Intersection of two sets that contain other sets as elements How would the intersection of $A=\{a, b, e, \{a, b, c, d\}, \{d, e\}\}$ and $B=\{a, b, c, f, \{a, d\}, \{d, e\}\}$ be defined? I've searched quite a few books but no luck so far.
 A: The intersection of two sets, say $A$ and $B$, is defined to be the set of elements in both $A$ and $B$. So if $A=\{1,2,3\}$ and $B=\{2,3,6,8\}$, then their intersection, denoted $A\cap B$ is
$$
A\cap B = \{2,3\}.
$$
Note that $2$ and $3$ are the only elements common to both $A$ and $B$.
Now, the only real difference for your particular $A$ and $B$ is that some of the elements are sets. Let's change our example. Say $A=\{1,2,\{3,4\}\}$ and $B=\{2,\{3,4\},\{6,8\}\}$. Note here that $\{3,4\}$ is an element of both $A$ and $B$. So $\{3,4\}$ is in their intersection. However, neither $3$ nor $4$ is an element of $A$ or $B$. Indeed, $3$ and $4$ are elements of the set $\{3,4\}$, which itself is an element of $A$ and $B$. So we see that
$$
A\cap B = \{2,\{3,4\}\}.
$$
See if you can identify the intersection of your sets, and feel free to ask if you need clarification.
A: Just to elaborate on my comment and put it into an answer:
The definition of intersection is always the same.  The intersection of two sets $X$ and $Y$ is the set of all things that are in both $X$ and $Y$.
In other words
$z \in X \cap Y \iff z \in X \wedge z \in Y$.  It may be that $z$ is a set itself, but from $X$'s perspective it is just an element or not.
So in this case we can just look at each element in $A$ and ask if it is in $B$ and if it is, it will be in $A \cap B$.


*

*$a \in A$ and $a \in B$ so $a \in A \cap B$

*$b \in A$ and $b \in B$ so $b \in A \cap B$

*$e \in A$ but $b \notin B$ so $b \notin A \cap B$

*$\{a,b,c,d\} \in A$ but $\{a,b,c,d\} \notin B$ so $\{a,b,c,d\} \notin A \cap B$

*$\{d,e\} \in A$ and $\{d,e\} \in B$ so $\{d,e\} \in A \cap B$


At this point we are done.  We have checked which elements of $A$ are in $B$, it doesn't matter that $B$ has extra elements; for something to be in $A \cap B$ it must be in both.
Thus, we see that 
$A \cap B = \{a,b, \{d,e\}\}$
A: What elements do the sets $A$ and $B$ have in common with each other? Remember that if $X=\{a\}$ and $Y=\{\{a\}\}$, then $X\cap Y=\emptyset$. This is because the set $X$ contains the element $a$ while the set $Y$ contains the element $\{a\}$. Thus the elements $a$ and $\{a\}$ are distinct.
Thus $A\cap B = {\{a,b,\{d,e\}}$. The intersection of the two sets that you have defined has cardinality $3$. Ryan displayed a method for finding the elements $A$ and $B$ have in common with each other. I hope this helps.
