Editing Ordered pair (section)

==Generalities==
Let <math>(a_1, b_1)</math> and <math>(a_2, b_2)</math> be ordered pairs. Then the ''characteristic'' (or ''defining'') ''property'' of the ordered pair is:
<math display="block">(a_1, b_1) = (a_2, b_2) \text{  if and only if  } a_1 = a_2 \text{ and } b_1 = b_2.</math>

The [[Set (mathematics)|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''. A [[binary relation]] between sets ''A'' and ''B'' is a [[subset]] of ''A'' × ''B''.

The {{math|(''a'', ''b'')}} notation may be used for other purposes, most notably as denoting [[open interval]]s on the [[real number line]]. In such situations, the context will usually make it clear which meaning is intended.<ref>{{citation | first=Steven R.|last=Lay | title=Analysis / With an Introduction to Proof | edition=4th | publisher=Pearson / Prentice Hall | isbn=978-0-13-148101-5 | year=2005 | page=50}}</ref><ref>{{citation | first=Keith|last=Devlin | title=Sets, Functions and Logic / An Introduction to Abstract Mathematics| edition=3rd | publisher=Chapman & Hall / CRC |year=2004 | isbn=978-1-58488-449-1 |page=79}}</ref> For additional clarification, the ordered pair may be denoted by the variant notation <math display="inline"> \langle a, b\rangle</math>, but this notation also has other uses.

The left and right {{vanchor|projection}} of a pair ''p'' is usually denoted by {{pi}}<sub>1</sub>(''p'') and {{pi}}<sub>2</sub>(''p''), or by {{pi}}<sub>''ℓ''</sub>(''p'') and {{pi}}<sub>''r''</sub>(''p''), respectively.
In contexts where arbitrary ''n''-tuples are considered, {{pi}}{{su|p=''n''|b=''i''}}(''t'') is a common notation for the ''i''-th component of an ''n''-tuple ''t''.