Definitions

# 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 (Aj)jJ.

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.

## Examples

• An enumeration of a set S gives an index set $J sub mathbb\left\{N\right\}$, where $f:J rarr mathbb\left\{N\right\}$ is the particular enumeration of S.
• Any countably infinite set can be indexed by $mathbb\left\{N\right\}$.
• For $r in mathbb\left\{R\right\}$, the indicator function on r, is the function $mathbf\left\{1\right\}_rcolon mathbb\left\{R\right\} rarr mathbb\left\{R\right\}$ given by

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

The set of all the $mathbf\left\{1\right\}_r$ functions is an uncountable set indexed by $mathbb\left\{R\right\}$.