Loop Bounds vs. Iteration Domain in Polyhedral optimization

Question

Context:

I was reading a tutorial on polyhedral optimization. But got confused while trying to translate the iteration domain (i.e. loop bound) to set builder notation.

Problem Description:

A code snippet from the article;

for (int i = 0; i <= 2*N; ++i)
    S: Z[i] = 0.;

The iteration domain for the statement 'S' is described with the set-builder notation as

{S[i] : 0 ≤ i ≤ N}. But the code says that i should go up to or equal to 2*N.

Example Case:

Let’s consider the code given above. And also assume N = 5

Then this loop will iterate from i = 0 to 10 (where, 2xN = 2x5 = 10).

But from the set builder notation, we can see that the iteration domain should be in the range of

0 <= i <= N, i.e, 0 <= i <= 5.

But considering the real code snippet,

for (int i = 0; i <= 2*N; ++i)
    S: Z[i] = 0;

i go up to 10.

So my doubts are…

1. Then should I write the set builder as {S[i] : 0 ≤ i ≤ 2N} or {S[i] : 0 ≤ i ≤ N}?
2. For now, even if both of them are correct; later are there any implications/issues that would arise? That I should be concerned on now? I am asking, because I have seen, that those notations are heavily used in forming systems of equations (To solve what, I don’t know yet )

Screenshots are attached from the tutorial for quick review. Thank you.