indexing set (original) (raw)
Let Λ and S be sets such that there exists a surjection f:Λ→S. 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 λ∈Λ. 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 λ∈Λ. If, however, the surjection from Λ to S were called s, this notation would be quite to the function notation: s(λ)=sλ.
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:
- For every x∈X, there exist i∈A and j∈B such that f(i,j)=x;
- For every i∈A, the map fi:B→X defined by fi(j)=f(i,j) is a partial function
;
- For every i∈A, the map fi:B→X defined by fi(j)=f(i,j) is a partial function
- For every j∈B, the map fj:A→X 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 set |
| Defines | subscript |
| Defines | index |
| Defines | indices |
| Defines | indexed by |
| Defines | double indices |
| Defines | multiple indices |