Editing Naive set theory (section)

== Ordered pairs and Cartesian products ==
Intuitively, an '''[[ordered pair]]''' is simply a collection of two objects such that one can be distinguished as the ''first element'' and the other as the ''second element'', and having the fundamental property that, two ordered pairs are equal if and only if their ''first elements'' are equal and their ''second elements'' are equal.

Formally, an ordered pair with '''first coordinate''' ''a'', and '''second coordinate''' ''b'', usually denoted by (''a'', ''b''), can be defined as the set <math>\{\{a\}, \{a, b\}\}.</math>

It follows that, two ordered pairs (''a'',''b'') and (''c'',''d'') are equal if and only if {{math|1=''a'' = ''c''}} and {{math|1=''b'' = ''d''}}.

Alternatively, an ordered pair can be formally thought of as a set {a,b} with a [[total order]].

(The notation (''a'', ''b'') is also used to denote an [[open interval]] on the [[real number line]], but the context should make it clear which meaning is intended. Otherwise, the notation ]''a'', ''b''[ may be used to denote the open interval whereas (''a'', ''b'') is used for the ordered pair).

If ''A'' and ''B'' are sets, then the '''[[Cartesian product]]''' (or simply '''product''') is defined to be:
:{{math|1=''A'' × ''B'' = {{mset|(''a'',''b'') | ''a'' &isin; ''A'' and ''b'' &isin; ''B''}}.}}
That is, {{math|1=''A'' × ''B''}} is the set of all ordered pairs whose first coordinate is an element of ''A'' and whose second coordinate is an element of ''B''.

This definition may be extended to a set {{math|1=''A'' × ''B'' × ''C''}} of ordered triples, and more generally to sets of ordered [[n-tuple]]s for any positive integer ''n''.
It is even possible to define infinite [[Cartesian product]]s, but this requires a more recondite definition of the product.

Cartesian products were first developed by [[René Descartes]] in the context of [[analytic geometry]]. If '''R''' denotes the set of all [[real number]]s, then {{math|1='''R'''<sup>2</sup> := '''R''' × '''R'''}} represents the [[Euclidean plane]] and {{math|1='''R'''<sup>3</sup> := '''R''' × '''R''' × '''R'''}} represents three-dimensional [[Euclidean space]].