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In order to define what it means to have a relation between sets elements, we have to provide a strict set-theoretical idea of putting the elements in a specific order. For instance, if we want to relate two elements with each other, we want to speak about “the first” (or “the left”) and “the second” (or “the right”) element. This leads us to the concept of an ordered pair. A strict, purely set-theoretical definition was first proposed by Kazimierz Kuratowski:

Definition: Ordered Pair, n-Tuple

Let \(X\) be a set and for $n\ge 1$, $i=1,\ldots n$ let \(x_i\in X\) be some of its elements.

We set for $n=1$ the tuple $(x_1)$ to be the singleton:

and in general, for $n > 2$ we set recursively the ordered \(n\)-tuple (or just \(n\)-tuple) $(x_1,\ldots,x_n)$ recursively as the ordered pair of $(x_1,\ldots,x_{n-1})$ and $x_n, $ formally

We denote \(x_i\) as the \(i\)-th element of the $n$-tuple.

Notes

Occasionally, $n$-tuples are also called lists.

Sometimes, objects denoted by $(x_1,x_2,\ldots)$ might be said to have infinite length. By definition, an $n$-tuple has a finite length and such objects are not $n$-tuples.

$n$-tuples differ from sets, since order matters and repetitions have meaning, while in sets, order and repetition are not important.