Mixing
Scalar Multiplication of a Set
$begingroup$
I am aware of how one can represent the Cartesian product of two sets, say $A$ and $B$. However, is there are standard way to represent the scalar product of a value and a set/multiset? As a simple example (with a multiset), let $P = {1, 2, 3, 4, 2}$ and $q = 2$. Then $$q cdot P := {(1 cdot 2),(2 cdot 2),(3 cdot 2), (4 cdot 2), (2 cdot 2)} = {2, 4, 6, 8, 4 }. $$
Could this be the appropriate notation for a scalar product? I'm not entirely certain scalar multiplication of a value and a set exists as I haven't been able to find it anywhere in books or online––if this is the case, is it because set are immutable? In case it is asked, I am unfortunately unable in this example to make $P$ a vector and do the same operation.
elementary-set-theory notation
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
$endgroup$
add a comment |
$begingroup$
I am aware of how one can represent the Cartesian product of two sets, say $A$ and $B$. However, is there are standard way to represent the scalar product of a value and a set/multiset? As a simple example (with a multiset), let $P = {1, 2, 3, 4, 2}$ and $q = 2$. Then $$q cdot P := {(1 cdot 2),(2 cdot 2),(3 cdot 2), (4 cdot 2), (2 cdot 2)} = {2, 4, 6, 8, 4 }. $$
Could this be the appropriate notation for a scalar product? I'm not entirely certain scalar multiplication of a value and a set exists as I haven't been able to find it anywhere in books or online––if this is the case, is it because set are immutable? In case it is asked, I am unfortunately unable in this example to make $P$ a vector and do the same operation.
elementary-set-theory notation
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
$endgroup$
1
$begingroup$
Well, if scalar product exists on the elements of the set (e.g. they themselves are scalars, or vectors, or matrices, or scalar valued functions, etc), then your definition [notation] is the natural one we use.
$endgroup$
– Berci
Jan 19 at 0:15
$begingroup$
I have seen the same notation in convex analysis. Here's something: en.wikipedia.org/wiki/Minkowski_functional Note the $lambda K$.
$endgroup$
– lightxbulb
Jan 19 at 0:15
add a comment |
$begingroup$
I am aware of how one can represent the Cartesian product of two sets, say $A$ and $B$. However, is there are standard way to represent the scalar product of a value and a set/multiset? As a simple example (with a multiset), let $P = {1, 2, 3, 4, 2}$ and $q = 2$. Then $$q cdot P := {(1 cdot 2),(2 cdot 2),(3 cdot 2), (4 cdot 2), (2 cdot 2)} = {2, 4, 6, 8, 4 }. $$
Could this be the appropriate notation for a scalar product? I'm not entirely certain scalar multiplication of a value and a set exists as I haven't been able to find it anywhere in books or online––if this is the case, is it because set are immutable? In case it is asked, I am unfortunately unable in this example to make $P$ a vector and do the same operation.
elementary-set-theory notation
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
$endgroup$
I am aware of how one can represent the Cartesian product of two sets, say $A$ and $B$. However, is there are standard way to represent the scalar product of a value and a set/multiset? As a simple example (with a multiset), let $P = {1, 2, 3, 4, 2}$ and $q = 2$. Then $$q cdot P := {(1 cdot 2),(2 cdot 2),(3 cdot 2), (4 cdot 2), (2 cdot 2)} = {2, 4, 6, 8, 4 }. $$
Could this be the appropriate notation for a scalar product? I'm not entirely certain scalar multiplication of a value and a set exists as I haven't been able to find it anywhere in books or online––if this is the case, is it because set are immutable? In case it is asked, I am unfortunately unable in this example to make $P$ a vector and do the same operation.
elementary-set-theory notation
elementary-set-theory notation
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
edited Jan 19 at 0:32
Xander Henderson
14.7k103555
14.7k103555
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
asked Jan 19 at 0:05
Luke PoeppelLuke Poeppel
212
212
1
$begingroup$
Well, if scalar product exists on the elements of the set (e.g. they themselves are scalars, or vectors, or matrices, or scalar valued functions, etc), then your definition [notation] is the natural one we use.
$endgroup$
– Berci
Jan 19 at 0:15
$begingroup$
I have seen the same notation in convex analysis. Here's something: en.wikipedia.org/wiki/Minkowski_functional Note the $lambda K$.
