Families

Why

We often use functions to keep track of several objects by the objects of some well-known set with which they correspond. In this case, we use specific language and notation.

Definition

Let $I$ and $X$ denote sets. A family is a function from $I$ to $X$. We call an element of $I$ an index and we call $I$ the index set. Of course, the letter $I$ was picked here to be a mnemonic for “index”. We call the range of the family the indexed set and we call the value of the family at an index $i$ a term of the family at $i$ or the $i$th term of the family.

Experience shows that it is useful to discuss sets using indices, especially when discussing a set of sets. If the values of the family are sets, we speak of a family of sets. Indeed, we often speak of a family of whatever object the values of the function are. So for instance, a family of subsets of $X$ is understood to be a function from some index set into $\powerset{X}$.

Notation

Let $x: I \to X$ be a family. We denote the $i$th term of $x$ by $x_i$. We sometimes denote the family by $\set{x_i}_{i \in I}$.

Other terminology

Many authors informally use the term family to refer to a set whose members are themselves sets. It is easy to identify this concept with our terminology. To do so, index the family by itself. In other words, take the identity mapping the family to itself. Throughout these sheets, by family we will always mean a function, not a set.