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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 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.

Examples

An enumeration of a set S gives an index set \( J \sub \mathbb{N} \) , where f : J → S is the particular enumeration of S.

Any countably infinite set can be indexed by \mathbb{N}.

For t \( r \in \mathbb{R}, the indicator function on r is the function t \( \mathbf{1}_r\colon \mathbb{R} \rarr \{0,1\} \) given by

t \( \mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if } x \ne r \\ 1, & \mbox{if } x = r. \end{cases} \)

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

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

Friendly-index set
Indexed family

References

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

Mathematics Encyclopedia

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