Indexed family

Definition. Let ${\displaystyle I}$ and ${\displaystyle X}$ be sets and ${\displaystyle x}$ a surjective function, such that

${\displaystyle {\begin{aligned}x\colon I&\to X\\i&\mapsto x_{i}=x(i),\end{aligned}}}$

then this establishes a family of elements in ${\displaystyle X}$ indexed by ${\displaystyle I}$ , which is denoted by ${\displaystyle (x_{i})_{i\in I}}$ or simply ${\displaystyle (x_{i})}$, when the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, the latter with the risk of mixing-up families with sets.

An indexed family can be turned into a set by considering the set ${\displaystyle {\mathcal {X}}=\{x_{i}:i\in I\}}$, that is, the image of I under x. Since the mapping x is not required to be injective, there may exist ${\displaystyle i,j\in I}$ with ${\displaystyle i\neq j}$ such that ${\displaystyle x_{i}=x_{j}}$. Thus, ${\displaystyle |{\mathcal {X}}|\leq |I|,}$ where |A| denotes the cardinality of the set A.

The index set is not restricted to be countable, and, of course, a subset of a powerset may be indexed, resulting in an indexed family of sets. For the important differences in sets and families see below.

Surjective functions and families are formally equivalent, as any function f with domain I induces a family (f(i))_i∈I. In practice, however, a family is viewed as a collection, not as a function: being an element of a family is equivalent with being in the range of the corresponding function. A family contains any element exactly once, if and only if the corresponding function is injective.

Like a set, a family is a container and any set X gives rise to a family (x_x)_x∈X. Thus any set naturally becomes a family. For any family (A_i)_i∈I there is the set of all elements {A_i | i∈I}, but this does not carry any information on multiple containment or the structure given by I. Hence, by using a set instead of the family, some information might be lost.

Let n be the finite set {1, 2, ..., n}, where n is a positive integer.

• An ordered pair is a family indexed by the two element set 2 = {1, 2}.
• An n-tuple is a family indexed by n.
• An infinite sequence is a family indexed by the natural numbers.
• A list is an n-tuple for an unspecified n, or an infinite sequence.
• An n×m matrix is a family indexed by the cartesian product n×m.
• A net is a family indexed by a directed set.

Index sets are often used in sums and other similar operations. For example, if (a_i)_i∈I is a family of numbers, the sum of all those numbers is denoted by

${\displaystyle \sum _{i\in I}a_{i}.}$

When (A_i)_i∈I is a family of sets, the union of all those sets is denoted by

${\displaystyle \bigcup _{i\in I}A_{i}.}$

Likewise for intersections and cartesian products.

A family (B_i)_i∈J is a subfamily of a family (A_i)_i∈I, if and only if J is a subset of I and for all i in J

B_i = A_i

The analogous concept in category theory is called a diagram. A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices.

• Sequence
• Coproduct
• Disjoint union
• Tagged union
• Index notation
• Array data type
• Net (mathematics)
• Diagram (category theory)
• Parametric family

• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).