In

_{1}(''p'') and _{2}(''p''), or by _{''ℓ''}(''p'') and _{''r''}(''p''), respectively.
In contexts where arbitrary ''n''-tuples are considered, (''t'') is a common notation for the ''i''-th component of an ''n''-tuple ''t''.

instead of to make the definition compatible with

_{1} ≠ ''Y''_{2} is never the case.
This is how we can extract the first coordinate of a pair (using the notation for arbitrary intersection and arbitrary union):
:$\backslash pi\_1(p)\; =\; \backslash bigcup\backslash bigcap\; p.$
This is how the second coordinate can be extracted:
:$\backslash pi\_2(p)\; =\; \backslash bigcup\backslash left\backslash .$

_{short}. 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'' = ''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

''If''. If ''a = c'' and ''b = d'', then = . Thus (''a, b'')_{K} = (''c, d'')_{K}.
''Only if''. Two cases: ''a'' = ''b'', and ''a'' ≠ ''b''.
If ''a'' = ''b'':
:(''a, b'')_{K} = = = .
:(''c, d'')_{K} = = .
:Thus = = , which implies ''a'' = ''c'' and ''a'' = ''d''. By hypothesis, ''a'' = ''b''. Hence ''b'' = ''d''.
If ''a'' ≠ ''b'', then (''a, b'')_{K} = (''c, d'')_{K} implies = .
:Suppose = . Then ''c = d = a'', and so = = = . But then would also equal , so that ''b = a'' which contradicts ''a'' ≠ ''b''.
:Suppose = . Then ''a = b = c'', which also contradicts ''a'' ≠ ''b''.
:Therefore = , so that ''c = a'' and = .
:If ''d = a'' were true, then = = ≠ , a contradiction. Thus ''d = b'' is the case, so that ''a = c'' and ''b = d''.
Reverse:

(''a, b'')_{reverse} = = = (''b, a'')_{K}.
''If''. If (''a, b'')_{reverse} = (''c, d'')_{reverse},
(''b, a'')_{K} = (''d, c'')_{K}. Therefore, ''b = d'' and ''a = c''.
''Only if''. If ''a = c'' and ''b = d'', then = .
Thus (''a, b'')_{reverse} = (''c, d'')_{reverse}.
Short:
''If'': If ''a = c'' and ''b = d'', then = . Thus (''a, b'')_{short} = (''c, d'')_{short}.
''Only if'': Suppose = .
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'' = must be the case.
:If ''a'' = , then by similar reasoning as above, is in the right hand side, so = ''c'' or = .
::If = ''c'' then ''c'' is in = ''a'' and ''a'' is in ''c'', and this combination contradicts the axiom of regularity, as has no minimal element under the relation "element of."
::If = , then ''a'' is an element of ''a'', from ''a'' = = , again contradicting regularity.
:Hence ''a = c'' must hold.
Again, we see that = ''c'' or = .
:The option = ''c'' and ''a = c'' implies that ''c'' is an element of ''c'', contradicting regularity.
:So we have ''a = c'' and = , and so: = \ = \ = , so ''b'' = ''d''.

, ∈ (a,b), if Kuratowski's definition (a,b) = was used.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, 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 pairIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

equals the unordered pair .)
Ordered pairs are also called 2-tuples, or sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors
Vector may refer to:
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. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.)
The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s and binary relation
Binary may refer to:
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Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

s (and hence functions
Function or functionality may refer to:
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* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

) are defined in terms of ordered pairs.
Generalities

Let $(a\_1,\; b\_1)$ and $(a\_2,\; b\_2)$ be ordered pairs. Then the ''characteristic'' (or ''defining'') ''property'' of the ordered pair is: :$(a\_1,\; b\_1)\; =\; (a\_2,\; b\_2)\backslash text\; a\_1\; =\; a\_2\backslash textb\_1\; =\; b\_2.$ 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 theCartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of ''A'' and ''B'', and written ''A'' × ''B''. A binary relation
Binary may refer to:
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* Binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

between sets ''A'' and ''B'' is a subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''A'' × ''B''.
The notation may be used for other purposes, most notably as denoting open interval
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s on the real number line
Real may refer to:
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The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil ...

