Question about Set within Set I got the following sets:
A= { { 1 , 3 , 5 , 7 , 9 } , { 1 , 2 , 3 } , 1 , 2 , 3 , 4 , 5 }
B= { 1 , 3 , 5 , 7 , 9 }
I'm a bit confused. What would A ∩ B be?  
 A: $A$ is an unusual set in that some of its elements are numbers and others are sets of numbers. This kind of sets are rarely used in mathematics -- except sometimes in exercises that confuse beginning students ...
$\{1,3,5,7,9\}$ is one of the elements of $A$. This set is not an element of $B$ -- it happens to be $B$ itself, but that doesn't make it an element of $B$ -- and therefore it is not an element of $A\cap B$.
$\{1,2,3\}$ is another element of $A$ that is not an element of $B$; it is not an element of $A\cap B$ either.
On the other hand $7$ is not an element of $A$. It's an element of an element of $A$, but that's a different thing.
A: When sets are elements, they are elements.  The elements within them are irrelevant.
We know $B = \{1,3,5,7,9\}$.  Let $L = \{1,2,3\}$.
Then $A =\{\{1,3,5,7,9\}, \{1,2,3\},1,2,3,4,5\}= \{B,L,1,2,3,4,5\}$
So what is $A\cap B$?  It is, as always, the elements that $A$ and $B$ common.
The first element of $A$, is the  element $B$.  Is $B$ an element of $B$?  No, it is not.  The elements of $B$ are $1,3,5,7,9$ and not a single one of them is the same thing as $B$.
That's all we are concerned with.  We don't need to think  about "well, the elements of $B$ are within $B$ so...".  $B$, the entire set itself, is the first element.  Is the entire set itself an actual element of $B$?  No, it is not.  That's it.  $B \not \in B$.  Period.
The  second element of $A$, is the  element $L$.  Is $L$ an element of $B$?  No, it is not.  The elements of $B$ are $1,3,5,7,9$ and not a single one of them is the same thing as $L$.
The  third element of $A$, is the  element $1$.  Is $1$ an element of $B$? Yees, it is.  The elements of $B$ are $1,3,5,7,9$ and one of them is the same thing as $1$.  So $1\in A\cap B$
The  fourth element of $A$, is the  element $2$.  Is $2$ an element of $B$?  No, it is not.  The elements of $B$ are $1,3,5,7,9$ and not a single one of them is the same thing as $2$.
The  fifth element of $A$, is the  element $3$.  Is $3$ an element of $B$? Yees, it is.  The elements of $B$ are $1,3,5,7,9$ and one of them is the same thing as $3$.  So $3\in A\cap B$
The  sixth element of $A$, is the  element $4$.  Is $4$ an element of $B$?  No, it is not.  The elements of $B$ are $1,3,5,7,9$ and not a single one of them is the same thing as $4$.
The  last element of $A$, is the  element $5$.  Is $5$ an element of $B$? Yees, it is.  The elements of $B$ are $1,3,5,7,9$ and one of them is the same thing as $5$.  So $5\in A\cap B$
So $A\cap B =\{1,3,5\}$
A: To answer your question, $A\cap B=\{1,3,5\} $.
And, for extra credit, appreciate the distinction between these two true statements:
$$\{1,2,3,4,5\}\subset A $$
$$\{1,3,5,7,9\}\in A $$
