I have a simple question, if we have two collection of sets, instead of sets, A and B, how is the intersection of A and B defined, is it defined as
{a intersection b:a a set in A and b a set in B}
thx
$\displaystyle A \cap B$={$\displaystyle P \cap U$,$\displaystyle P \cap V$,...}
So can we define this as
$\displaystyle A \cap B$={$\displaystyle I_{1} \cap I_{2}$;for $\displaystyle I_{1} \in A$ and $\displaystyle I_{2} \in B$}
I have no idea what you mean by that question.
This is a collection of sets $\displaystyle \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$
$\displaystyle J\in\mathcal{A}\cap\mathcal{B}\iff J\in\mathcal{A}\text{ and }J\in\mathcal{B}$.
$\displaystyle \bigcup\limits_{G \in A\;\& \;H \in B} {\left\{ {G \cap H} \right\}} $ is the set of elements which are in the intersection of a set in $\displaystyle \mathcal{A}$ and some set in $\displaystyle \mathcal{B}$.