Explain why (A × B) × (C × D) and A × (B × C) × D are not the same. Explain why $(A × B) × (C × D)$ and $A × (B × C) × D$ are not the same.
My thinking is that you need to use the cardinality in some way to show that they have different cardinality, thus the sets are not the same. If this is correct, how could I solve it this way?
 A: You wouldn't use cardinality because they have the same cardinality. 
In general..
What does an element of $(A \times B) \times(C \times D)$ look like?
It looks like $((a,b),(c,d))$.
What does an element of $A \times (B \times C) \times D$ look like?
It looks like $(a,(b,c),d)$. 
It actually might look a little different depending on how your book defines multiple direct products.
My suggestion is using your book's definition of a tuple, determine what is the actual set $((a,b),(c,d))$ vs the actual set $(a,(b,c),d)$.
Or specifically...
You can actually construct a pretty easy counter-example with $A=B=C=D=\{\{\}\}$. 
 $(A \times B) \times(C \times D)$ and $A \times (B \times C) \times D$ both only have one element: $((\{\},\{\}),(\{\},\{\}))$ and $(\{\},(\{\},\{\}),\{\})$ respectively. So when you show those two elements aren't equal, then you've immediately shown  $(A \times B) \times(C \times D)$ is not equal to $A \times (B \times C) \times D$.
A: No, this sets are of equal cardinality.
Use the definition of cartesian product of two sets (the set of all ordered pairs), and show that these two sets contain elements of different 'types' (something like 'pairs of type (A,B)' vs 'pairs of type (pairs of type (B,C), pairs of type (A, D))').
Actually, although these sets are not strictly equal, there always exists a natural bijection between them.
A: an element of $(A\times B)\times(C\times D)$ is of the form
$$ \Big((a,b),(c,d)\Big) $$
Because "$\times$" is a binary operation on sets, $A\times(B\times C)\times D$ doesn't make sense UNLESS you defined it as ordered triples; in which case an element is of the form
$$\Big(a,(b,c),d\Big)$$
There is a one-to-one onto function from one set to the other however. Namely
$$ \Big((a,b),(c,d)\Big)\mapsto \Big(a,(b,c),d\Big) $$
Recall $(a,b)$ is notation for $\{\{a\},\{a,b\}\}$.
Following Endertons' Elements of Set Theory textbook, $n$-tuples are defined as follows
$$ (x,y,z)\overset{\mathrm{def}}{=} ((x,y),z) $$
$$ (x,y,z,u)\overset{\mathrm{def}}{=} ((x,y,z),u) $$
$$\vdots$$