. In such situations, the context will usually make it clear which meaning is intended. For additional clarification, the ordered pair may be denoted by the variant notation $\backslash langle\; a,b\backslash rangle$, but this notation also has other uses.
The left and right projection of a pair ''p'' is usually denoted by Informal and formal definitions

In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such asFor any two objects and , the ordered pair is a notation specifying the two objects and , in that order.This is usually followed by a comparison to a set of two elements; pointing out that in a set and 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. 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 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. 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 thatset theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below( see also ).
Wiener's definition

Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

proposed the first set theoretical definition of the ordered pair in 1914:
:$\backslash left(\; a,\; b\; \backslash right)\; :=\; \backslash left\backslash .$
He observed that this definition made it possible to define the types
Type may refer to:
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* Typing
Typing is the process of writing or inputting text by pressing keys on a typewriter, computer keyboard, cell phone, or calculator. It can be distinguished from other means of text inpu ...

of ''Principia Mathematica
Image:Principia Mathematica 54-43.png, 500px, ✸54.43:
"From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...

'' as sets. ''Principia Mathematica'' had taken types, and hence relations
Relation or relations may refer to:
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of all arities, as primitive
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.
Wiener used type theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

where all elements in a class must be of the same "type". With ''b'' nested within an additional set, its type is equal to $\backslash $'s.
Hausdorff's definition

About the same time as Wiener (1914),Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

proposed his definition:
: $(a,\; b)\; :=\; \backslash left\backslash $
"where 1 and 2 are two distinct objects different from a and b."
Kuratowski's definition

In 1921Kazimierz Kuratowski
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish
Polish may refer to:
* Anything from or related to Poland
Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a cou ...

offered the now-accepted definition
of the ordered pair (''a'', ''b''):
:$(a,\; \backslash \; b)\_K\; \backslash ;\; :=\; \backslash \; \backslash .$
Note that this definition is used even when the first and the second coordinates are identical:
: $(x,\backslash \; x)\_K\; =\; \backslash \; =\; \backslash \; =\; \backslash $
Given some ordered pair ''p'', the property "''x'' is the first coordinate of ''p''" can be formulated as:
:$\backslash forall\; Y\backslash in\; p:x\backslash in\; Y.$
The property "''x'' is the second coordinate of ''p''" can be formulated as:
:$(\backslash exist\; Y\backslash in\; p:x\backslash in\; Y)\backslash land(\backslash forall\; Y\_1,Y\_2\backslash in\; p:Y\_1\backslash ne\; Y\_2\backslash rarr\; (x\backslash notin\; Y\_1\backslash lor\; x\; \backslash notin\; Y\_2)).$
In the case that the left and right coordinates are identical, the right conjunct
{{For, the linguistic and logical operation of conjunction, Logical conjunction
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Ling ...

$(\backslash forall\; Y\_1,Y\_2\backslash in\; p:Y\_1\backslash ne\; Y\_2\backslash rarr\; (x\backslash notin\; Y\_1\; \backslash lor\; x\; \backslash notin\; Y\_2))$ is trivially true, since ''Y''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 $(a,b)\; =\; (x,y)\; \backslash leftrightarrow\; (a=x)\; \backslash land\; (b=y)$. In particular, it adequately expresses 'order', in that $(a,b)\; =\; (b,a)$ is false unless $b\; =\; a$. There are other definitions, of similar or lesser complexity, that are equally adequate: * $(\; a,\; b\; )\_\; :=\; \backslash ;$ * $(\; a,\; b\; )\_\; :=\; \backslash ;$ * $(\; a,\; b\; )\_\; :=\; \backslash .$ 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. Proving that short satisfies the characteristic property requires theZermelo–Fraenkel set theory
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...

axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoi ...

. Moreover, if one uses von Neumann's set-theoretic construction of the natural numbers, then 2 is defined as the set = , which is indistinguishable from the pair (0, 0)Mizar system
The Mizar system consists of a formal language
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...

is founded.)
Proving that definitions satisfy the characteristic property

Prove: (''a'', ''b'') = (''c'', ''d'')if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

''a'' = ''c'' and ''b'' = ''d''.
Kuratowski:''If''. If ''a = c'' and ''b = d'', then = . Thus (''a, b'')

(''a, b'')

