Index set - Alchetron, The Free Social Encyclopedia

In mathematics, an index set is a set whose members label (or index) members of another set. 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 (A_j)_j∈J.

Examples

An enumeration of a set S gives an index set J ⊂ N , where f : J → S is the particular enumeration of S.

Any countably infinite set can be indexed by N .

For r ∈ R , the indicator function on r is the function 1 r : R → { 0 , 1 } given by

1 r ( x ) := { 0 , if x ≠ r 1 , if x = r .

The set of all the 1 r functions is an uncountable set indexed by R .

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.

References

Index set Wikipedia

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Examples

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References