Index set

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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 the elements 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.

Examples[edit]

  • 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 uses[edit]

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 also[edit]

References[edit]

  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.