Math Proofs: Definitions of Intersection and Union
Solution 1:
$x\in\bigcup F$ means there is some $A\in F$ for which $x\in A.$
"Some" means at least one, but possibly more than one.
$x\in\bigcap F$ means for every $A\in F,$ $x\in A.$
That is how this notation is used in set theory, but most mathematicians write $\displaystyle \bigcup_{A\,\in\,F} A$ or a similar thing rather than $\bigcup F.$ (And similarly for intersections.)