The set-theoretical meaning of ordered tuples shows that being an $n$-tuple is a predicate of a logical calculus. Therefore, we can use the axiom of separation to define what today's mathematicians call the Cartesian product. RenĂ© Descartes (1596 - 1650) invented this concept in the 17th century, long before the set theory was born in its strict form but with our new set-theoretical tools we can provide a foundation for this concept:
Let \(A_1\ldots A_n\) be sets. Using the set-theoretical meaning of ordered tuples and the axiom of separation we define the set \[A_1\times A_2\times\cdots\times A_n :=\{(x_1,\ldots,x_n)\mid x_i\in A_i, 1\le i\le n\}.\] as the Cartesian product of the sets \(A_1\ldots A_n\). In other words, the Cartesian product is the set of ordered \(n\)-tuples \((a_1,a_2,\ldots,a_n)\), such that \(a_i\in A_i\) for $i=1,\ldots,n.$^{1}.
If all of the \(A_i\) denote the same set \(A\), then we use the short form
\[A^n:=\underbrace{A\times A\times\cdots\times A}_{n\text{ times}}.\]
Corollaries: 1
Definitions: 2 3 4 5 6 7
Examples: 8
Explanations: 9
Motivations: 10 11
Parts: 12
Proofs: 13 14 15 16 17 18
Propositions: 19 20
Please note that, in general, the Cartesian product is not commutative, i.e. \(A\times B\neq B\times A\), because of the order or the elements would change, in contradiction to the definition of ordered tuples. ↩