| context | $ X,Y $ |
| definiendum | $ \langle X,Y \rangle \equiv \{\{X\},\{X,Y\}\} $ |
The ordered pair is a model of
| $ \langle x,y \rangle = \langle z,u \rangle\ \Leftrightarrow\ x=z \land y=u $ |
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See also Wikipedia: Ordered pair, Kuratowski defintion. In calculations, only this property of it should be used.
Now Let $x_i$ be indexed sets and define $p \equiv \langle x_1,x_2\rangle$.
Canonical projection: Using arbitrary union and arbitrary intersection, we can extract the first and second component of the pair:
| $\pi_1(p) \equiv p_1 \equiv \bigcup\bigcap p $ |
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| $\pi_2(p) \equiv p_2 \equiv \bigcup\{y \in \bigcup p \mid \bigcup p \not= \bigcap p \implies y \notin \bigcap p \} $ |
These expressions highly depend on the chosen model. But it's clear that any proper theory of the ordered pair should enable us to project out it's components, and so we will not usually come back to this representation of $\pi_1,\pi_2$.
For the hereby defined n-tuples, we write
| $ \langle x_1,x_2,x_3\rangle \equiv \langle \langle x_1,x_2\rangle,x_3\rangle $ |
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| $ \langle x_1,x_2,x_3,x_4\rangle \equiv \langle \langle \langle x_1,x_2\rangle,x_3\rangle,x_4\rangle $ |
| $ \langle x_1,x_2,x_3,x_4,x_5\rangle \equiv \ \dots$ |
This induces the n'th projection $\pi_n$ alla $\pi_3(\langle x_1,x_2,x_3,x_4,x_5\rangle)=\pi_2(\pi_1(\pi_1(\langle x_1,x_2,x_3,x_4,x_5\rangle)))$.
There are bijections between $\mathbb N$ and ${\mathbb N}^2$ and so one can encode pairs of numbers as numbers.
Wikipedia: Ordered pair