Section 5.4 Ordered pairs
We can also create ordered pairs. For sets \(X\) and \(Y\text{,}\) we write
\begin{equation*}
\angs{X,Y}\coloneq\left\{\{X\},\{X,Y\}\right\}\text{.}
\end{equation*}
Now for any two sets \(X\) and \(Y\text{,}\) we define the product as the set of all ordered pairs:
\begin{equation*}
X\times Y\coloneq\left\{S\in\PP(\PP(X\cup Y)) : (\exists x\in X)(\exists y\in Y)(\angs{x,y}=S)\right\}\text{.}
\end{equation*}
We can also define an ordered triple by the rule
\begin{equation*}
\angs{X,Y,Z}\coloneq\angs{\angs{X,Y},Z}\text{.}
\end{equation*}
We can keep going with this to build ordered quadruples and ordered quintuples, etc., but once we have the concept of map up and running, we'll find more efficient and intuitive ways to talk about these things.
\​begin{exr} How many elements does the ordered pair \(\angs{x,x}\) have? How about the ordered triple \(\angs{x,x,x}\text{?}\) \end{exr}