1. ## union of collections

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

2. ## Re: union of collections

Originally Posted by sung
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}
From that description it is very hard to understand your question.
Say you have two collections $\mathcal{A}=\{P,Q,R\}~\&~\mathcal{B}=\{U,V,X,Y\}$ of sets.
Show what you are saying about $\mathcal{A}\cap\mathcal{B}$.

3. ## Re: union of collections

$A \cap B$={ $P \cap U$, $P \cap V$,...}

So can we define this as
$A \cap B$={ $I_{1} \cap I_{2}$;for $I_{1} \in A$ and $I_{2} \in B$}

4. ## Re: union of collections

Originally Posted by sung
$A \cap B$={ $P \cap U$, $P \cap V$,...}
So can we define this as
$A \cap B$={ $I_{1} \cap I_{2}$;for $I_{1} \in A$ and $I_{2} \in B$}
So you want a collection of set $\{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$

5. ## Re: union of collections

yes is that the correct definition for the intersection of two collections?
So how is the intersection of two collections defined?

6. ## Re: union of collections

Originally Posted by Plato
So you want a collection of set $\{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$
Originally Posted by sung
So how is the intersection of two collections defined?
I have no idea what you mean by that question.
This is a collection of sets $\{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$

$J\in\mathcal{A}\cap\mathcal{B}\iff J\in\mathcal{A}\text{ and }J\in\mathcal{B}$.

$\bigcup\limits_{G \in A\;\& \;H \in B} {\left\{ {G \cap H} \right\}}$ is the set of elements which are in the intersection of a set in $\mathcal{A}$ and some set in $\mathcal{B}$.

7. ## Re: union of collections

The definition of the intersection of two sets is the same regardless of the nature of the elements of those two sets: whether they are numbers, other sets, etc.

8. ## Re: union of collections

Originally Posted by emakarov
The definition of the intersection of two sets is the same regardless of the nature of the elements of those two sets: whether they are numbers, other sets, etc.
While I agree that is the case and said so when I posted
Originally Posted by Plato
$J\in\mathcal{A}\cap\mathcal{B}\iff J\in\mathcal{A}\text{ and }J\in\mathcal{B}$.
I think that the OP had something more in mind.
Such as a generalize intersection.
That is why I asked for clarification.
But I did not really get what I thought I would.