Definitions

index-set

Index set

In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called 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.

In 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.

Examples

  • An enumeration of a set S gives an index set J sub mathbb{N}, where f:J rarr mathbb{N} is the particular enumeration of S.
  • Any countably infinite set can be indexed by mathbb{N}.
  • For r in mathbb{R}, the indicator function on r, is the function mathbf{1}_rcolon mathbb{R} rarr mathbb{R} given by

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}.

References

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