Indexed family

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from 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 (possibly the same) as indexing.

More formally, an indexed family is a mathematical function together with its domain $$I$$ and image $$X$$ (that is, indexed families and mathematical functions are technically identical, just points of view are different). Often the elements of the set $$X$$ are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set $$I$$ is called the index set of the family, and $$X$$ is the indexed set.

Sequences are one type of families indexed by natural numbers. In general, the index set $$I$$ 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.

Formal definition
Let $$I$$ and $$X$$ be sets and $$f$$ a function such that $$\begin{align} f ~:~ &I \to X \\ &i \mapsto x_i = f(i), \end{align}$$ where $$i$$ is an element of $$I$$ and the image $$f(i)$$ of $$i$$ under the function $$f$$ is denoted by $$x_i$$. For example, $$f(3)$$ is denoted by $$x_3.$$ The symbol $$x_i$$ is used to indicate that $$x_i$$ is the element of $$X$$ indexed by $$i \in I.$$ The function $$f$$ thus establishes a family of elements in $$X$$ indexed by $$I,$$ which is denoted by $$\left(x_i\right)_{i \in I},$$ or simply $$\left(x_i\right)$$ if the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.

Functions and indexed families are formally equivalent, since any function $$f$$ with a domain $$I$$ induces a family $$(f(i))_{i \in I}$$ 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 $$X$$ gives rise to a family $$\left(x_x\right)_{x \in X},$$ where $$X$$ is indexed by itself (meaning that $$f$$ is the identity 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.

An indexed family $$\left(x_i\right)_{i \in I}$$ defines a set $$\mathcal{X} = \{x_i : i \in I\},$$ that is, the image of $$I$$ under $$f.$$ Since the mapping $$f$$ is not required to be injective, there may exist $$i, j \in I$$ with $$i \neq j$$ such that $$x_i = x_j.$$ Thus, $$| \mathcal{X}| \leq |I|$$, where $$|A|$$ denotes the cardinality of the set $$A.$$ For example, the sequence $$\left( (-1)^i \right)_{i\in \N} $$ indexed by the natural numbers $$\N = \{1, 2, 3, \ldots\}$$ has image set $$\left\{(-1)^i : i \in \N\right\} = \{-1,1\}.$$ In addition, the set $$\{ x_i : i \in I \}$$ does not carry information about any structures on $$I.$$ 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 $$\left(B_i\right)_{i \in J}$$ is a subfamily of an indexed family $$\left(A_i\right)_{i \in I},$$ if and only if $$J$$ is a subset of $$I$$ and $$B_i = A_i$$ holds for all $$i \in J.$$

Indexed vectors
For example, consider the following sentence: "The vectors $v_1, \ldots, v_n$ are linearly independent."

Here $$\left(v_i\right)_{i \in \{1, \ldots, n\}}$$ denotes a family of vectors. The $$i$$-th vector $$v_i$$ only makes sense with respect to this family, as sets are unordered so there is no $$i$$-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 $$n = 2$$ and $$v_1 = v_2 = (1, 0)$$ 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).

Matrices
Suppose a text states the following: "A square matrix $A$ is invertible, if and only if the rows of $A$ are linearly independent."

As in the previous example, it is important that the rows of $$A$$ are linearly independent as a family, not as a set. For example, consider the matrix $$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}.$$ The set of the rows consists of a single element $$(1, 1)$$ 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 hands, the family of the rows contains two elements indexed differently such as the 1st row $$(1, 1)$$ and the 2nd row $$(1, 1)$$ 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. (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.)

Other examples
Let $$\mathbf{n}$$ be the finite set $$\{1, 2, \ldots n\},$$ where $$n$$ is a positive integer.
 * An ordered pair (2-tuple) is a family indexed by the set of two elements, $$\mathbf{2} = \{1, 2\};$$ each element of the ordered pair is indexed by each element of the set $$\mathbf{2}.$$
 * An $n$-tuple is a family indexed by the set $$\mathbf{n}.$$
 * 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 $$n \times m$$ matrix is a family indexed by the Cartesian product $$\mathbf{n} \times \mathbf{m}$$ which elements are ordered pairs; for example, $$(2, 5)$$ indexing the matrix element at the 2nd row and the 5th column.
 * A net is a family indexed by a directed set.

Operations on indexed families
Index sets are often used in sums and other similar operations. For example, if $$\left(a_i\right)_{i \in I}$$ is an indexed family of numbers, the sum of all those numbers is denoted by $$\sum_{i \in I} a_i.$$

When $$\left(A_i\right)_{i \in I}$$ is a family of sets, the union of all those sets is denoted by $$\bigcup_{i \in I} A_i.$$

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