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\begin{document}

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\section*{ordered pair}

\hypertarget{ordered_pairs}{}\section*{{Ordered pairs}}\label{ordered_pairs}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{formalisations}{Formalisations}\dotfill \pageref*{formalisations} \linebreak
\noindent\hyperlink{generalisations}{Generalisations}\dotfill \pageref*{generalisations} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given any things $a$ and $b$, the \textbf{ordered pair} of $a$ and $b$ is a thing, usually written $(a,b)$, sometimes $[a,b]$ or $\langle{a,b}\rangle$. The important property is

\begin{equation}
(a,b) = (c,d) \;\Leftrightarrow\; a = c,\; b = d .
\label{basic}\end{equation}
The things $a$ and $b$ are called the \textbf{components} of the ordered pair $(a,b)$. Given any two sets $X$ and $Y$, their [[Cartesian product]] is a set $X \times Y$ whose [[elements]] are precisely the ordered pairs whose components are respectively elements of $X$ and elements of $Y$.

Note that nothing is \textbf{ordered} in an ordered pair other than how it is written out, so sometimes just the word \textbf{pair} is used for this concept.

In terms of [[category theory]] an ordered pair is an element in a [[Cartesian product]].

The concept of ``pair'' meaning a set containing just one or two members as in the [[axiom of pairing]] of [[ZFC|Zermelo–Frankel Set Theory]] is now usually distinguished as an \textbf{[[unordered pair]]}. A pair in which the components are ordered is basically an arrow between the components, which is sometimes called or analyzed as an [[interval object|interval]] within a larger context.

\hypertarget{formalisations}{}\subsection*{{Formalisations}}\label{formalisations}

One may wish to declare ordered pairs to exist by fiat, which was done, for example, by both [[Bourbaki]] and [[Bill Lawvere]]. In Bourbaki's foundational [[set theory]], $\langle{-,-}\rangle$ is a fundamental binary operation on mathematical objects satisfying two axioms: \eqref{basic} and the existence (as a set) of the Cartesian product of any two sets. In Lawvere's foundational set theory, [[ETCS]], one axiom is the existence of [[products]] in the category of sets; when applied to [[global elements]], this gives their ordered pair (with the product itself being the Cartesian product), and \eqref{basic} can be proved. Other [[structural set theories]] should contain an axiom similar to Lawvere's axiom of products.

I need to check that I remembered Bourbaki correctly; it varies with the edition. ---Toby

Instead, one may construct ordered pairs out of some more basic operation. In a [[material set theory]], one may use \emph{[[Kuratowski pairs]]}

\begin{displaymath}
(a,b) \coloneqq \big\{\,\{a\},\{a,b\}\,\big\} ;
\end{displaymath}
it is straightforward (using the [[axiom of extensionality]]) to prove that \eqref{basic} holds. Sometimes one sees the alternative

\begin{displaymath}
(a,b) \coloneqq \big\{a, \{a,b\}\,\big\} ;
\end{displaymath}
but now the [[axiom of foundation]] is also needed to prove \eqref{basic}, so the first form is usually preferred. To prove that the cartesian product of two sets is a set, one may use the axiom of separation ([[bounded separation]] is enough) to construct $X \times Y$ as a [[subset]] of the [[power set]] of the power set of the [[union]] of $X$ and $Y$, or else use the axiom of replacement ([[restricted replacement]] is enough) to construct it directly, since its elements are indexed by the sets $X$ and $Y$.

\begin{remark}
\label{class}\hypertarget{class}{}
Again in the context of material set theory, there are other options for defining ordered pairs that may offer technical advantages. For example, assuming we have the natural numbers $\mathbb{N}$, and given a set $x$, let $\varphi(x)$ be the set $(x \setminus \mathbb{N}) \cup \{n+1: n \in x \cap \mathbb{N}\}$. Then define

\begin{displaymath}
(A, B) \coloneqq \{\varphi(a): a \in A\} \cup \{\varphi(b) \cup \{0\}: b \in B\}
\end{displaymath}
and observe that the first entry $A$ may be retrieved as the set of elements of $(A, B)$ that do not contain $0$, and $B$ as the set of elements that do. One advantage is that even for classes $A, B$ (defined in ZFC as formulas), we may define a class $(A, B)$ as another formula patterned on this construction. This solves a technical exercise posed by Andrej Bauer in this \href{https://mathoverflow.net/a/63268/2926}{MO post}; compare \href{https://ncatlab.org/nlab/show/function#for_classes}{class functions}. Thus, a function between classes $f: A \to B$ may be defined as an ordered triple $(A, f, B)$ where $f$ is a class of pairs $(a, b)$ defined by a formula that is [[entire relation|entire]] and [[functional relation|functional]].

\end{remark}
In a foundational [[type theory]], ordered pairs are usually also given by fiat, but \eqref{basic} may not hold, depending on the type theory used. Now Bourbaki's binary operation of pairing becomes a typed operation; given $a$ of type $X$ and $b$ of type $Y$, the ordered pair $(x,y)$ has type $A \times B$. There are also two typed operations (either basic or definable, depending on the style of type theory used) $\pi\colon X \times Y \to X$ and $\rho\colon X \times Y \to Y$, satisfying the [[beta-rule]]s $\pi(x,y) = x$ and $\rho(x,y) = y$. Then we can either add the [[eta-rule]] $z = (\pi z,\rho z)$, which will allow \eqref{basic} to be proved; or else take \eqref{basic} as the \emph{definition} of equality on the product type $X \times Y$, which will then allow the eta-rule to be proved. (Or you can do neither, and then \eqref{basic} and the eta-rule will fail.)

\hypertarget{generalisations}{}\subsection*{{Generalisations}}\label{generalisations}

\begin{itemize}%
\item [[tuples]]


\item [[families]]


\item [[pairings]]


\item [[dependent sums]]



\end{itemize}
[[!redirects ordered pair]] [[!redirects ordered pairs]]



\end{document}