Index set

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Not to be confused with Indexed set.

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.

Examples[edit]

  • An enumeration of a set S gives an index set J \sub \mathbb{N}, where f : JS is the particular enumeration of S.
\mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if }  x \ne r  \\ 1,  & \mbox{if } x = r. \end{cases}

The set of all the \mathbf{1}_r functions is an uncountable set indexed by \mathbb{R}.

Other uses[edit]

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; i.e., on input 1n, I 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. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
  3. ^ Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3.