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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, indexed by the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s'' 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 functi ...
together with its
domain and
image (that is, indexed families and mathematical functions are technically identical, just point of views are different.) Often the
elements of the set
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
is called the ''index set'' of the family, and
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 t ...
are one type of families indexed by
natural numbers. In general, the index set
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 numbers ...
. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.
Formal definition
Let
and
be sets and
a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
such that
where
is an element of
and the image
of
under the function
is denoted by
. For example,
is denoted by
The symbol
is used to indicate that
is the element of
indexed by
The function
thus establishes a family of elements in
indexed by
which is denoted by
or simply
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
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 (meaning that
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
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
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 are ...
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 (as a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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
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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
is 0. On the other hands, 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. (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
be the finite set
where
is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
.
* An
ordered pair (2-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
) is a family indexed by the set of two elements,
each element of the ordered pair is indexed by each element of the set
* An
-tuple is a family indexed by the set
* 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.
* 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 unio ...
is an
-tuple for an unspecified
or an infinite sequence.
* An
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is a family indexed by the
Cartesian product which elements are ordered pairs; for example,
indexing the matrix element at the 2nd row and the 5th column.
* A
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
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 ha ...
.
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
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
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
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
morphisms depending on two indices.
See also
*
*
*
*
*
*
*
*
*
*
References
{{reflist
*
Mathematical Society of Japan
The Mathematical Society of Japan (MSJ, ja, 日本数学会) is a learned society for mathematics in Japan.
In 1877, the organization was established as the ''Tokyo Sugaku Kaisha'' and was the first academic society in Japan. It was re-organized ...
, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).
Basic concepts in set theory
Mathematical notation