View source for Ordered pair

In [[mathematics]], an '''ordered pair''' (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''.  (In contrast, the [[unordered pair]] {''a'', ''b''} equals the unordered pair {''b'', ''a''}.)

Ordered pairs are also called [[tuple|2-tuples]], or [[sequence]]s (sometimes, lists in a computer science context) of length 2; ordered pairs of [[scalar (mathematics)|scalars]] are also called 2-dimensional [[Vector (mathematics and physics)|vectors]].
The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered [[n-tuple|''n''-tuple]]s (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another.

In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ''second entry'' of the pair. 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 product]]s and [[binary relation]]s (and hence [[function (mathematics)|functions]]) are defined in terms of ordered pairs.

==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>(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.

<span id="projection"></span>
The left and right 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''.

==Informal and formal definitions==

In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as
<blockquote>
For any two objects {{mvar|a}} and {{mvar|b}}, the ordered pair {{math|(''a'', ''b'')}} is a notation specifying the two objects {{mvar|a}} and {{mvar|b}}, in that order.<ref name=Wolf>{{citation|first=Robert S.|last=Wolf|title=Proof, Logic, and Conjecture / The Mathematician's Toolbox|publisher=W. H. Freeman and Co.|year=1998|isbn=978-0-7167-3050-7|page=164}}</ref>
</blockquote>
This is usually followed by a comparison to a set of two elements; pointing out that in a set {{mvar|a}} and {{mvar|b}} must be different, but in an ordered pair they may be equal and that while the order of listing the elements of a set doesn't matter, in an ordered pair changing the order of distinct entries changes the ordered pair.

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.<ref>{{citation|first1=Peter|last1=Fletcher|first2=C. Wayne|last2=Patty|title=Foundations of Higher Mathematics|publisher=PWS-Kent|year=1988|isbn=0-87150-164-3|page=80}}</ref>

A more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics. 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 [[Nicolas Bourbaki|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.<ref name=Wolf />

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.

==Defining the ordered pair using set theory==

If one agrees that [[set theory]] is an appealing [[foundations of mathematics|foundation of mathematics]], then all mathematical objects must be defined as [[Set (mathematics)|sets]] of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set.<ref>[[Willard van Orman Quine|Quine]] has argued that the set-theoretical implementations of the concept of the ordered pair   is a paradigm for the clarification of philosophical  ideas (see  "[[Word and Object]]", section 53).
The general notion of such definitions or implementations  are discussed in Thomas Forster "Reasoning about theoretical entities".
</ref> Several set-theoretic definitions of the ordered pair are given below.

===Wiener's definition===
[[Norbert Wiener]] proposed the first set theoretical definition of the ordered pair in 1914:<ref>Wiener's paper "A Simplification of the logic of relations" is reprinted, together with a valuable commentary on pages 224ff in van Heijenoort, Jean (1967), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1979–1931'', Harvard University Press, Cambridge MA, {{isbn|0-674-32449-8}} (pbk.). van Heijenoort states the simplification this way: "By giving a definition of the ordered pair of two elements in terms of class operations, the note reduced the theory of relations to that of classes".</ref>
:<math>\left( a, b \right) := 
\left\{\left\{ \left\{a\right\},\, \emptyset \right\},\,  \left\{\left\{b\right\}\right\}\right\}.</math>
He observed that this definition made it possible to define the [[type theory|types]] of ''[[Principia Mathematica]]'' as sets. ''Principia Mathematica'' had taken types, and hence [[relation (mathematics)|relations]] of all arities, as [[primitive notion|primitive]].

Wiener used <nowiki>{{</nowiki>''b''}} instead of {''b''} to make the definition compatible with [[type theory]] where all elements in a class must be of the same "type". With ''b'' nested within an additional set, its type is equal to <math>\{\{a\}, \emptyset\}</math>'s.

