Ordered pair

Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914:[7]

$${\displaystyle \left(a,b\right):=\left\{\left\{\left\{a\right\},\,\emptyset \right\},\,\left\{\left\{b\right\}\right\}\right\}.}$$

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.

Wiener used {{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 ${\displaystyle \{\{a\},\emptyset \}}$'s.

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

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

"where 1 and 2 are two distinct objects different from a and b."[8]

In 1921 Kazimierz Kuratowski offered the now-accepted definition[9][10] of the ordered pair (a, b):

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

Note that this definition is used even when the first and the second coordinates are identical:

$${\displaystyle (x,\ x)_{K}=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}}$$

Given some ordered pair p, the property "x is the first coordinate of p" can be formulated as:

$${\displaystyle \forall Y\in p:x\in Y.}$$

The property "x is the second coordinate of p" can be formulated as:

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

In the case that the left and right coordinates are identical, the right conjunct ${\displaystyle (\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2}))}$ is trivially true, since Y₁ ≠ Y₂ is never the case.

This is how we can extract the first coordinate of a pair (using the iterated-operation notation for arbitrary intersection and arbitrary union):

$${\displaystyle \pi _{1}(p)=\bigcup \bigcap p.}$$

This is how the second coordinate can be extracted:

$${\displaystyle \pi _{2}(p)=\bigcup \left\{\left.x\in \bigcup p\,\right|\,\bigcup p\neq \bigcap p\rightarrow x\notin \bigcap p\right\}.}$$

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 ${\displaystyle (a,b)=(x,y)\leftrightarrow (a=x)\land (b=y)}$. In particular, it adequately expresses 'order', in that ${\displaystyle (a,b)=(b,a)}$ is false unless ${\displaystyle b=a}$. There are other definitions, of similar or lesser complexity, that are equally adequate:

• ${\displaystyle (a,b)_{\text{reverse}}:=\{\{b\},\{a,b\}\};}$
• ${\displaystyle (a,b)_{\text{short}}:=\{a,\{a,b\}\};}$
• ${\displaystyle (a,b)_{\text{01}}:=\{\{0,a\},\{1,b\}\}.}$[11]

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 axiom of regularity.[12] Moreover, if one uses 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)_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".

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}, {c, d}} = (c, d)_K = (a, b)_K = {{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:[13]

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

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.

Rosser (1953)[14] employed a definition of the ordered pair due to Quine which requires a prior definition of the natural numbers. Let ${\displaystyle \mathbb {N} }$ be the set of natural numbers and define first

$${\displaystyle \sigma (x):={\begin{cases}x,&{\text{if }}x\notin \mathbb {N} ,\\x+1,&{\text{if }}x\in \mathbb {N} .\end{cases}}}$$

The function ${\displaystyle \sigma }$ 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 ${\displaystyle \sigma }$. As ${\displaystyle x\setminus \mathbb {N} }$ is the set of the elements of ${\displaystyle x}$ not in ${\displaystyle \mathbb {N} }$ go on with

$${\displaystyle \varphi (x):=\sigma [x]=\{\sigma (\alpha )\mid \alpha \in x\}=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.}$$

This is the set image of a set ${\displaystyle x}$ under ${\displaystyle \sigma }$, sometimes denoted by ${\displaystyle \sigma ''x}$ as well. Applying function ${\displaystyle \varphi }$ to a set x simply increments every natural number in it. In particular, ${\displaystyle \varphi (x)}$ does never contain the number 0, so that for any sets x and y,

$${\displaystyle \varphi (x)\neq \{0\}\cup \varphi (y).}$$

Further, define

$${\displaystyle \psi (x):=\sigma [x]\cup \{0\}=\varphi (x)\cup \{0\}.}$$

By this, ${\displaystyle \psi (x)}$ does always contain the number 0.

Finally, define the ordered pair (A, B) as the disjoint union

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

(which is ${\displaystyle \varphi ''A\cup \psi ''B}$ in alternate notation).

Extracting all the elements of the pair that do not contain 0 and undoing ${\displaystyle \varphi }$ yields A. Likewise, B can be recovered from the elements of the pair that do contain 0.[15]

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

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).[16]

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:[17]

$${\displaystyle (x,y)=\{R:xRy\}.}$$

This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the cardinal of a set as the class of all sets equipotent with the given set.[18]

Morse–Kelley set theory makes free use of proper classes.[19] 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

$${\displaystyle (x,y)=(\{0\}\times s(x))\cup (\{1\}\times s(y))}$$

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

$${\displaystyle s(x)=\{\emptyset \}\cup \{\{t\}\mid t\in x\}}$$

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

$${\displaystyle (x,y,z)=(\{0\}\times s(x))\cup (\{1\}\times s(y))\cup (\{2\}\times s(z))}$$

The use of the singleton set ${\displaystyle s(x)}$ 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.

Ordered pairs can also be introduced in Zermelo–Fraenkel set theory (ZF) axiomatically by just adding to ZF a new function symbol ${\displaystyle f}$ of arity 2 (it is usually omitted) and a defining axiom for ${\displaystyle f}$:

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

This definition is acceptable because this extension of ZF is a conservative extension.

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