a set family indexed by an empty set

Suppose we have a set family
A = { A_1, A_2, ... A_n }, and i \in \emptyset. Consider the union and
intersection of A indexed by i.

\bigcup_i A

and

\bigcap_i A

The textbook says that (quote) "if you read the definitions of union
and intersection carefully, you'll see that"

\bigcup_i A = \emptyset
whereas
\bigcap_i A = A

I'm a bit stumbled here, and try to find the way to reason. So far I
got something like that.
(i) union over A: to belong to the resulting set, a member m must
belong to *at least* one A_i. So we have start checking whether m is
A_1, if yes, we stop, put m into resulting set and proceed with next m.
But since A is indexed by i \in \emptyset, we effectively fail on the
very first check. So non of the members m meet get deposited into the
resulting set, and so the resulting set is empty.

(ii) intersection over A: to belong to the resulting set, a member m
must belong to *all* sets in A. So in principle we have to check for
each set. However since A is indexed by i \in emptyset, we, like, can
say, "yes, m is in every set of this indexed family --- the family is
empty, hence we don't even have to check anything." and deposit it into
resulting set, and this holds for every m. So the resulting set is A.

And yet somehow I'm not entirely convinced in my logic, especially in
case of (ii), so I would appreciate if anyone provide some kind of
convincing line of reasoning on both results. TIA.

.

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