indexing set
Let and be sets such that there exists a surjection . Then is an indexing set for . Also, is indexed by .
In such situations, the elements of could be referenced by using the indexing set , such as for some . On the other hand, quite often, indexing sets are used without explicitly defining a surjective function. When this occurs, the elements of are referenced by using subscripts (also called indices) which are elements of , such as for some . If, however, the surjection from to were called , this notation would be quite to the function notation: .
Indexing sets are quite useful for describing sequences, nets, summations, products, unions, and intersections.
Multiple indices are possible. For example, consider the set . Some people would consider the indexing set for to be . Others would consider the indexing set to be . (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 be a surjection. On the other hand, must be a partial surjection. For example, if a set is indexed by , the following must hold:
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1.
For every , there exist and such that ;
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2.
For every , the map defined by is a partial function;
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3.
For every , the map defined by 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 |