Definition 3.5.7 (Ordered \(n\)-tuple and \(n\)-fold Cartesian product). Let \(n\) be a natural number. An ordered \(n\)-tuple \((x_i)_{1\leq i\leq n}\) (also denoted \((x_1,\cdots,x_n)\)) is a collection of objects \(x_i\), one for every natural number \(i\) between \(1\) and \(n\); we refer to \(x_i\) as the \(i^{th}\) component of the ordered \(n\)-tuple. Two ordered \(n\)-tuples \((x_i)_{1\leq i\leq n}\) and \((y_i)_{1\leq 1\leq n}\) are said to be equal iff \(x_i=y_i\) for all \(1\leq i\leq n\). If \((X_i)_{1\leq i\leq n}\) is an ordered \(n\)-tuple of sets, we define their Cartesian product \(\prod_{1\leq i\leq n}X_i\) (also denoted \(\prod_{i=1}^n X_i\) or \(X_1\times\cdots X_n\)) by $$ \prod_{1\leq i\leq n}X_i:=\{(x_i)_{1\leq i\leq n}:x_i\in X_i\text{ for all }1\leq i\leq n\} $$ Again, this definition simply postulates that an ordered \(n\)-tuple and a Cartesian product always exists when needed, but using the axioms of set theory one can explicitly construct these objects (Exercise 3.5.2)