===Hausdorff's definition===
About the same time as Wiener (1914), [[Felix Hausdorff]] proposed his definition:
: <math>(a, b) := \left\{ \{a, 1\}, \{b, 2\} \right\}</math>
"where 1 and 2 are two distinct objects different from a and b."<ref>cf introduction to Wiener's paper in van Heijenoort 1967:224</ref>

===Kuratowski's definition===
In 1921 [[Kazimierz Kuratowski]] offered the now-accepted definition<ref>cf introduction to Wiener's paper in van Heijenoort 1967:224. van Heijenoort observes that the resulting set that represents the ordered pair "has a type higher by 2 than the elements (when they are of the same type)"; he offers references that show how, under certain circumstances, the type can be reduced to 1 or 0.</ref><ref>{{cite journal|title=Sur la notion de l'ordre dans la Théorie des Ensembles|first=Casimir|last=Kuratowski|authorlink=Kazimierz Kuratowski|year=1921|journal=[[Fundamenta Mathematicae]]|pages=161–171|volume=2|number=1|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2122.pdf}}</ref>
of the ordered pair (''a'', ''b''):
:<math>(a, \ b)_K \; := \  \{ \{ a \}, \ \{ a, \ b \} \}.</math>
Note that this definition is used even when the first and the second coordinates are identical:
: <math>(x,\ x)_K = \{\{x\},\{x, \ x\}\} = \{\{x\},\ \{x\}\} = \{\{x\}\}</math>

Given some ordered pair ''p'', the property "''x'' is the first coordinate of ''p''" can be formulated as:
:<math>\forall Y\in p:x\in Y.</math>
The property "''x'' is the second coordinate of ''p''" can be formulated as:
:<math>(\exist Y\in p:x\in Y)\land(\forall Y_1,Y_2\in p:Y_1\ne Y_2\rarr (x\notin Y_1\lor x \notin Y_2)).</math>
In the case that the left and right coordinates are identical, the right [[conjunct]] <math>(\forall Y_1,Y_2\in p:Y_1\ne Y_2\rarr (x\notin Y_1 \lor x \notin Y_2))</math> is trivially true, since ''Y''<sub>1</sub> ≠ ''Y''<sub>2</sub> is never the case.

This is how we can extract the first coordinate of a pair (using the notation for [[Intersection (set theory)#Arbitrary intersections|arbitrary intersection]] and [[Union (set theory)#Arbitrary unions|arbitrary union]]):

:<math>\pi_1(p) = \bigcup\bigcap p.</math>

This is how the second coordinate can be extracted:

:<math>\pi_2(p) = \bigcup\{x \in \bigcup p \mid \bigcup p \neq \bigcap p \rarr x \notin \bigcap p \}.</math>

====Variants====
The above Kuratowski definition of the ordered pair is "adequate" in that it satisfies the characteristic property that an ordered pair must satisfy, namely that <math>(a,b) = (x,y) \leftrightarrow (a=x) \land (b=y)</math>. In particular, it adequately expresses 'order', in that <math>(a,b) = (b,a)</math> is false unless <math>b = a</math>. There are other definitions, of similar or lesser complexity, that are equally  adequate:
* <math>( a, b )_{\text{reverse}} :=  \{ \{ b \}, \{a, b\}\};</math>
* <math>( a, b )_{\text{short}} :=  \{ a, \{a, b\}\};</math>
* <math>( a, b )_{\text{01}} := \{\{0, a \}, \{1, b \}\}.</math><ref>This differs from Hausdorff's definition in not requiring the two elements 0 and 1 to be distinct from ''a'' and ''b''.</ref>
The '''reverse''' definition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest. The definition '''short''' is so-called because it requires two rather than three pairs of [[braces (punctuation)|braces]]. Proving that '''short''' satisfies the characteristic property requires the [[Zermelo–Fraenkel set theory]] [[axiom of regularity]].<ref>Tourlakis, George (2003) ''Lectures in Logic and Set Theory. Vol. 2: Set Theory''. Cambridge Univ. Press. Proposition III.10.1.</ref> Moreover, if one uses [[von Neumann ordinal|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)<sub>short</sub>. Yet another disadvantage of the '''short''' pair is the fact, that even if ''a'' and ''b'' are of the same type, the elements of the '''short''' pair are not. (However, if ''a''&nbsp;=&nbsp;''b'' then the '''short''' version keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair". Also note that the '''short''' version is used in [[Tarski–Grothendieck set theory]], upon which the [[Mizar system]] is founded.)

