Relation between $ \bigcap_{i \in I}A_i $ and $ \bigcap_{i=1}^{n}A_i $

Question

Let I be a nonempty set and a family of sets such that every element of the family is a subset of U.

$\mathcal F = \{A_i | i \in I\}$

I understand the meaning of this operation:

$$ \bigcap_{i \in I}A_i $$

That's the intersection of all the elements of the sets of the family $\mathcal F$.

But I don't truly understand what this other operation means and what's the relation between the above one. $$ \bigcap_{i=1}^{n}A_i $$

I understand that this operation is also an intersection but what I am trying to understand is the relation between $\bigcap_{i \in I}A_i$ and $\bigcap_{i=1}^{n}A_i$

$$ \bigcap_{i=1}^{n}A_i \subseteq\bigcap_{i \in I}A_i $$

Is the relation above true?

Answer 1

The notation $$\bigcap _{i=1}^nA_i$$ denotes the same as $$ \bigcap_{i\in I}A_i$$ for the special case $I=\{1,2,\ldots, n\}$.