at 9:31
hkBst
33817
add a comment |
up vote
0
down vote
up vote
0
down vote
I think this is an example of a function being implicitly applied to a set of its possible inputs in a point-wise fashion. Namely, since it is clear what it means for an element s of span(S) to be contained in V, we can (by point-wise extension) also give meaning to the statement that span(S) is contained in V.
The function this is applied to in this case is simply:
$$in_V : S to mathrm{Bool} : s mapsto [s in V].$$
answered Nov 26 at 9:31
hkBst
33817
I think this is an example of a function being implicitly applied to a set of its possible inputs in a point-wise fashion. Namely, since it is clear what it means for an element s of span(S) to be contained in V, we can (by point-wise extension) also give meaning to the statement that span(S) is contained in V.
The function this is applied to in this case is simply:
$$in_V : S to mathrm{Bool} : s mapsto [s in V].$$
answered Nov 26 at 9:31
hkBst
33817
answered Nov 26 at 9:31
hkBst
33817
answered Nov 26 at 9:31
hkBst
33817
answered Nov 26 at 9:31
hkBst
33817
33817
add a comment |
add a comment |
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