Scalar Multiplication of a Set 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.
 A: 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 $b_j\in\mathbb{R}^N$.  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
$$ q\cdot P = qP = \{ qp : p \in P \}. $$
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 $p\in P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation
$$ \mathbb{Z} / p\mathbb{Z} \qquad\text{(where $p$ is prime)} $$
in a typical undergraduate course in abstract algebra.  This is the quotient of $\mathbb{Z}$ (the integers) with $p\mathbb{Z}$ (multiples of $p$, i.e. the set $\{np : n\in\mathbb{Z} \}$).

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
$$ A\pm B := \{ a\pm b : a\in A \land b\in B \}. $$
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
$$ CD := \{ cd : c\in C \land d\in D \}, $$
where $C, D \subseteq \mathbb{R}$.  That same paper also uses the notation
$$ 1/\mathbb{N} := \{ \tfrac{1}{n} : n\in\mathbb{N} \}, $$
which is consistent with the multiplicative notation.
