Does there exist an operation or function which takes an operation and a set and returns the set with the operation applied to all of its elements?
I could just define the function $$F\left( n,\circ, \bigcup_i\left\{ x_i \right\}\right)=\bigcup_i\left\{n\circ x_i \right\}$$
but I imagine that there already exists some similar notation. For example, one can scale a vector $$n\langle a,b,c\rangle=\langle na,nb,nc\rangle$$ so why couldn’t one “scale” a set $$n\{a,b,c\}=\{na,nb,nc\}$$ in a similar manner?
I have seen some authors notate this kind of math by simply replacing the argument of a binary relation on two numbers with a set and a number (e.g., $\mathbb{Q}+1$).