indexing set

Let Λ and S be sets such that there exists a surjection f:ΛS. Then Λ is an indexing set for S. Also, S is indexed by Λ.

In such situations, the elements of S could be referenced by using the indexing set Λ, such as f(λ) for some λΛ. On the other hand, quite often, indexing sets are used without explicitly defining a surjective function. When this occurs, the elements of S are referenced by using subscripts (also called indices) which are elements of Λ, such as sλ for some λΛ. If, however, the surjection from Λ to S were called s, this notation would be quite to the function notation: s(λ)=sλ.

Indexing sets are quite useful for describing sequences, nets, summations, productsPlanetmathPlanetmath, unions, and intersectionsMathworldPlanetmath.

Multiple indices are possible. For example, consider the set X={xaa,xab,xac,xbb,xbc,xcc}. Some people would consider the indexing set for X to be {aa,ab,ac,bb,bc,cc}. Others would consider the indexing set to be {a,b,c}×{a,b,c}. (The double indices can be considered as ordered pairs.) Thus, in the case of multiple indices, it need not be the case that the underlying function f be a surjection. On the other hand, f must be a partial surjection. For example, if a set X is indexed by A×B, the following must hold:

  1. 1.

    For every xX, there exist iA and jB such that f(i,j)=x;

  2. 2.

    For every iA, the map fi:BX defined by fi(j)=f(i,j) is a partial functionMathworldPlanetmath;

  3. 3.

    For every jB, the map fj:AX defined by fj(i)=f(i,j) is a partial function.

Title indexing set
Canonical name IndexingSet
Date of creation 2013-03-22 16:07:51
Last modified on 2013-03-22 16:07:51
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 9
Author Wkbj79 (1863)
Entry type Definition
Classification msc 03E99
Synonym index setMathworldPlanetmath
Defines subscript
Defines index
Defines indices
Defines indexed by
Defines double indices
Defines multiple indices