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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). More formally, an indexed family is a
mathematical function In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the funct ...
together with its domain $I$ and
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
$X.$ (that is, indexed families and mathematical functions are technically identical, just point of views 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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
are one type of families indexed by
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
. In general, the index set $I$ is not restricted to be
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
. 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 $\begin f ~:~ &I \to X \\ &i \mapsto x_i = f(i), \end$ where $i$ is an element of $I$ and the image $f\left(i\right)$ of $i$ under the function $f$ is denoted by $x_i$. For example, $f\left(3\right)$ 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 $\left\left(x_i\right\right)_,$ or simply $\left\left(x_i\right\right)$ 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 $\left(f\left(i\right)\right)_$ 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 $\left\left(x_x\right\right)_,$ 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 $\left\left(x_i\right\right)_$ defines a set $\mathcal = \,$ 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, $, \mathcal, \leq , I,$, where $, A,$ denotes the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalize ...
of the set $A.$ For example, the sequence $\left\left( \left(-1\right)^i \right\right)_$ indexed by the natural numbers $\N = \$ has image set $\left\ = \.$ In addition, the set $\$ 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 $\left\left(B_i\right\right)_$ is a subfamily of an indexed family $\left\left(A_i\right\right)_,$ if and only if $J$ is a subset of $I$ and $B_i = A_i$ holds for all $i \in J.$

# Examples

## Indexed vectors

For example, consider the following sentence: Here $\left\left(v_i\right\right)_$ 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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
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 $v_1 = v_2 = \left(1, 0\right)$ 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: 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 $A = \begin 1 & 1 \\ 1 & 1 \end.$ The ''set'' of the rows consists of a single element $\left(1, 1\right)$ 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 $\left(1, 1\right)$ and the 2nd row $\left(1, 1\right)$ 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 mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
, in which the elements are also kept distinct but which lacks some of the structure of an indexed family.)

## Other examples

Let $\mathbf$ be the finite set $\,$ where $n$ is a positive integer. * An
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
(2- tuple) is a family indexed by the set of two elements, $\mathbf = \;$ each element of the ordered pair is indexed by each element of the set $\mathbf.$ * An $n$-tuple is a family indexed by the set $\mathbf.$ * An infinite
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
is a family indexed by the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
. * A
list A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...
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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
$\mathbf \times \mathbf$ which elements are ordered pairs; for example, $\left(2, 5\right)$ indexing the matrix element at the 2nd row and the 5th column. * A net is a family indexed by a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
.

# Operations on indexed families

Index sets are often used in sums and other similar operations. For example, if $\left\left(a_i\right\right)_$ is an indexed family of numbers, the sum of all those numbers is denoted by $\sum_ a_i.$ When $\left\left(A_i\right\right)_$ is a
family of sets In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
, the union of all those sets is denoted by $\bigcup_ A_i.$ Likewise for
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
s and
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
s.

# Usage in category theory

The analogous concept in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
is called a
diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...
. A diagram is a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
giving rise to an indexed family of objects in a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
, indexed by another category , and related by
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s depending on two indices.