Index set explained

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 may be indexed or labeled by means of the elements of a set, then is an index set. The indexing consists of a surjective function from onto, and the indexed collection is typically called an indexed family, often written as .

Examples

• An enumeration of a set gives an index set

J \sub \N , where is the particular enumeration of .\N .

• For

r \in \R , the indicator function on is the function 1 _{r\colon} \R \to \{ 0 , 1 \} given by

The set of all such indicator functions,

\{ 1 _{r} \} _{r\in\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 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

Notes and References

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