Set Theory and Beyond

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 $$s:I⟶X.$$ 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...