Say I have two sets of arbitrary numbers, A and B. How do I notate, for example, multiplying each member of ${A_i}$ by ${B_i}$, to produce set C with each member, $C_i$ equal to ${A_i*B_i}$(not cartesian product). E.g $C\in\{a_1*b_1, a_2*b_2, ..., a_i*b_i\}$.
At first I thought sigma would demonstrate the idea, however at second thought I believe ${\sum A_i*B_i }$ would not imply my intentions per se, rather that I was adding the product of each pair to single value C.
So how exactly would be the best way to go about notating this? Note that taking the product was just an example, I am looking for a general method.
If $A$ and $B$ are two subsets of a ring (some algebraic structure with a multiplication such as the set of integers), then $$A \cdot B = \{a\cdot b\mid a\in A,b\in B\}$$ is the socalled complex product of $A$ and $B$.