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

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.

• Maybe $\{ (x^1, l^1), \ldots, (x^N, l^N) \}$ – Mauro ALLEGRANZA Dec 16 '18 at 16:54
That means that $$D$$ is the set of all $$(x^n,l^n)$$, where $$n$$ varies from $$1$$ to $$N$$.
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.