For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.
More formally, an indexed family is a mathematical function together with its domain
(that is, indexed families and mathematical functions are technically identical, just points of view are different).
In this view, indexed families are interpreted as collections of indexed elements instead of functions.
is called the index set of the family, and
Sequences are one type of families indexed by natural numbers.
For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.
Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.
Being an element of a family is equivalent to being in the range of the corresponding function.
In practice, however, a family is viewed as a collection, rather than a function.
However, families differ from sets in that the same object can appear multiple times with different indices in a family, whereas a set is a collection of distinct objects.
A family contains any element exactly once if and only if the corresponding function is injective.
is not required to be injective, there may exist
denotes the cardinality of the set
Hence, by using a set instead of the family, some information might be lost.
denotes a family of vectors.
only makes sense with respect to this family, as sets are unordered so there is no
Furthermore, linear independence is defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family.
as the same vector, then the set of them consists of only one element (as a set is a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent).
Suppose a text states the following: A square matrix
are linearly independent.As in the previous example, it is important that the rows of
are linearly independent as a family, not as a set.
The set of the rows consists of a single element
as a set is made of unique elements so it is linearly independent, but the matrix is not invertible as the matrix determinant is 0.
The statement is therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows.
(The statement is also correct when "the rows" is interpreted as referring to a multiset, in which the elements are also kept distinct but which lacks some of the structure of an indexed family.)
Index sets are often used in sums and other similar operations.
Likewise for intersections and Cartesian products.
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.