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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)jJ.

Contents

ExamplesEdit

  • An enumeration of a set S gives an index set  , where f : JS is the particular enumeration of S.
  • Any countably infinite set can be indexed by the set of natural numbers  .
  • For  , the indicator function on r is the function   given by
 

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

Other usesEdit

In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm   that can sample the set efficiently; e.g., on input  ,   can efficiently select a poly(n)-bit long element from the set.[3]

See alsoEdit

ReferencesEdit

  1. ^ Weisstein, Eric. "Index Set". Wolfram MathWorld. Wolfram Research. Retrieved 30 December 2013. 
  2. ^ Munkres, James R. (2000). Topology. 2. Upper Saddle River: Prentice Hall. 
  3. ^ Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3.