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 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 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.
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Proof and Problem Solving  Family of Sets and Indexed Family of Sets

Indexed Sets

Indexed Families of Sets

[Discrete Mathematics] Indexed Sets and Well Ordering Principle

Set Theory and Logic Lecture 6 Indexed Sets
Transcription
Formal definition
Let and be sets and a function such that
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 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:
The vectors are linearly independent.
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 (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 is invertible, if and only if the rows of 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. For example, consider the matrix
Other examples
Let be the finite set where is a positive integer.
 An ordered pair (2tuple) 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 is a family indexed by the natural numbers.
 A list is an tuple for an unspecified or an infinite sequence.
 An matrix 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 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 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 C, indexed by another category J, and related by morphisms depending on two indices.
See also
 Array data type – Data type that represents a collection of elements (values or variables)
 Coproduct – Categorytheoretic construction
 Diagram (category theory) – Indexed collection of objects and morphisms in a category
 Disjoint union – In mathematics, operation on sets
 Family of sets – Any collection of sets, or subsets of a set
 Index notation – Manner of referring to elements of arrays or tensors
 Net (mathematics) – A generalization of a sequence of points
 Parametric family – family of objects whose definitions depend on a set of parameters
 Sequence – Finite or infinite ordered list of elements
 Tagged union – Data structure used to hold a value that could take on several different, but fixed, types
References
 Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).