====Proving that definitions satisfy the characteristic property====
Prove: (''a'', ''b'') = (''c'', ''d'') [[if and only if]] ''a'' = ''c'' and ''b'' = ''d''.

'''Kuratowski''':<br>
''If''. If ''a = c'' and ''b = d'', then {{''a''}, {''a, b''}} = {{''c''}, {''c, d''}}. Thus (''a, b'')<sub>K</sub> = (''c, d'')<sub>K</sub>.

''Only if''. Two cases: ''a'' = ''b'', and ''a'' ≠ ''b''.

If ''a'' = ''b'':
:(''a, b'')<sub>K</sub> = {{''a''}, {''a, b''}}  = {{''a''}, {''a, a''}} = <nowiki>{{</nowiki>''a''}}.
:(''c, d'')<sub>K</sub> = {{''c''}, {''c, d''}} = <nowiki>{{</nowiki>''a''}}.
:Thus {''c''} = {''c, d''} = {''a''}, which implies ''a'' = ''c'' and ''a'' = ''d''. By hypothesis, ''a'' = ''b''. Hence ''b'' = ''d''.

If ''a'' ≠ ''b'', then (''a, b'')<sub>K</sub> = (''c, d'')<sub>K</sub> implies {{''a''}, {''a, b''}} = {{''c''}, {''c, d''}}.

:Suppose {''c, d''} = {''a''}. Then ''c = d = a'', and so {{''c''}, {''c, d''}} = {{''a''}, {''a, a''}} = {{''a''}, {''a''}} = <nowiki>{{</nowiki>''a''}}. But then {{''a''}, {''a, b''}} would also equal <nowiki>{{</nowiki>''a''}}, so that ''b = a'' which contradicts ''a'' ≠ ''b''.

:Suppose {''c''} = {''a, b''}. Then ''a = b = c'', which also contradicts ''a'' ≠ ''b''.

:Therefore {''c''} = {''a''}, so that ''c = a'' and {''c, d''} = {''a, b''}.

:If ''d = a'' were true, then {''c, d''} = {''a, a''} = {''a''} ≠ {''a, b''}, a contradiction. Thus ''d = b'' is the case, so that ''a = c'' and ''b = d''.

'''Reverse''':<br>
(''a, b'')<sub>reverse</sub> = {{''b''}, {''a, b''}} = {{''b''}, {''b, a''}} = (''b, a'')<sub>K</sub>.

''If''. If (''a, b'')<sub>reverse</sub> = (''c, d'')<sub>reverse</sub>,
(''b, a'')<sub>K</sub> = (''d, c'')<sub>K</sub>. Therefore, ''b = d'' and ''a = c''.

''Only if''. If ''a = c'' and ''b = d'', then {{''b''}, {''a, b''}} = {{''d''}, {''c, d''}}.
Thus (''a, b'')<sub>reverse</sub> = (''c, d'')<sub>reverse</sub>.

'''Short:'''<ref>For a formal [[Metamath]] proof of the adequacy of '''short''', see [http://us.metamath.org/mpegif/opthreg.html here (opthreg).] Also see Tourlakis (2003), Proposition III.10.1.</ref>

''If'': If ''a = c'' and ''b = d'', then {''a'', {''a, b''}} = {''c'', {''c, d''}}. Thus (''a, b'')<sub>short</sub> = (''c, d'')<sub>short</sub>.

