Ordered pair

Analytic geometryassociates to each point in theEuclidean planean ordered pair. The redellipseis associated with the set of all pairs (x,y) such that⁠x²/4⁠+y²= 1.

Inmathematics,anordered 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) unlessa=b.(In contrast, theunordered pair{a,b} equals the unordered pair {b,a}.)

Ordered pairs are also called2-tuples,orsequences(sometimes, lists in a computer science context) of length 2. Ordered pairs ofscalarsare sometimes called 2-dimensionalvectors.(Technically, this is an abuse ofterminologysince an ordered pair need not be an element of avector space.) The entries of an ordered pair can be other ordered pairs, enabling therecursivedefinition of orderedn-tuples(ordered lists ofnobjects). 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 objectais called thefirst entry,and the objectbthesecond entryof the pair. Alternatively, the objects are called the first and secondcomponents,the first and secondcoordinates,or the left and rightprojectionsof the ordered pair.

Cartesian productsandbinary relations(and hencefunctions) are defined in terms of ordered pairs, cf. picture.

Generalities

Let$(a_{1},b_{1})$and$(a_{2},b_{2})$be ordered pairs. Then thecharacteristic(ordefining)propertyof the ordered pair is: $$(a_{1},b_{1})=(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.$$

Thesetof all ordered pairs whose first entry is in some setAand whose second entry is in some setBis called theCartesian productofAandB,and writtenA×B.Abinary relationbetween setsAandBis asubsetofA×B.

The(a,b)notation may be used for other purposes, most notably as denotingopen intervalson thereal 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${\textstyle \langle a,b\rangle }$,but this notation also has other uses.

The left and rightprojectionof a pairpis usually denoted byπ₁(p) andπ₂(p), or byπ_ℓ(p) andπ_r(p), respectively. In contexts where arbitraryn-tuples are considered,πⁿ
_i(t) is a common notation for thei-th component of ann-tuplet.

Informal and formal definitions

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

For any two objectsaandb,the ordered pair(a,b)is a notation specifying the two objectsaandb,in that order.

This is usually followed by a comparison to a set of two elements; pointing out that in a setaandbmust 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 oforder.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 aprimitive notion,whose associated axiom is the characteristic property. This was the approach taken by theN. Bourbakigroup in itsTheory 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'sTheory 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 theoryis an appealingfoundation of mathematics,then all mathematical objects must be defined assetsof 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 Wienerproposed the first set theoretical definition of the ordered pair in 1914: $$\left(a,b\right):=\left\{\left\{\left\{a\right\},\,\emptyset \right\},\,\left\{\left\{b\right\}\right\}\right\}.$$ He observed that this definition made it possible to define thetypesofPrincipia Mathematicaas sets.Principia Mathematicahad taken types, and hencerelationsof all arities, asprimitive.

Wiener used {{b}} instead of {b} to make the definition compatible withtype theorywhere all elements in a class must be of the same "type". Withbnested within an additional set, its type is equal to$\{\{a\},\emptyset \}$'s.

Hausdorff's definition

About the same time as Wiener (1914),Felix Hausdorffproposed his definition: $$(a,b):=\left\{\{a,1\},\{b,2\}\right\}$$ "where 1 and 2 are two distinct objects different from a and b."

Kuratowski's definition

In 1921Kazimierz Kuratowskioffered the now-accepted definition of the ordered pair (a,b): $$(a,\ b)_{K}\;:=\ \{\{a\},\ \{a,\ b\}\}.$$ When the first and the second coordinates are identical, the definition obtains: $$(x,\ x)_{K}=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}$$

Given some ordered pairp,the property "xis the first coordinate ofp"can be formulated as: $$\forall Y\in p:x\in Y.$$ The property "xis the second coordinate ofp"can be formulated as: $$(\exists Y\in p:x\in Y)\land (\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2})).$$ In the case that the left and right coordinates are identical, the rightconjunct$(\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2}))$is trivially true, sinceY₁≠Y₂is never the case.

