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:

\[\begin{array}{rcl}

(x_1)&:=&\{x_1\}

\end{array}\]

We define the **ordered pair** as

\[\begin{array}{rcl}

(x_1,x_2)&:=&\{\{x_1,x_2\},\{x_1\}\}

\end{array}\]

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

\[\begin{array}{rcl}

(x_1,\ldots,x_n)&:=&((x_1,\ldots,x_{n-1}),x_n)\end{array}.\]

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

| | | | | created: 2014-07-05 10:42:08 | modified: 2019-01-19 06:20:36 | by: *bookofproofs*

## 1.**Proposition**: Set-Theoretical Meaning of Ordered Tuples

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