Quine–Rosser definition

Rosser (1953) employed a definition of the ordered pair due to Quine which requires a prior definition of thenatural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. Let $\backslash N$ be the set of natural numbers and define first
:$\backslash sigma(x)\; :=\; \backslash begin\; x,\; \&\; \backslash textx\; \backslash not\backslash in\; \backslash N,\; \backslash \backslash \; x+1,\; \&\; \backslash textx\; \backslash in\; \backslash N.\; \backslash end$
The function $\backslash sigma$ 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 $\backslash sigma$.
As $x\; \backslash smallsetminus\; \backslash N$ is the set of the elements of $x$ not in $\backslash N$ go on with
:$\backslash varphi(x)\; :=\; \backslash sigma;\; href="/html/ALL/s/.html"\; ;"title="">$
This is the set image of a set $x$ under $\backslash sigma$, sometimes denoted by $\backslash sigma\text{'}\text{'}x$ as well. Applying function $\backslash varphi$ to a set ''x'' simply increments every natural number in it. In particular, $\backslash varphi(x)$ does never contain the number 0, so that for any sets ''x'' and ''y'',
:$\backslash varphi(x)\; \backslash neq\; \backslash \; \backslash cup\; \backslash varphi(y).$
Further, define
:$\backslash psi(x)\; :=\; \backslash sigma;\; href="/html/ALL/s/.html"\; ;"title="">$
By this, $\backslash psi(x)$ does always contain the number 0.
Finally, define the ordered pair (''A'', ''B'') as the disjoint union
:$(A,\; B)\; :=\; \backslash varphi;\; href="/html/ALL/s/.html"\; ;"title="">$
(which is $\backslash varphi\text{'}\text{'}A\; \backslash cup\; \backslash psi\text{'}\text{'}B$ in alternate notation).
Extracting all the elements of the pair that do not contain 0 and undoing $\backslash varphi$ yields ''A''. Likewise, ''B'' can be recovered from the elements of the pair that do contain 0.
For example, the pair $(\; \backslash \; ,\; \backslash \; )$ is encoded as $\backslash $ provided $a,b,c,d,e,f\backslash notin\; \backslash N$.
In type theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

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. Hence this definition has the advantage of enabling a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

, 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 NF, but not in type theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

or in NFU. J. Barkley Rosser
John Barkley Rosser Sr. (December 6, 1907 – September 5, 1989) was an United States, American logician, a student of Alonzo Church, and known for his part in the Church–Rosser theorem, in lambda calculus. He also developed what is now called t ...

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
In axiomatic set theory and the branches of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

. For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998).
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: :$(x,\; y)\; =\; \backslash .$ This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining thecardinal
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of a set as the class of all sets equipotent with the given set.
Morse definition

Morse–Kelley set theory makes free use ofproper class
Proper may refer to:
Mathematics
* Proper map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...

es. 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
:$(x,\; y)\; =\; (\backslash \; \backslash times\; s(x))\; \backslash cup\; (\backslash \; \backslash times\; s(y))$
where the component Cartesian products are Kuratowski pairs of sets and where
:$s(x)\; =\; \backslash \; \backslash cup\; \backslash $
This renders possible pairs whose projections are proper classes. The Quine–Rosser definition above also admits proper class
Proper may refer to:
Mathematics
* Proper map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...

es as projections. Similarly the triple is defined as a 3-tuple as follows:
:$(x,\; y,\; z)\; =\; (\backslash \; \backslash times\; s(x))\; \backslash cup\; (\backslash \; \backslash times\; s(y))\; \backslash cup\; (\backslash \; \backslash times\; s(z))$
The use of the singleton set $s(x)$ 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.
Axiomatic definition

Ordered pairs can also be introduced inZermelo–Fraenkel set theory
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...

(ZF) axiomatically by just adding to ZF a new function symbol $f$ of arity 2 (it is usually omitted) and a defining axiom for $f$:
:$f(a\_1,\; b\_1)\; =\; f(a\_2,\; b\_2)\backslash text\; a\_1\; =\; a\_2\backslash textb\_1\; =\; b\_2.$
This definition is acceptable because this extension of ZF is a conservative extensionIn mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

.
The definition helps to avoid so called accidental theorems like (a,a) = Category theory

A category-theoretic product ''A'' × ''B'' in acategory of sets In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

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
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the Category (mathematics), categories invo ...

.
References

{{Mathematical logicBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...

Order theory
Type theory