If$p=(x,y)=\{\{x\},\{x,y\}\}$then:

$\bigcap p=\bigcap {\bigg \{}\{x\},\{x,y\}{\bigg \}}=\{x\}\cap \{x,y\}=\{x\},$

$\bigcup p=\bigcup {\bigg \{}\{x\},\{x,y\}{\bigg \}}=\{x\}\cup \{x,y\}=\{x,y\}.$

This is how we can extract the first coordinate of a pair (using theiterated-operation notationforarbitrary intersectionandarbitrary union): $$\pi _{1}(p)=\bigcup \bigcap p=\bigcup \{x\}=x.$$

This is how the second coordinate can be extracted: $$\pi _{2}(p)=\bigcup \left\{\left.a\in \bigcup p\,\right|\,\bigcup p\neq \bigcap p\rightarrow a\notin \bigcap p\right\}=\bigcup \left\{\left.a\in \{x,y\}\,\right|\,\{x,y\}\neq \{x\}\rightarrow a\notin \{x\}\right\}=\bigcup \{y\}=y.$$

(if$x\neq y$,then the set {y} could be obtained more simply:$\{y\}=\{\left.a\in \{x,y\}\,\right|\,a\notin \{x\}\}$,but the previous formula also takes into account the case when x=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$(a,b)=(x,y)\leftrightarrow (a=x)\land (b=y)$.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:

• $(a,b)_{\text{reverse}}:=\{\{b\},\{a,b\}\};$
• $(a,b)_{\text{short}}:=\{a,\{a,b\}\};$
• $(a,b)_{\text{01}}:=\{\{0,a\},\{1,b\}\}.$

Thereversedefinition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest. The definitionshortis so-called because it requires two rather than three pairs ofbraces.Proving thatshortsatisfies the characteristic property requires theZermelo–Fraenkel set theoryaxiom of regularity.Moreover, if one usesvon 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)_short.Yet another disadvantage of theshortpair is the fact that, even ifaandbare of the same type, the elements of theshortpair are not. (However, ifa=bthen theshortversion keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair".)

Proving that definitions satisfy the characteristic property

Prove: (a,b) = (c,d)if and only ifa=candb=d.

Kuratowski:
If.Ifa=candb=d,then {{a}, {a,b}} = {{c}, {c,d}}. Thus (a, b)_K= (c,d)_K.

Only if.Two cases:a=b,anda≠b.

Ifa=b:

(a, b)_K= {{a}, {a,b}} = {{a}, {a,a}} = {{a}}.

{{c}, {c,d}} = (c,d)_K= (a,b)_K= {{a}}.

Thus {c} = {c,d} = {a}, which impliesa=canda=d.By hypothesis,a=b.Henceb=d.

Ifa≠b,then (a,b)_K= (c,d)_Kimplies {{a}, {a,b}} = {{c}, {c,d}}.

Suppose {c,d} = {a}. Thenc=d=a,and so {{c}, {c,d}} = {{a}, {a,a}} = {{a}, {a}} = {{a}}. But then {{a}, {a, b}} would also equal {{a}}, so thatb=awhich contradictsa≠b.

Suppose {c} = {a,b}. Thena=b=c,which also contradictsa≠b.

Therefore {c} = {a}, so thatc = aand {c,d} = {a,b}.

Ifd=awere true, then {c,d} = {a,a} = {a} ≠ {a,b}, a contradiction. Thusd=bis the case, so thata=candb=d.

Reverse:
(a, b)_reverse= {{b}, {a, b}} = {{b}, {b, a}} = (b, a)_K.

If.If (a, b)_reverse= (c, d)_reverse, (b, a)_K= (d, c)_K.Therefore,b = danda = c.

Only if.Ifa = candb = d,then {{b}, {a, b}} = {{d}, {c, d}}. Thus (a, b)_reverse= (c, d)_reverse.

Short:

If:Ifa = candb = d,then {a,{a, b}} = {c,{c, d}}. Thus (a, b)_short= (c, d)_short.

Only if:Suppose {a,{a, b}} = {c,{c, d}}. Thenais in the left hand side, and thus in the right hand side. Because equal sets have equal elements, one ofa = cora= {c, d} must be the case.

Ifa= {c, d}, then by similar reasoning as above, {a, b} is in the right hand side, so {a, b} =cor {a, b} = {c, d}.

