# Definition:Ordered Pair/Kuratowski Formalization

## Definition

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

That is, $\left\{{a, b}\right\} = \left\{{b, a}\right\}$, 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:

- $\left({a, b}\right) = \left\{{\left\{{a}\right\}, \left\{{a, b}\right\}}\right\}$

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

### Coordinates

Let $\left({a, b}\right)$ 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 coordinates.

## Motivation

The only reason for the Kuratowski formalization of ordered pairs:

- $\left({a, b}\right) = \left\{{\left\{{a}\right\}, \left\{{a, b}\right\}}\right\}$

is so their existence can be justified in the strictures of the axiomatic set theory, in particular Zermelo-Fraenkel set theory. Once that has been demonstrated, there is no need to invoke it again.

The fact that this formulation allows that:

- $\left({a, b}\right) = \left({c, d}\right) \iff a = c, b = d$

is its stated aim.

The fact that $\left\{{a, b}\right\} \in \left({a, b}\right)$ 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 $a = b$:

\(\displaystyle \left({a, a}\right)\) | \(=\) | \(\displaystyle \left\{ {\left\{ {a}\right\}, \left\{ {a, a}\right\} }\right\}\) | $\quad$ Definition of Kuratowski Formalization of Ordered Pair | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {\left\{ {a}\right\}, \left\{ {a}\right\} }\right\}\) | $\quad$ Definition of Uniqueness of Set Elements | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {\left\{ {a}\right\} }\right\}\) | $\quad$ Definition of Uniqueness of Set Elements | $\quad$ |

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

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

## Also see

## Source of Name

This entry was named for Kazimierz Kuratowski.

This formalization was established in collaboration with Norbert Wiener.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 6$: Ordered Pairs - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): Exercise $1.2: \ 12$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 3$ - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.4$: Sets of Sets - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $9$: Definition $1.2$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Ordered Pairs