Ordered pair
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2.
(Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.)
Alternatively, the objects are called the first and second components, the first and second coordinates, or the left and right projections of the ordered pair.
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs, cf.
The (a, b) notation may be used for other purposes, most notably as denoting open intervals on the real number line.
In such situations, the context will usually make it clear which meaning is intended.
[1][2] For additional clarification, the ordered pair may be denoted by the variant notation
The left and right projection of a pair p is usually denoted by π1(p) and π2(p), or by πℓ(p) and πr(p), respectively.
In contexts where arbitrary n-tuples are considered, πni(t) is a common notation for the i-th component of an n-tuple t. In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as For any two objects a and b, the ordered pair (a, b) is a notation specifying the two objects a and b, in that order.
This "definition" is unsatisfactory because it is only descriptive and is based on an intuitive understanding of order.
However, as is sometimes pointed out, no harm will come from relying on this description and almost everyone thinks of ordered pairs in this manner.
Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property.
This was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954.
However, this approach also has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed.
[3] 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.
[6] Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914:[7]
He observed that this definition made it possible to define the types of Principia Mathematica as sets.
Principia Mathematica had taken types, and hence relations of all arities, as primitive.
About the same time as Wiener (1914), Felix Hausdorff proposed his definition:
"[8] In 1921 Kazimierz Kuratowski offered the now-accepted definition[9][10] of the ordered pair (a, b):
are generalized functions, in the sense that their domains and codomains are proper classes.
Proving that short satisfies the characteristic property requires the Zermelo–Fraenkel set theory axiom of regularity.
[12] Moreover, if one uses von Neumann's set-theoretic construction of the natural numbers, then 2 is defined as the set {0, 1} = {0, {0}}, which is indistinguishable from the pair (0, 0)short.
In type theory and in outgrowths thereof such as the axiomatic set theory NF, the Quine–Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair.
For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998).
[18] Morse–Kelley set theory makes free use of proper classes.
[19] Morse defined the ordered pair so that its projections could be proper classes as well as sets.
He first defined ordered pairs whose projections are sets in Kuratowski's manner.
In this context the characteristic property above is a consequence of the universal property of the product and the fact that elements of a set X can be identified with morphisms from 1 (a one element set) to X.
