Families
Everything I said about \(n\)-tuples can also be said about "\(I\)-tuples", where \(I\) is now an arbitrary set. What I mean by this is that we can define an \(I\)-tuple to be a function with domain \(I\). If we set \(I=n\), we just get the notion of an \(n\)-tuple! However, in this case we usually don't use the term "tuple", but rather a family. So, a family indexed by set \(I\) is just a function with domain \(I\). An \(n\)-tuple is then a family indexed by \(n\). A family of elements of \(X\) (indexed by \(I\)) is a function \[s:I\longrightarrow X.\] We sometimes write \((a_i : i\in I)\) to denote the family \(s\) indexed by \(I\) such that \(s(i)=a_i\) for all \(i\in I\). If \((A_i : i\in I)\) is a family (of sets), we define its product \[\prod_{i\in I}A_i\] to be the class consisting of all families \(s\) indexed by \(I\) which satisfy that \(s(i)\in A_i\) for all \(i\in I\). If we have that \(A_i=X\) for all \(i\in I\), we use the notation \[X^I:=\pr...