# Operate on all the elements of a set simultaneously

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., $\mathbb{Q}+1$).

• If $A$ is the algebraic thing and $a \in A$ with $B \subseteq A$ you often see something like $a*B$ for $\{a*b: b \in B\}$. Replace the star by $+$, juxtaposition, $\circ$, etc. for your needs. Sep 2, 2017 at 22:34
• Beware of drawing bad conclusions like $|B| = |a*B|$ without actually proving them. In your example, what if $na=nb$? Then that set actually has (at most) two elements. Sep 2, 2017 at 22:37

• So sorry—I meant without elsewhere defining $f$. Sep 3, 2017 at 6:08
• Or is it okay to just say something like $2+\{1,2,3\}$? Sep 3, 2017 at 6:09