Index set
From Infogalactic: the planetary knowledge core
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In mathematics, an index set is a set whose members label (or index) members of another set.[1][2] For instance, if the elements of a set A may be indexed or labeled by means of a set J, then J is an index set. The indexing consists of a surjective function from J onto A and the indexed collection is typically called an (indexed) family, often written as (Aj)j∈J.
Contents
Examples
- An enumeration of a set S gives an index set
, where f : J → S is the particular enumeration of S.
- Any countably infinite set can be indexed by
.
- For
, the indicator function on r is the function
given by
The set of all the functions is an uncountable set indexed by
.
Other uses
In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm I that can sample the set efficiently; e.g., on input 1n, I can efficiently select a poly(n)-bit long element from the set.[3]