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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an index set is a set whose members label (or index) members of another set. For instance, if the elements of 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 ...
may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists of a
surjective function In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
from onto , and the indexed collection is typically called an '' (indexed) family'', often written as .


Examples

*An
enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (fo ...
of a set gives an index set J \sub \N, where is the particular enumeration of . *Any countably infinite set can be (injectively) indexed by the set of natural numbers \N. *For r \in \R, the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
on is the function \mathbf_r\colon \R \to \ given by \mathbf_r (x) := \begin 0, & \mbox x \ne r \\ 1, & \mbox x = r. \end The set of all such indicator functions, \_ , is an
uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
indexed by \mathbb.


Other uses

In
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by ...
and
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
, an index set is a set for which there exists an algorithm that can sample the set efficiently; e.g., on input , can efficiently select a poly(n)-bit long element from the set.


See also

*
Friendly-index set In graph theory, a friendly-index set is a finite set of integers associated with a given undirected graph and generated by a type of graph labeling called a friendly labeling. A friendly labeling of an -vertex undirected graph is defined to be ...
*
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, whe ...


References

{{Reflist Mathematical notation Basic concepts in set theory