# Index 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 (*A*_{j})_{j∈J}.

## Examples

- An enumeration of a set
*S*gives an index set , where*f*:*J*→*S*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

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

- Friendly-index set
- Indexed family

## References

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

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