Definition:Ordered Pair/Kuratowski Formalization

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Definition

The definition of a set does not take any account of the order in which the elements are listed.

That is, $\set {a, b} = \set {b, a}$, and the elements $a$ and $b$ have the same status - neither is distinguished above the other as being more "important".


The concept of an ordered pair can be formalized by the definition:

$\tuple {a, b} := \set {\set a, \set {a, b} }$

This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory.


Coordinates

Let $\tuple {a, b}$ be an ordered pair.

The following terminology is used:

$a$ is called the first coordinate
$b$ is called the second coordinate.

This definition is compatible with the equivalent definition in the context of Cartesian coordinate systems.


Motivation

The only reason for the Kuratowski formalization of ordered pairs:

$\tuple {a, b} = \set {\set a, \set {a, b} }$

is so their existence can be justified in the strictures of the axiomatic set theory, in particular Zermelo-Fraenkel set theory (ZF).

Once that has been demonstrated, there is no need to invoke it again.

The fact that this formulation allows that:

$\tuple {a, b} = \tuple {c, d} \iff a = c, b = d$

is its stated aim.

The fact that $\set {a, b} \in \tuple {a, b}$ is an unfortunate side-effect brought about by means of the definition.

It would be possible to add another axiom to ZF or ZFC specifically to allow for ordered pairs to be defined, and in some systems of axiomatic set theory this is what is done.


Warning

The weakness of the Kuratowski formalization of the ordered pair shows up when $a = b$:

\(\ds \tuple {a, a}\) \(=\) \(\ds \set {\set a, \set {a, a} }\) Definition of Kuratowski Formalization of Ordered Pair
\(\ds \) \(=\) \(\ds \set {\set a, \set a}\) Definition of Uniqueness of Set Elements
\(\ds \) \(=\) \(\ds \set {\set a}\) Definition of Uniqueness of Set Elements

Thus the ordered pair degenerates into the set $\set {\set a}$.

Most works on this subject gloss over this point, and indeed, completely fail to mention it.


Notation

In the field of symbolic logic and modern treatments of set theory, the notation $\sequence {a, b}$ is often seen to denote an ordered pair.

In sources where the possibility of confusion is only minor, one can encounter $a \times b$ for $\tuple {a, b}$ on an ad hoc basis.

These notations are not used on $\mathsf{Pr} \infty \mathsf{fWiki}$, where $\tuple {a, b}$ is used exclusively.


Also see


Source of Name

This entry was named for Kazimierz Kuratowski.


Historical Note

The Kuratowski formalization of the concept of the ordered pair was established in $1914$ in collaboration with Norbert Wiener.


Sources

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