How should superscript N and subscript n=1 be thought of in set theory?

Question

I'm learning set theory while reading a research paper and they use $$D = {( x^n, l^n)}^N_{n=1}$$

How should this be read? I'm know it would indicate ${( x<^1, l^1)}$ But with the $N$ being a capital $N$ I'm not entirely sure what that would represent.

Thanks for your help!

Answer 1

That means that $D$ is the set of all $(x^n,l^n)$, where $n$ varies from $1$ to $N$.

Answer 2

That index notation whether in the form $\sum_{n=1}^Na_n$ or $\prod_{n=1}^Na_n$ or $\{a_n\}_{n=1}^N a_n$ usually means to evaluate for $a_1,a_2,....$ upto $a_{N-2},a_{N-1},a_N$.

So $\{(x^n,l^n)\}_{n=1}^N$ probably (but might not) means $\{(x^1, l^1),(x^2,l^2),......,(x^N, l^N)\}$

At least that is my guess. Is $N$ used as a constant value elsewhere? Does the $N$ look like the symbol for the natural numbers, $\mathbb N$? It might mean something else but I doubt it.