What is the cartesian product of cartesian products? If I have four sets $A$, $B$, $C$ and $D$, what is the meaning of $(A \times B) \times (C \times D)$?
Can I define it to be a matrix, such that the first index of each element will be given by its order in the bigger cartesian product i.e.$(A \times B) \times (C \times D)$, and the second index will be given by its order within the smaller cartesian product i.e. $(A \times B)$ or $(C \times D)$? 

$(A \times B)\times (C \times D))=\{(u,v): u \in A \times B, v \in C \times D\}=\{((a,b),(c,d)):a \in A, b \in B, c \in C, d \in D\}$.