$endgroup$
– lightxbulb
Jan 19 at 0:15
add a comment |
1
$begingroup$
Well, if scalar product exists on the elements of the set (e.g. they themselves are scalars, or vectors, or matrices, or scalar valued functions, etc), then your definition [notation] is the natural one we use.
$endgroup$
– Berci
Jan 19 at 0:15
$begingroup$
I have seen the same notation in convex analysis. Here's something: en.wikipedia.org/wiki/Minkowski_functional Note the $lambda K$.
$endgroup$
– lightxbulb
Jan 19 at 0:15
1
1
$begingroup$
Well, if scalar product exists on the elements of the set (e.g. they themselves are scalars, or vectors, or matrices, or scalar valued functions, etc), then your definition [notation] is the natural one we use.
$endgroup$
– Berci
Jan 19 at 0:15
$begingroup$
Well, if scalar product exists on the elements of the set (e.g. they themselves are scalars, or vectors, or matrices, or scalar valued functions, etc), then your definition [notation] is the natural one we use.
$endgroup$
– Berci
Jan 19 at 0:15
$begingroup$
I have seen the same notation in convex analysis. Here's something: en.wikipedia.org/wiki/Minkowski_functional Note the $lambda K$.
$endgroup$
– lightxbulb
Jan 19 at 0:15
$begingroup$
I have seen the same notation in convex analysis. Here's something: en.wikipedia.org/wiki/Minkowski_functional Note the $lambda K$.
$endgroup$
– lightxbulb
Jan 19 at 0:15
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The notation you suggest is very common notation in contexts where scalar multiplication makes sense. For example, in the study of fractal geometry, fractal sets often display self-similarity. It is not uncommon to see a self-similar subset of $mathbb{R}^n$ described as a set $F$ having the property that
$$ F = bigcup_{j=1}^{N} c_j F + b_j, $$
for some collection of $c_j in mathbb{R}$ and $b_jinmathbb{R}^N$. Here, each $c_j$ scales the set $F$, then $b_j$ translates the scaled copy. Both the scalar multiplication and the addition make perfect sense, and the notation is exactly what you suggest.
Slightly more generally, let $V$ be a vector space over a field $k$ (for example, $V = mathbb{R}^n$ is a vector space over $k = mathbb{R}$), let $P subseteq V$, and let $q in k$. Then
$$ qcdot P = qP = { qp : p in P }. $$
Even more generally, if $P$ is any set with a structure which supports some kind of multiplication, and $q$ is any object such that $qp$ makes sense for $pin P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation
$$ mathbb{Z} / pmathbb{Z} qquadtext{(where $p$ is prime)} $$
in a typical undergraduate course in abstract algebra. This is the quotient of $mathbb{Z}$ (the integers) with $pmathbb{Z}$ (multiples of $p$, i.e. the set ${np : ninmathbb{Z} }$).
As an addendum, there are also notions of sums and differences of sets with this kind of notation. If $A$ and $B$ are two subsets of some space in which addition and subtraction are defined, then we can define the Minkowski sum and difference of $A$ and $B$ as
$$ Apm B := { apm b : ain A land bin B }. $$
A similar notation could easily be adopted for a "Minkowski product" or "Minkowski quotient" (though I would be careful with those terms, as they may have other meanings). Indeed, I recently came across a paper which uses the notation
$$ CD := { cd : cin C land din D }, $$
where $C, D subseteq mathbb{R}$. That same paper also uses the notation
$$ 1/mathbb{N} := { tfrac{1}{n} : ninmathbb{N} }, $$
which is consistent with the multiplicative notation.
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
$endgroup$
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
add a comment |
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$begingroup$
The notation you suggest is very common notation in contexts where scalar multiplication makes sense. For example, in the study of fractal geometry, fractal sets often display self-similarity. It is not uncommon to see a self-similar subset of $mathbb{R}^n$ described as a set $F$ having the property that
$$ F = bigcup_{j=1}^{N} c_j F + b_j, $$
for some collection of $c_j in mathbb{R}$ and $b_jinmathbb{R}^N$. Here, each $c_j$ scales the set $F$, then $b_j$ translates the scaled copy. Both the scalar multiplication and the addition make perfect sense, and the notation is exactly what you suggest.
