Indexed family


In mathematics, a family, or indexed family, is a collection of objects, each associated with an element, known as an index, that belongs to some index set.
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 as indexing.
More formally, an indexed family is a mathematical function together with its domain and image . Often the elements of the set are referred to as making up the family. In this view, an indexed family is interpreted as a collection of indexed elements, instead of a function. The set is called the index set of the family, and is the indexed set.
Sequences are one type of families indexed by natural numbers. In general, the index set is not restricted to be countable. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.

Definition

Let and be sets. A family of elements of indexed by , denoted, is a function such that
for all. The element is known as the term of index.
Notation using different brackets, such as is also valid.
Functions and indexed families are formally equivalent, since any function with a domain induces a family and conversely. 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.
Any set gives rise to a family where is indexed by itself. 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.
An indexed family defines a set that is, the image of under Since the mapping is not required to be injective, there may exist with such that Thus,, where denotes the cardinality of the set For example, the sequence indexed by the natural numbers has image set In addition, the set does not carry information about any structures on Hence, by using a set instead of the family, some information might be lost. For example, an ordering on the index set of a family induces an ordering on the family, but no ordering on the corresponding image set.

Indexed subfamily

An indexed family is a subfamily of an indexed family if and only if is a subset of and holds for all

Examples

Indexed vectors

For example, consider the following sentence:
Here denotes a family of vectors. The -th vector only makes sense with respect to this family, as sets are unordered so there is no -th vector of a set. 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. For example, if we consider and as the same vector, then the set of them consists of only one element and is linearly independent, but the family contains the same element twice and is linearly dependent.

Matrices

Suppose a text states the following:
As in the previous example, it is important that the rows of are linearly independent as a family, not as a set. For example, consider the matrix
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. On the other hand, the family of the rows contains two elements indexed differently such as the 1st row and the 2nd row so it is linearly dependent. The statement is therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows.

Other examples

Let be the finite set where is a positive integer.

Operations on indexed families

Index sets are often used in sums and other similar operations. For example, if is an indexed family of numbers, the sum of all those numbers is denoted by
When is a family of sets, the union of all those sets is denoted by
Likewise for intersections and Cartesian products.

Usage in category theory

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, indexed by another category, and related by morphisms depending on two indices.