An indexed family $(A_i)_{i \in I}$ is a function with domain $I$; $\{A_i : i \in I\}$ is its range

Let $I$ be a set. An indexed family with index set $I$ is a function $A$ (A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain) with $\operatorname{dom} A = I$. It is written $(A_i)_{i \in I}$, and $A_i$ abbreviates the value $A(i)$.

The set of its members is its range (Relation, $\operatorname{dom} R$, $\operatorname{ran} R$, $\operatorname{fld} R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$"):

$\{A_i : i \in I\} \;:=\; \operatorname{ran} A .$

• An indexed family is not the set of its members. Two different indices may carry the same member, and the family records that while the set $\{A_i : i \in I\}$ does not. The family with $I = \{a,b\}$ and $A_a = A_b = X$ has $\{A_i : i \in I\} = \{X\}$, and it is a different function from the family indexed by $\{a\}$ alone.

• Every set is the range of some family. For a set $F$, the identity relation $\Delta_F$ is a function with domain $F$ and range $F$, so $F$ is the set of members of the family it indexes. This is why the family forms of the distributive and De Morgan laws below say no less than the unindexed ones.