If {a, b} =cthencis in {c, d} =aandais inc,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}, thenais an element ofa,froma= {c, d} = {a, b}, again contradicting regularity.

Hencea = cmust hold.

Again, we see that {a, b} =cor {a, b} = {c, d}.

The option {a, b} =canda = cimplies thatcis an element ofc,contradicting regularity.

So we havea = cand {a, b} = {c, d}, and so: {b} = {a, b} \ {a} = {c, d} \ {c} = {d}, sob=d.

Quine–Rosser definition

Rosser(1953)employed a definition of the ordered pair due toQuinewhich requires a prior definition of thenatural numbers.Let$\mathbb {N}$be the set of natural numbers and define first $$\sigma (x):={\begin{cases}x,&{\text{if }}x\notin \mathbb {N},\\x+1,&{\text{if }}x\in \mathbb {N}.\end{cases}}$$ 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$x\setminus \mathbb {N}$is the set of the elements of$x$not in$\mathbb {N}$go on with $$\varphi (x):=\sigma [x]=\{\sigma (\ Alpha )\mid \ Alpha \in x\}=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.$$ This is theset imageof a set$x$under$\sigma$,sometimes denotedby$\sigma ''x$as well. Applying function$\varphi$to a setxsimply increments every natural number in it. In particular,$\varphi (x)$does never contain the number 0, so that for any setsxandy, $$\varphi (x)\neq \{0\}\cup \varphi (y).$$ Further, define $$\psi (x):=\sigma [x]\cup \{0\}=\varphi (x)\cup \{0\}.$$ By this,$\psi (x)$does always contain the number 0.

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

Extracting all the elements of the pair that do not contain 0 and undoing$\varphi$yieldsA.Likewise,Bcan be recovered from the elements of the pair that do contain 0.

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

Intype theoryand in outgrowths thereof such as the axiomatic set theoryNF,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 afunction,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 inNF,but not intype theoryor inNFU.J. Barkley Rossershowed that the existence of such a type-level ordered pair (or even a "type-raising by 1" ordered pair) implies theaxiom of infinity.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: $$(x,y)=\{R:xRy\}.$$

This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining thecardinalof a set as the class of all sets equipotent with the given set.

Morse definition

Morse–Kelley set theorymakes free use ofproper classes.Morsedefined 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 thenredefinedthe pair $$(x,y)=(\{0\}\times s(x))\cup (\{1\}\times s(y))$$ where the component Cartesian products are Kuratowski pairs of sets and where $$s(x)=\{\emptyset \}\cup \{\{t\}\mid t\in x\}$$

This renders possible pairs whose projections are proper classes. The Quine–Rosser definition above also admitsproper classesas projections. Similarly the triple is defined as a 3-tuple as follows: $$(x,y,z)=(\{0\}\times s(x))\cup (\{1\}\times s(y))\cup (\{2\}\times s(z))$$

The use of the singleton set$s(x)$which has an inserted empty set allows tuples to have the uniqueness property that ifais ann-tuple and b is anm-tuple anda=bthenn=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(ZF) axiomatically by just adding to ZF a new function symbol$f$of arity 2 (it is usually omitted) and a defining axiom for$f$: $$f(a_{1},b_{1})=f(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.$$

This definition is acceptable because this extension of ZF is aconservative extension.

The definition helps to avoid so called accidental theorems like (a,a) = {{a}}, and {a} ∈ (a,b), if Kuratowski's definition (a,b) = {{a}, {a,b}} was used.

Category theory

Commutative diagramfor the set productX₁×X₂.

A category-theoreticproductA×Bin acategory of setsrepresents the set of ordered pairs, with the first element coming fromAand the second coming fromB.In this context the characteristic property above is a consequence of theuniversal propertyof the product and the fact that elements of a setXcan be identified with morphisms from 1 (a one element set) toX.While different objects may have the universal property, they are allnaturally isomorphic.

See also

• Cartesian product
• Tarski–Grothendieck set theory
• Trybulec, Andrzej, 1989, "Tarski–Grothendieck Set Theory",Journal of Formalized Mathematics(definition Def5 of "ordered pairs" as { { x,y }, { x } })