''Only if'': Suppose {''a'', {''a, b''}} = {''c'', {''c, d''}}.
Then ''a'' is in the left hand side, and thus in the right hand side.
Because equal sets have equal elements, one of ''a = c'' or ''a'' = {''c, d''} must be the case.
:If ''a'' = {''c, d''}, then by similar reasoning as above, {''a, b''} is in the right hand side, so {''a, b''} = ''c'' or {''a, b''} = {''c, d''}.
::If {''a, b''} = ''c'' then ''c'' is in {''c, d''} = ''a'' and ''a'' is in ''c'', and this combination contradicts the axiom of regularity, as {''a, c''} has no minimal element under the relation "element of."
::If {''a, b''} = {''c, d''}, then ''a'' is an element of ''a'', from ''a'' = {''c, d''} = {''a, b''}, again contradicting regularity.
:Hence ''a = c'' must hold.

Again, we see that {''a, b''} = ''c'' or {''a, b''} = {''c, d''}.
:The option {''a, b''} = ''c'' and ''a = c'' implies that ''c'' is an element of ''c'', contradicting regularity.
:So we have ''a = c'' and {''a, b''} = {''c, d''}, and so: {''b''} = {''a, b''} \ {''a''} = {''c, d''} \ {''c''} = {''d''}, so ''b'' = ''d''.

===Quine–Rosser definition===
[[J. Barkley Rosser|Rosser]] (1953)<ref>[[J. Barkley Rosser]], 1953. ''Logic for Mathematicians''. McGraw–Hill.</ref> employed a definition of the ordered pair due to [[Willard van Orman Quine|Quine]] which requires a prior definition of the [[natural number]]s. Let <math>\N</math> be the set of natural numbers and define first
:<math>\sigma(x) := \begin{cases}
      x, & \text{if }x \not\in \N, \\
      x+1, & \text{if }x \in \N.
    \end{cases}</math>
The function <math>\sigma</math> increments its argument if it is a natural number and leaves it as is otherwise; the number 0 does not appear as functional value of <math>\sigma</math>.
As <math>x \smallsetminus \N</math> is the set of the elements of <math>x</math> not in <math>\N</math> go on with
:<math>\varphi(x) := \sigma[x] = \{\sigma(\alpha)\mid\alpha \in x\} = (x \smallsetminus \N) \cup \{n+1 : n \in (x \cap \N) \}.</math>
This is the [[Image (mathematics)#Image of a subset|set image]] of a set <math>x</math> under <math>\sigma</math>, [[Image (mathematics)#Other terminology|sometimes denoted]] by <math>\sigma''x</math> as well. Applying function <math>\varphi</math> to a set ''x'' simply increments every natural number in it. In particular, <math>\varphi(x)</math> does never contain the number 0, so that for any sets ''x'' and ''y'',
:<math>\varphi(x) \neq \{0\} \cup \varphi(y).</math>
Further, define
:<math>\psi(x) := \sigma[x] \cup \{0\} = \varphi(x) \cup \{0\}.</math>
By this, <math>\psi(x)</math> does always contain the number 0.

Finally, define the ordered pair (''A'', ''B'') as the disjoint union
:<math>(A, B) := \varphi[A] \cup \psi[B] = \{\varphi(a) : a \in A\} \cup \{\varphi(b) \cup \{0\} : b \in B \}.</math>
(which is <math>\varphi''A \cup \psi''B</math> in alternate notation).

