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 $\mathrm{dom}A=I$. 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, $\mathrm{dom}R$, $\mathrm{ran}R$, $\mathrm{fld}R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$"):

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

• 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 ${\mathrm{\Delta }}_F$ 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.