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 ("A""j")"j"∈"J".

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. [cite book
title= Foundations of Cryptography: Volume 1, Basic Tools
last= Goldreich
first= Oded
year= 2001
publisher= Cambridge University Press
isbn= 0-521-79172-3
]

Examples

*An enumeration of a set "S" gives an index set J sub mathbb{N}, where f:J arr 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} arr mathbb{R} given by

:mathbf{1}_r (x) := egin{cases} 0, & mbox{if } x e 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

ee also

* Index
* Indexed family


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