Ordered pair

In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as

Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory. This can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of this definition is due to Kuratowski (see below) and his definition was used in the second edition of Bourbaki's Theory of Sets, published in 1970. Even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of Kuratowski in an exercise.

About the same time as Wiener (1914), Felix Hausdorff proposed his definition:

Note that this definition is used even when the first and the second coordinates are identical:

where the component Cartesian products are Kuratowski pairs of sets and where

This renders possible pairs whose projections are proper classes. The Quine–Rosser definition above also admits proper classes as projections. Similarly the triple is defined as a 3-tuple as follows: