Cartesian Product in mathematics

When using the cartesian product, and you have three collections, do you take two collections at a time OR all three together to calculate the product?

My question is if you have more than two collections, let's say A, B and C

A = {1,2,3}
B = {4,5,6}
C = {7,8}

A x B x C
{1,2,3} x {4,5,6} x {7,8}


Do you with the cartesian product calculate A x B, then B x C? And maybe A x C? Which means you take only two collections at a time.

OR

Do you take all three collections at the same time A x B x C?

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A X B X C = {(1,4,7), (1,4,8), ... , (3,6,8)} – Alin Purcaru Jan 30 '12 at 15:56
A Cartesian Product is usually defined as a product of two sets, but when you are going to find the n-ary product, the product set will contain all possible triplets. – 0605002 Jan 30 '12 at 15:58
Do you want the fully formal answer, or just a simple, easy-to-us answer? – Arturo Magidin Jan 30 '12 at 16:44
In the usual formal buildup, it is in principle two at a time. However, let $I_n$ be a fixed $n$-element set, say the set $\{1,2, \dots,n\}$. We could define the $n$-fold Cartesian product $A_1\times\cdots\times A_n$ as the set of all functions $f$ from $I_n$ to $\cup A_i$ such that $f(i)\in A_i$ for $i=1$ to $n$. – André Nicolas Jan 30 '12 at 17:20

For $n \in \mathbb{N}$, the $n$-ary Cartesian product of $n$ sets $A_1, \dots, A_n$, denoted $A_1 \times \cdots \times A_n$, is defined to be the set of all $n$-tuples $(a_1, \dots, a_n)$ for which $a_i \in A_i$ for each $i$.

So in particular

$$A \times B \times C = \{ (a,b,c)\, :\, a \in A,\ b \in B,\ c \in C \}$$

This is distinct from

$$(A \times B) \times C = \{ ((a,b),c)\, :\, a \in A,\ b \in B,\ c \in C \}$$

each of whose elements is an ordered pair, the first 'coordinate' of which is itself an ordered pair.

Nonetheless, there is a very natural bijection

\begin{align} A \times B \times C & \to (A \times B) \times C \\ (a,b,c) &\mapsto ((a,b),c) \end{align}

and similarly for $A \times (B \times C)$.

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The cartesian product is an operation defined on two sets. Given the sets A and B the product A x B is not equal to the product B x A. So you will have to use two sets at a time and you will need to define an order, you want to apply the operation in, since (A x B) x C is not equal to A x (B x C).

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all three sets at the same time.

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