# union of collections

• Sep 13th 2011, 12:57 PM
sung
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
• Sep 13th 2011, 01:11 PM
Plato
Re: union of collections
Quote:

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 $\displaystyle \mathcal{A}=\{P,Q,R\}~\&~\mathcal{B}=\{U,V,X,Y\}$ of sets.
Show what you are saying about $\displaystyle \mathcal{A}\cap\mathcal{B}$.
• Sep 13th 2011, 01:24 PM
sung
Re: union of collections
$\displaystyle A \cap B$={$\displaystyle P \cap U$,$\displaystyle P \cap V$,...}

So can we define this as
$\displaystyle A \cap B$={$\displaystyle I_{1} \cap I_{2}$;for $\displaystyle I_{1} \in A$ and $\displaystyle I_{2} \in B$}
• Sep 13th 2011, 01:29 PM
Plato
Re: union of collections
Quote:

Originally Posted by sung
$\displaystyle A \cap B$={$\displaystyle P \cap U$,$\displaystyle P \cap V$,...}
So can we define this as
$\displaystyle A \cap B$={$\displaystyle I_{1} \cap I_{2}$;for $\displaystyle I_{1} \in A$ and $\displaystyle I_{2} \in B$}

So you want a collection of set $\displaystyle \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$
• Sep 13th 2011, 01:41 PM
sung
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?
• Sep 13th 2011, 02:04 PM
Plato
Re: union of collections
Quote:

Originally Posted by Plato
So you want a collection of set $\displaystyle \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$

Quote:

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 $\displaystyle \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}$

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

$\displaystyle \bigcup\limits_{G \in A\;\& \;H \in B} {\left\{ {G \cap H} \right\}}$ is the set of elements which are in the intersection of a set in $\displaystyle \mathcal{A}$ and some set in $\displaystyle \mathcal{B}$.
• Sep 13th 2011, 02:47 PM
emakarov
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.
• Sep 13th 2011, 05:37 PM
Plato
Re: union of collections
Quote:

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
Quote:

Originally Posted by Plato
$\displaystyle J\in\mathcal{A}\cap\mathcal{B}\iff J\in\mathcal{A}\text{ and }J\in\mathcal{B}$.

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.