Cartesian product of indexed sets - am I using the notation correctly?

Question

I want to express a set that evolves choosing members from a non-finite collection of sets and I think this should be possible, using the notion of Cartesian products for indexed sets.

Now if I write $I=\{1,2,3\}$ then would this equality be correct:

$$\{x_1+x_2+x_3:(x_1,x_2,x_3)\in X_1\times X_2\times X_3\}=\left\{\sum_{i\in I}f(i):f\in \prod_{i\in I}X_i\right\}$$

Can someone comment on if I am using the notation right? I'm working under the idea that the indexed product should be the set of all choice functions for my index but I'm unsure if thats correct. Also I am using a finite example here just to make things simpler.

Answer 1

An indexed product can be understood as a function, but this needs to be pointed out. It is more easy to understand what you're talking about if you write $f_i$ instead, or clarify the notation first.

Answer 2

You’re answer is correct, although it immediately raises a question in my mind: what do you mean by the sum when $I$ is infinite? If you have some notion of the sum of an infinite number of elements, valid for any choice function $f$, you’re fine.

However, I agree with the other answer stylistically. You use subscripts for the $X_i$’s, so I’d do that also for the $f_i$’s. But from a purely logical standpoint, both notations work.