Extracting all the elements of the pair that do not contain 0 and undoing <math>\varphi</math> yields ''A''. Likewise, ''B'' can be recovered from the elements of the pair that do contain 0.<ref>[http://math.boisestate.edu/~holmes/ Holmes, M. Randall]: ''[http://math.boisestate.edu/~best/best18/Talks/holmes_best18.pdf On Ordered Pairs]'', on: Boise State, March 29, 2009. The author uses <math>\sigma_1</math> for <math>\varphi</math> and <math>\sigma_2</math> for <math>\psi</math>.</ref>

For example, the pair <math>( \{\{a,0\},\{b,c,1\}\} , \{\{d,2\},\{e,f,3\}\} )</math> is encoded as <math>\{\{a,1\},\{b,c,2\},\{d,3,0\},\{e,f,4,0\}\}</math> provided <math>a,b,c,d,e,f\notin \N</math>.

In [[type theory]] and in outgrowths thereof such as the axiomatic set theory [[New Foundations|NF]], the Quine–Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair. Hence this definition has the advantage of enabling a [[function (mathematics)|function]], defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments. This definition works only if the set of natural numbers is infinite. This is the case in [[New Foundations|NF]], but not in [[type theory]] or in [[New Foundations|NFU]]. [[J. Barkley Rosser]] showed that the existence of such a type-level ordered pair (or even a "type-raising by 1" ordered pair) implies the [[axiom of infinity]]. For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998).<ref>Holmes, M. Randall (1998) ''[http://math.boisestate.edu/~holmes/holmes/head.pdf  Elementary Set Theory with a Universal Set]''. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this monograph via the web.</ref>

===Cantor–Frege definition===
Early in the development of the set theory, before paradoxes were discovered, Cantor followed Frege by defining the ordered pair of two sets as the class of all relations that hold between these sets, assuming that the notion of relation is primitive:<ref>{{cite book |last=Frege |first=Gottlob |year=1893 |title=Grundgesetze der Arithmetik |url=https://korpora.zim.uni-duisburg-essen.de/Frege/PDF/gga1_o_corr.pdf |location=Jena |publisher=Verlag Hermann Pohle }} §144</ref>

:<math>(x, y) = \{R : x R y \}.</math>

This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the [[Cardinality|cardinal]] of a set as the class of all sets equipotent with the given set.<ref>{{cite book |last=Kanamori |first=Akihiro |year=2007 |title=Set Theory From Cantor to Cohen |url=http://math.bu.edu/people/aki/16.pdf |publisher=Elsevier BV }} p. 22, footnote 59</ref>

===Morse definition===
[[Morse–Kelley set theory]] makes free use of [[proper class]]es.<ref>{{cite book |last=Morse |first=Anthony P. |year=1965 |title=A Theory of Sets |url=https://archive.org/details/theoryofsets0000mors |url-access=registration |location= |publisher=Academic Press }}</ref> [[Anthony Morse|Morse]] defined the ordered pair so that its projections could be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then ''redefined'' the pair

:<math> (x, y) = (\{0\} \times s(x)) \cup (\{1\} \times s(y))</math>

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

:<math> s(x) = \{\emptyset \} \cup \{\{t\} \mid t \in x\} </math>

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

:<math> (x, y, z) = (\{0\} \times s(x)) \cup (\{1\} \times s(y)) \cup (\{2\} \times s(z))</math>

The use of the singleton set <math> s(x) </math> which has an inserted empty set allows tuples to have the uniqueness property that if ''a'' is an ''n''-tuple and b is an ''m''-tuple and ''a'' = ''b'' then ''n'' = ''m''.  Ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs.

==Category theory==
[[File:CategoricalProduct-03.svg|thumb|[[Commutative diagram]] for the set product ''X''<sub>1</sub>×''X''<sub>2</sub>.]]
A category-theoretic [[Product (category theory)|product]] ''A'' × ''B'' in a [[category of sets]] represents the set of ordered pairs, with the first element coming from ''A'' and the second coming from ''B''. 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''. While different objects may have the universal property, they are all [[naturally isomorphic]].

==References==
{{Reflist|30em}}

{{Mathematical logic}}

[[Category:Basic concepts in set theory]]
[[Category:Order theory]]
[[Category:Type theory]]