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Let (,) and (,) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is: (,) = (,) = =.. The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B.
By 1914 Norbert Wiener, using Whitehead and Russell's symbolism, eliminated axiom *12.11 (the "two-variable" (relational) version of the axiom of reducibility) by expressing a relation as an ordered pair using the null set. At approximately the same time, Hausdorff (1914, p. 32) gave the definition of the ordered pair (a, b) as {{a,1}, {b, 2
An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes , [ 5 ] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product .
Kazimierz Kuratowski (Polish pronunciation: [kaˈʑimjɛʂ kuraˈtɔfskʲi]; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician.He was one of the leading representatives of the Warsaw School of Mathematics.
2. Kripke–Platek set theory consists roughly of the predicative parts of set theory Kuratowski 1. Kazimierz Kuratowski 2. A Kuratowski ordered pair is a definition of an ordered pair using only set theoretical concepts, specifically, the ordered pair (a, b) is defined as the set {{a}, {a, b}}. 3.
In NFU, these two definitions have a technical disadvantage: the Kuratowski ordered pair is two types higher than its projections, while the Wiener ordered pair is three types higher. It is common to postulate the existence of a type-level ordered pair (a pair (,) which is the same type as its projections) in NFU. It is convenient to use the ...
Using Kuratowski's representation for an ordered pair, the second definition above can be reformulated in terms of pure set theory: The 0-tuple (i.e. the empty tuple) is represented by the empty set ∅ {\displaystyle \emptyset } ;
In the case of the Kuratowski ordered pair, the existence and uniqueness of the values of the projection functions can be proven a couple ways, such as by a messy explicit intersection like this one, or by application of the axiom of extensionality to the set representing the pair.