Some context: I'm working through Velleman's $\textit{How to Prove it}$ and am currently at section 2.3.5.

I'm having some trouble grasping the definitions of $\cap\mathcal{F}$ and $\cup\mathcal{F}$. On an intuitive level, it's fairly obvious what each one means:

* The intersection of a family of sets are the elements which are common to all the sets in $\mathcal{F}$.
* The union of a family of sets are all the elements which make up the sets in $\mathcal{F}$.

This intuitive understanding doesn't seem to match up with the formal definitions provided in the book:

* $\cap\mathcal{F}=\{x|\forall x \in \mathcal{F}(x \in A)\}=\{x|\forall A(A \in \mathcal{F} \to x \in A)\}$
* $\cup\mathcal{F}=\{x|\exists A \in \mathcal{F}(x \in A)\}=\{x|\exists A(A \in \mathcal{F} \land x \in A)\}$

Here's how I'm parsing (obviously incorrectly) each statement in english:

* $\cap\mathcal{F}=$ "For every set A, if A is in $\mathcal{F}$ then the element x is in A?" This would make more sense to me if it was something along the lines of $(A \in \mathcal{F} \land x \in A)$... i.e "every set A is in $\mathcal{F}$ and a given element x is in each set A."
* $\cup\mathcal{F}=$ "For some set A, A is in $\mathcal{F}$ and the element x is in that set A." Again, this is doesn't seem to match up with my intuitive understanding of the union concept.

I would appreciate some guidance on how to correctly interpret these formal definitions! Thanks in advance.