Question about the union of an empty set I would like some clarification regarding the following text from a textbook:

There is no problem with these definitions if one of the elements of $\mathscr{A}$ happens to be the empty set. But it is a bit tricky to decide what (if anything) these definitions mean if we allow $\mathscr{A}$ to be the empty collection. Applying the definitions literally, we see that no element of $x$ satisfies the defining property for the union of the elements of $\mathscr{A}$. So it is reasonable to say that $$ \bigcup_{A \in \mathscr{A}}A=\emptyset$$
If $\mathscr{A}$ is empty. On the other hand, every $x$ satisfies (vacuously) the defining property for the intersection of the elements of $\mathscr{A}$.

I wanted to know why we don't say that every $x$ vacuously satisfies the defining property for the union of the elements of $\mathscr{A}$? Does it just come down to convention?
 A: Where's the witness?  It can be helpful to think of these definitions asserting the existence (or absence) of a particular object which satisfies (i.e., is a witness for) some property.
For the union of the empty collection, for $x$ to be in the union, there must be an $A \in \mathscr{A}$ with $x \in A$.  That is, $A$ is a witness of the membership of $x$ in the union.  Since there is no candidate $A$ in $\mathscr{A}$, there is no witness to the inclusion of $x$ in the union.  Consequently, $x$ is not in the union.
For the intersection of the empty collection, an element $x$ is in the intersection if $x$ is in every member of $\mathscr{A}$.  This means that a single $A \in \mathscr{A}$ such that $x \not\in A$ is a witness to $x$ not in the intersection.  However, there is no candidate $A$ in $\mathscr{A}$ to act as a witness of the absence of $x$ in the intersection.  Consequently, $x$ must be in the intersection.
A: $x\in\bigcup\mathscr{A}$ means you have to actually exhibit an $A\in\mathscr{A}$ such that $x\in A$.
On the other hand, $x\in\bigcap\mathscr{A}$ means $x\in A$ for every $A\in\mathscr{A}$, so this is vacuous.
