How to define a function with summation? Suppose we have two sets: $A=\{a_1,a_2,a_3\}$ and $B=\{b_1,b_2,b_3\}$.
Is there a way to define a function that simply adds/subtracts the elements of these two sets? For example,
$$\mu(\cdot)=\sum_{i\in\mathbb{N}:\;a_i\in A}a_i+\sum_{j\in\mathbb{N}:\;b_j\in B}b_j.$$
I am not sure if I am allowed to say that $\mu(\cdot)$ is a function with domain $A\times B$.
Basically, my question is about rigorously defining a function that simply adds elements from two different sets. Is that possible? If yes, then how?

You can define $\mu$ as a function with two variables (both finite subsets of a set of numbers) as
$$\mu(A,B) :=\sum_{a\in A}a\, +\,\sum_{b\in B}b$$
