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 $(C \times D)$?
4 Answers
$\begingroup$By definition $(A\times B)\times(C\times D)$ is the set of all ordered pairs of ordered pairs of the form
$$\big\langle\langle a,b\rangle,\langle c,d\rangle\big\rangle\;,$$
where $a\in A,b\in B,c\in C$, and $d\in D$. You’ve described the set of $2\times 2$ matrices of the form
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}\;,$$
where $a\in A,b\in B,c\in C$, and $d\in D$. There is a natural bijection between these two sets given by
$$\big\langle\langle a,b\rangle,\langle c,d\rangle\big\rangle\mapsto\begin{bmatrix}a&b\\c&d\end{bmatrix}\;,$$
but the two sets are not equal, simply because a $2\times 2$ matrix is not an ordered pair of ordered pairs: the set of matrices is not the same thing as the Cartesian product.
$\endgroup$ 8 $\begingroup$Cartesian product of cartesian products is a set of tuples of tuples. $$(A \times B) \times (C \times D) = \left\{((a,b),(c,d))|a\in A, b\in B, c\in C, d\in D\right\}$$
$\endgroup$ $\begingroup$$(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\}$.
$\endgroup$ $\begingroup$The meaning of $(A\times B)\times(C\times D)$ is that of a set of ordered pairs of ... ordered pairs, the first pair with the first element from $A$ and the second from $B$, and the second pair with the first element from $C$ and the second from $D$.
An example would be this: if $A=B=C=D=\mathbb{R}$, then $(A\times B)\times (C\times D)$ can represent the set of all segments on the cartesian plane, each segment being represented as a pair of points, with each point in turn represented as a pair of coordinates: $(x_1,y_1),(x_2,y_2)$.
You can certainly represent it as a matrix as long as all four sets are at most countably infinite - so not in the segments example above! What's really important is not to confuse $((A\times B)\times(C \times D))$ with $A \times B \times C \times D$; the former is a set of pairs of pairs, the second is a set of $4$-uples.
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