Slightly more generally, let $V$ be a vector space over a field $k$ (for example, $V = mathbb{R}^n$ is a vector space over $k = mathbb{R}$), let $P subseteq V$, and let $q in k$. Then
$$ qcdot P = qP = { qp : p in P }. $$
Even more generally, if $P$ is any set with a structure which supports some kind of multiplication, and $q$ is any object such that $qp$ makes sense for $pin P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation
$$ mathbb{Z} / pmathbb{Z} qquadtext{(where $p$ is prime)} $$
in a typical undergraduate course in abstract algebra. This is the quotient of $mathbb{Z}$ (the integers) with $pmathbb{Z}$ (multiples of $p$, i.e. the set ${np : ninmathbb{Z} }$).
As an addendum, there are also notions of sums and differences of sets with this kind of notation. If $A$ and $B$ are two subsets of some space in which addition and subtraction are defined, then we can define the Minkowski sum and difference of $A$ and $B$ as
$$ Apm B := { apm b : ain A land bin B }. $$
A similar notation could easily be adopted for a "Minkowski product" or "Minkowski quotient" (though I would be careful with those terms, as they may have other meanings). Indeed, I recently came across a paper which uses the notation
$$ CD := { cd : cin C land din D }, $$
where $C, D subseteq mathbb{R}$. That same paper also uses the notation
$$ 1/mathbb{N} := { tfrac{1}{n} : ninmathbb{N} }, $$
which is consistent with the multiplicative notation.
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
$endgroup$
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
add a comment |
$begingroup$
The notation you suggest is very common notation in contexts where scalar multiplication makes sense. For example, in the study of fractal geometry, fractal sets often display self-similarity. It is not uncommon to see a self-similar subset of $mathbb{R}^n$ described as a set $F$ having the property that
$$ F = bigcup_{j=1}^{N} c_j F + b_j, $$
for some collection of $c_j in mathbb{R}$ and $b_jinmathbb{R}^N$. Here, each $c_j$ scales the set $F$, then $b_j$ translates the scaled copy. Both the scalar multiplication and the addition make perfect sense, and the notation is exactly what you suggest.
Slightly more generally, let $V$ be a vector space over a field $k$ (for example, $V = mathbb{R}^n$ is a vector space over $k = mathbb{R}$), let $P subseteq V$, and let $q in k$. Then
$$ qcdot P = qP = { qp : p in P }. $$
Even more generally, if $P$ is any set with a structure which supports some kind of multiplication, and $q$ is any object such that $qp$ makes sense for $pin P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation
$$ mathbb{Z} / pmathbb{Z} qquadtext{(where $p$ is prime)} $$
in a typical undergraduate course in abstract algebra. This is the quotient of $mathbb{Z}$ (the integers) with $pmathbb{Z}$ (multiples of $p$, i.e. the set ${np : ninmathbb{Z} }$).
As an addendum, there are also notions of sums and differences of sets with this kind of notation. If $A$ and $B$ are two subsets of some space in which addition and subtraction are defined, then we can define the Minkowski sum and difference of $A$ and $B$ as
$$ Apm B := { apm b : ain A land bin B }. $$
A similar notation could easily be adopted for a "Minkowski product" or "Minkowski quotient" (though I would be careful with those terms, as they may have other meanings). Indeed, I recently came across a paper which uses the notation
$$ CD := { cd : cin C land din D }, $$
where $C, D subseteq mathbb{R}$. That same paper also uses the notation
$$ 1/mathbb{N} := { tfrac{1}{n} : ninmathbb{N} }, $$
which is consistent with the multiplicative notation.
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
$endgroup$
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
add a comment |
$begingroup$
The notation you suggest is very common notation in contexts where scalar multiplication makes sense. For example, in the study of fractal geometry, fractal sets often display self-similarity. It is not uncommon to see a self-similar subset of $mathbb{R}^n$ described as a set $F$ having the property that
$$ F = bigcup_{j=1}^{N} c_j F + b_j, $$
for some collection of $c_j in mathbb{R}$ and $b_jinmathbb{R}^N$. Here, each $c_j$ scales the set $F$, then $b_j$ translates the scaled copy. Both the scalar multiplication and the addition make perfect sense, and the notation is exactly what you suggest.
