Does “V contains S” have two different meanings?
$begingroup$
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 '18 at 2:08
cb7cb7
1476
$endgroup$
add a comment |
$begingroup$
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 '18 at 2:08
cb7cb7
1476
$endgroup$
add a comment |
$begingroup$
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 '18 at 2:08
cb7cb7
1476
$endgroup$
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 '18 at 2:08
cb7cb7
1476
asked Nov 25 '18 at 2:08
cb7cb7
1476
asked Nov 25 '18 at 2:08
cb7cb7
1476
asked Nov 25 '18 at 2:08
cb7cb7
1476
asked Nov 25 '18 at 2:08
cb7cb7
1476
1476
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
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 '18 at 2:19
bofbof
52.7k559121
$endgroup$
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
4
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
1
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
1
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
3
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
|
show 1 more comment
$begingroup$
$S$ 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 '18 at 2:13
mathpadawanmathpadawan
2,216522
$endgroup$
add a comment |
$begingroup$
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 '18 at 9:31
hkBsthkBst
33817
$endgroup$
add a comment |
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3 Answers
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active
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votes
3 Answers
3
active
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$begingroup$
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 '18 at 2:19
bofbof
52.7k559121
$endgroup$
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
4
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
1
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
1
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
3
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
|
show 1 more comment
$begingroup$
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 '18 at 2:19
bofbof
52.7k559121
$endgroup$
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
4
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
1
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
1
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
3
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
|
show 1 more comment
$begingroup$
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 '18 at 2:19
bofbof
52.7k559121
$endgroup$
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 '18 at 2:19
bofbof
52.7k559121
edited Nov 26 '18 at 5:46
answered Nov 25 '18 at 2:19
bofbof
52.7k559121
answered Nov 25 '18 at 2:19
bofbof
52.7k559121
answered Nov 25 '18 at 2:19
bofbof
52.7k559121
52.7k559121
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
4
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
1
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
1
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
3
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
|
show 1 more comment
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
4
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
1
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
1
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
3
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
$begingroup$
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.)
$endgroup$
– Barry Cipra
Nov 25 '18 at 2:29
4
4
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
$begingroup$
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.
$endgroup$
– Mong H. Ng
Nov 25 '18 at 4:26
1
1
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
$begingroup$
@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.
$endgroup$
– bof
Nov 25 '18 at 7:01
1
1
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
$begingroup$
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.
$endgroup$
– Alex Vong
Nov 25 '18 at 10:31
3
3
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
$begingroup$
@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 $A$.
$endgroup$
– Ilmari Karonen
Nov 25 '18 at 14:03
|
show 1 more comment
$begingroup$
$S$ 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 '18 at 2:13
mathpadawanmathpadawan
2,216522
$endgroup$
add a comment |
$begingroup$
$S$ 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 '18 at 2:13
mathpadawanmathpadawan
2,216522
$endgroup$
add a comment |
$begingroup$
$S$ 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 '18 at 2:13
mathpadawanmathpadawan
2,216522
$endgroup$
$S$ 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 '18 at 2:13
mathpadawanmathpadawan
2,216522
answered Nov 25 '18 at 2:13
mathpadawanmathpadawan
2,216522
answered Nov 25 '18 at 2:13
mathpadawanmathpadawan
2,216522
answered Nov 25 '18 at 2:13
mathpadawanmathpadawan
2,216522
2,216522
add a comment |
add a comment |
$begingroup$
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 '18 at 9:31
hkBsthkBst
33817
$endgroup$
add a comment |
$begingroup$
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 '18 at 9:31
hkBsthkBst
33817
$endgroup$
add a comment |
$begingroup$
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 '18 at 9:31
hkBsthkBst
33817
$endgroup$
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 '18 at 9:31
hkBsthkBst
33817
answered Nov 26 '18 at 9:31
hkBsthkBst
33817
answered Nov 26 '18 at 9:31
hkBsthkBst
33817
answered Nov 26 '18 at 9:31
hkBsthkBst
33817
33817
add a comment |
add a comment |
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