Ordered pair

In mathematics, an ordered pair is a collection (of objects) having two coordinates (or entries or projections), such that it is distinguishable, which object is the first coordinate (or first entry or left projection) of the pair and which object is the second coordinate (or second entry or right projection) of the pair. If the first coordinate is a and the second is b, the usual notation for an ordered pair is (a, b). The pair is "ordered" in that (a, b) differs from (b, a) unless a = b.

Cartesian products and binary relations (and hence the ubiquitous functions) are defined in terms of ordered pairs.

Generalities

Let $(a_{1},b_{1})$ and $(a_{2},b_{2})$ be two ordered pairs. Then the characteristic (or defining) property of the ordered pair is:

$(a_{1},b_{1})=(a_{2},b_{2})\quad {\text{if and only if}}\quad a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.\!$

The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n terms). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another. This approach is mirrored in computer programming languages that enable constructing a list of elements by nesting cons cells. For example, the list (1 2 3 4 5) becomes (1, (2, (3, (4, (5, {}))))). The Lisp programming language employs such lists as its primary data structure.

The set of all ordered pairs whose first element is in some set X and whose second element is in some set Y is called the Cartesian product of X and Y, and written X×Y. A binary relation over the field X∪Y is a subset of X×Y.

If one wishes to employ the $\ (a,b)$ notation for a different purpose (such as denoting open intervals on the real number line) the ordered pair may be denoted by the variant notation $\left\langle a,b\right\rangle .$

Defining the ordered pair using set theory

The above characteristic property of ordered pairs 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, long after Kuratowski discovered his reduction (below). The Kuratowski definition was added in the second edition of Theory of Sets, published in 1970.

If one agrees that set theory 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.

Wiener's definition

Norbert Wiener proposed 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 the types of Principia Mathematica as sets. Principia Mathematica had taken types, and hence relations of all arities, as primitive.

Hausdorff's definition

About the same time as Wiener (1914), Felix Hausdorff proposed his definition:

$(a,b):=\left\{\{a,1\},\{b,2\}\right\}$

"where 1 and 2 are two distinct objects different from a and b" .

Kuratowski definition

In 1921 Kuratowski offered the now-accepted definition of the ordered pair (a, b):

$({\textit {a}},\ {\textit {b}})_{K}\ :=\ \{\{{\textit {a}}\},\ \{{\textit {a}},\ {\textit {b}}\}\}.$

Note that this definition remains valid when the first and the second coordinates are identical:

$(x,\ x)=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}$

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:

$(\exists {Y}{\in }{p}:{x}{\in }{Y})\land (\forall {Y_{1},Y_{2}}{\in }{p}:Y_{1}\neq Y_{2}\rightarrow ({x}{\notin }{Y_{1}}\lor {x}{\notin }{Y_{2}})).$

In the case that the left and right coordinates are identical, the right conjunct $(\forall {Y_{1},Y_{2}}{\in }{p}:Y_{1}\neq Y_{2}\rightarrow ({x}{\notin }{Y_{1}}\lor {x}{\notin }{Y_{2}}))$ is trivially true, since Y₁ ≠ Y₂ 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:

$\pi _{2}(p)=\bigcup \{x\in \bigcup p\mid \bigcup p\not =\bigcap p\rightarrow x\notin \bigcap p\}$

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 $(a,b)=(x,y)\leftrightarrow (a=x)\land (b=y)$. There are other definitions, of similar or lesser complexity, that are equally adequate:

• $(\{{\textit {a}},{\textit {b}}\})$_reverse := $(\{\{{\textit {b}}\},\{{\textit {a}},{\textit {b}}\}\}$;
• $(\{{\textit {a}},{\textit {b}}\})$_short := $(\{\{{\textit {a}}\},\{{\textit {a}},{\textit {b}}\}\}$;
• $(\{{\textit {a}},{\textit {b}}\})$₀₁ := $\{\{0,{\textit {a}}\},\{1,{\textit {b}}\}\}$.

reverse is merely a trivial variant of the Kuratowski definition, and as such is of no further interest. 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 axiom of regularity Moreover, if one accepts the standard 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.

Proving that definitions satisfy the characteristic property

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

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

Only if. Two cases: a = b, and a ≠ b.

If a = b:

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

(c, d)_K = {{c}, {c, d}} = {{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)_K = (c, d)_K 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}} = {{a}}. But then {{a}, {a, b}} would also equal {{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:
(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 = d and a = c.

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

Short:

If: Obvious.

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

Rosser (1953) employed a definition of the ordered pair, due to Quine and requiring a prior definition of the natural numbers. Let $\mathbb {N}$ be the set of natural numbers, and define

$\varphi (x)=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.$

Applying this function simply increments every natural number in x. In particular, $\varphi (x)$ does not contain the number 0, so that for any sets x and y,

$\varphi (x)\not =\{0\}\cup \varphi (y).$

Define the ordered pair (A, B) as

$(A,B)=\{\varphi (a):a\in A\}\cup \{\varphi (b)\cup \{0\}:b\in B\}.$

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.

In type theory 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, 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 or in 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).

Morse definition

Morse-Kelley set theory (Morse 1965) makes free use of proper classes. 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) as $(x\times \{0\})\cup (y\times \{1\})$, where the component Cartesian products are Kuratowski pairs on sets. This second step renders possible pairs whose projections are proper classes. The Quine-Rosser definition above also admits proper classes as projections.

Category theory

A category-theoretic product A x 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.