Slightly more generally, let $V$ be a vector space over a field $k$ (for example, $V = mathbb{R}^n$ is a vector space over $k = mathbb{R}$), let $P subseteq V$, and let $q in k$. Then
$$ qcdot P = qP = { qp : p in P }. $$
Even more generally, if $P$ is any set with a structure which supports some kind of multiplication, and $q$ is any object such that $qp$ makes sense for $pin P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation
$$ mathbb{Z} / pmathbb{Z} qquadtext{(where $p$ is prime)} $$
in a typical undergraduate course in abstract algebra. This is the quotient of $mathbb{Z}$ (the integers) with $pmathbb{Z}$ (multiples of $p$, i.e. the set ${np : ninmathbb{Z} }$).
As an addendum, there are also notions of sums and differences of sets with this kind of notation. If $A$ and $B$ are two subsets of some space in which addition and subtraction are defined, then we can define the Minkowski sum and difference of $A$ and $B$ as
$$ Apm B := { apm b : ain A land bin B }. $$
A similar notation could easily be adopted for a "Minkowski product" or "Minkowski quotient" (though I would be careful with those terms, as they may have other meanings). Indeed, I recently came across a paper which uses the notation
$$ CD := { cd : cin C land din D }, $$
where $C, D subseteq mathbb{R}$. That same paper also uses the notation
$$ 1/mathbb{N} := { tfrac{1}{n} : ninmathbb{N} }, $$
which is consistent with the multiplicative notation.
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
$endgroup$
The notation you suggest is very common notation in contexts where scalar multiplication makes sense. For example, in the study of fractal geometry, fractal sets often display self-similarity. It is not uncommon to see a self-similar subset of $mathbb{R}^n$ described as a set $F$ having the property that
$$ F = bigcup_{j=1}^{N} c_j F + b_j, $$
for some collection of $c_j in mathbb{R}$ and $b_jinmathbb{R}^N$. Here, each $c_j$ scales the set $F$, then $b_j$ translates the scaled copy. Both the scalar multiplication and the addition make perfect sense, and the notation is exactly what you suggest.
Slightly more generally, let $V$ be a vector space over a field $k$ (for example, $V = mathbb{R}^n$ is a vector space over $k = mathbb{R}$), let $P subseteq V$, and let $q in k$. Then
$$ qcdot P = qP = { qp : p in P }. $$
Even more generally, if $P$ is any set with a structure which supports some kind of multiplication, and $q$ is any object such that $qp$ makes sense for $pin P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation
$$ mathbb{Z} / pmathbb{Z} qquadtext{(where $p$ is prime)} $$
in a typical undergraduate course in abstract algebra. This is the quotient of $mathbb{Z}$ (the integers) with $pmathbb{Z}$ (multiples of $p$, i.e. the set ${np : ninmathbb{Z} }$).
As an addendum, there are also notions of sums and differences of sets with this kind of notation. If $A$ and $B$ are two subsets of some space in which addition and subtraction are defined, then we can define the Minkowski sum and difference of $A$ and $B$ as
$$ Apm B := { apm b : ain A land bin B }. $$
A similar notation could easily be adopted for a "Minkowski product" or "Minkowski quotient" (though I would be careful with those terms, as they may have other meanings). Indeed, I recently came across a paper which uses the notation
$$ CD := { cd : cin C land din D }, $$
where $C, D subseteq mathbb{R}$. That same paper also uses the notation
$$ 1/mathbb{N} := { tfrac{1}{n} : ninmathbb{N} }, $$
which is consistent with the multiplicative notation.
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
edited Jan 19 at 2:03
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
answered Jan 19 at 0:30
Xander HendersonXander Henderson
14.7k103555
14.7k103555
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
add a comment |
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
$begingroup$
Thanks so much for such a detailed and helpful response!
$endgroup$
– Luke Poeppel
Jan 19 at 1:10
add a comment |
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$begingroup$
Well, if scalar product exists on the elements of the set (e.g. they themselves are scalars, or vectors, or matrices, or scalar valued functions, etc), then your definition [notation] is the natural one we use.
$endgroup$
– Berci
Jan 19 at 0:15
$begingroup$
I have seen the same notation in convex analysis. Here's something: en.wikipedia.org/wiki/Minkowski_functional Note the $lambda K$.
$endgroup$
– lightxbulb
Jan 19 at 0:15