Why is it called a "family" of sets?

Question

If I'm not mistaken, a "family of sets" is just $f:I\mapsto X$. My question is: why they decided to call it a "family" of sets? What's the intuitive picture that made some guy call this a $\textbf{family}$?

Answer 1

A family of things is certainly different from a set of things.

If $A$ is a nonempty set then its elements $x$ are not "organized" in any way. Each thing $x$ you can think of either is an element of $A$, or is not an element of $A$, period. Given $A$ you can, e.g., form the set ${A\choose 3}$of all three-element subsets of $A$. This is a set of sets, but not a family of sets.

In order to obtain a family of things $x$ you need an index set $I$ and a function $f$ that produces for each $\iota\in I$ one such thing $x_\iota:=f(\iota)$. The resulting "data structure" is then called a family of things, and is denoted by $\bigl(x_\iota\bigr)_{\iota\in I}$.

As an example, if $X$ is an arbitrary nonempty set and ${\cal P}(X)$ its power set then an $$f:\>I\to{\cal P}(X),\quad\iota\to A_\iota$$ produces a family of subsets of $X$. This family is denoted by $\bigl(A_\iota\bigr)_{\iota\in I}$. Note that the same subset $A\subset X$ may occur several times in this family, whereas in a set of subsets it would occur at most once.

The functions $f$ appearing in this explanation are not interesting per se; therefore they do not appear in the notation. But every time we talk about a "family" such an $f$ is looming in the background.