Unions and Intersections Will I be correct in stating the following?

The intersection of a collection of sets need not be a subset of its union.

This is what I’m thinking while positing the above: $\bigcup \varnothing =\varnothing,$ but $\bigcap \varnothing$ is not even a set.

But I think that my proposition will be false for a nonempty collection of sets, correct?
 A: Facts first:


*

*If $A\ne \varnothing$, then $\bigcup A$ and $\bigcap A$ are both sets and $\bigcap A\subseteq \bigcup A$.

*If $A =\varnothing$, then $\bigcup A=\varnothing$, and $\bigcap A$ does not exist (as a set).


Whether your statement is a reasonable expression of this state of facts is more a question of language and communication, than one of hard mathematical truth. You certainly have a defensible position that it is not wrong. But still it sounds pretty odd, because in order to be right, it has to be speaking of "the" intersection of a collection of sets in a situation where there is no such intersection at all.
Other than winning bar bets because the claim tricks the listener into not considering the empty collection, I have doubts that your claim serves a useful communicative purpose, standing alone.
We could imagine uttering that claim while trying to develop an automatic proof verification system, where someone had implemented a rule that $\bigcap A\subseteq \bigcup A$ for general $A$. But then it would be much better communication to actually point at the concrete problem for $A=\varnothing$ than to merely deny that the rule is valid.
In any case, this is not a problem that is specific to set theory. We could get mostly the same discussion out of considering a statement in arithmetics such as

$\frac 1 x \cdot x$ is not necessarily $1$.

A: If we're talking about the subsets of a fixed "universal" set (these form a kind of structure called a complete atomic Boolean algebra), then every collection of sets has a union and an intersection, and the intersection is contained in the union, except when the collection is empty.
More generally, in a complete lattice, every set has a supremum and an infimum, and $\inf X\le\sup X$ holds except when $X=\emptyset$. For example, every set of real numbers has a supremum and an infimum in the extended real line, and $\inf X\le\sup X$ when $X\ne\emptyset$, but $\sup\emptyset=-\infty$ and $\inf\emptyset=+\infty$.
