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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 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 ...

sequence
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: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
. (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: Science and technology Mathematics * 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 ...
s (and hence
functions 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 ...
) 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)\text a_1 = a_2\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 the
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 ...
of ''A'' and ''B'', and written ''A'' × ''B''. A
binary relation Binary may refer to: Science and technology Mathematics * 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). ...

subset
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: Currencies * Brazilian real 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 ...

real number line
. 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 \langle a,b\rangle, but this notation also has other uses. The left and right projection of a pair ''p'' is usually denoted by 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''.


Informal and formal definitions

In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as
For 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 that
set 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 ...

Norbert Wiener
proposed the first set theoretical definition of the ordered pair in 1914: :\left( a, b \right) := \left\. He observed that this definition made it possible to define the
types Type may refer to: Science and technology Computing * 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: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
of all arities, as
primitive Primitive may refer to: Mathematics * Primitive element (field theory) * Primitive element (finite field) * Primitive cell (crystallography) * Primitive notion, axiomatic systems * Primitive polynomial (disambiguation), one of two concepts * Primit ...
. Wiener used instead of to make the definition compatible with
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 \'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) := \left\ "where 1 and 2 are two distinct objects different from a and b."


Kuratowski's definition

In 1921
Kazimierz 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 ...

Kazimierz Kuratowski
offered the now-accepted definition of the ordered pair (''a'', ''b''): :(a, \ b)_K \; := \ \. Note that this definition is used even when the first and the second coordinates are identical: : (x,\ x)_K = \ = \ = \ Given some ordered pair ''p'', the property "''x'' is the first coordinate of ''p''" can be formulated as: :\forall Y\in p:x\in Y. The property "''x'' is the second coordinate of ''p''" can be formulated as: :(\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)). 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 ...
(\forall Y_1,Y_2\in p:Y_1\ne Y_2\rarr (x\notin Y_1 \lor x \notin Y_2)) is trivially true, since ''Y''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): :\pi_1(p) = \bigcup\bigcap p. This is how the second coordinate can be extracted: :\pi_2(p) = \bigcup\left\.


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) \leftrightarrow (a=x) \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 )_ := \; * ( a, b )_ := \; * ( a, b )_ := \. 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 the
Zermelo–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)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
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'')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''.


Quine–Rosser definition

Rosser (1953) employed a definition of the ordered pair due to Quine which requires a prior definition of the
natural 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 \N be the set of natural numbers and define first :\sigma(x) := \begin x, & \textx \not\in \N, \\ x+1, & \textx \in \N. \end The function \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 \sigma. As x \smallsetminus \N is the set of the elements of x not in \N go on with :\varphi(x) := \sigma = \ = (x \smallsetminus \N) \cup \. This is the set image of a set x under \sigma, sometimes denoted by \sigma''x as well. Applying function \varphi to a set ''x'' simply increments every natural number in it. In particular, \varphi(x) does never contain the number 0, so that for any sets ''x'' and ''y'', :\varphi(x) \neq \ \cup \varphi(y). Further, define :\psi(x) := \sigma \cup \ = \varphi(x) \cup \. By this, \psi(x) does always contain the number 0. Finally, define the ordered pair (''A'', ''B'') as the disjoint union :(A, B) := \varphi \cup \psi = \ \cup \. (which is \varphi''A \cup \psi''B in alternate notation). Extracting all the elements of the pair that do not contain 0 and undoing \varphi yields ''A''. Likewise, ''B'' can be recovered from the elements of the pair that do contain 0. For example, the pair ( \ , \ ) is encoded as \ provided a,b,c,d,e,f\notin \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) = \. This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the
cardinal Cardinal or The Cardinal may refer to: Christianity * Cardinal (Catholic Church), a senior official of the Catholic Church * Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral Navigation * Cardina ...
of a set as the class of all sets equipotent with the given set.


Morse definition

Morse–Kelley set theory makes free use of
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. 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) = (\ \times s(x)) \cup (\ \times s(y)) where the component Cartesian products are Kuratowski pairs of sets and where : s(x) = \ \cup \ 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) = (\ \times s(x)) \cup (\ \times s(y)) \cup (\ \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 in
Zermelo–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)\text a_1 = a_2\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) = , ∈ (a,b), if Kuratowski's definition (a,b) = was used.


Category theory

A category-theoretic product ''A'' × ''B'' in a
category 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 logic
Basic 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