Index set

Short description: Mathematical term

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}j∈J.

## Contents

* 1 Examples
* 2 Other uses
* 3 See also
* 4 References

## 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 (injectively) indexed by the set of natural numbers ℕ.
* For r∈ℝ, the indicator function on r is the function 𝟏r:ℝ->{0,1} given by 𝟏r(x):={0,if x≠r1,if x=r.

The set of all such indicator functions, {𝟏r}r∈ℝ , 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.[3]

## See also

* Friendly-index set

## References

1. ↑ Weisstein, Eric. "Index Set". Wolfram MathWorld. Wolfram Research. http://mathworld.wolfram.com/IndexSet.html.
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.

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