# Index set

Short description: Mathematical term

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 the elements 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}jJ.

## Examples

• An enumeration of a set S gives an index set $\displaystyle{ J \sub \N }$, where f : JS is the particular enumeration of S.
• Any countably infinite set can be (injectively) indexed by the set of natural numbers $\displaystyle{ \N }$.
• For $\displaystyle{ r \in \R }$, the indicator function on r is the function $\displaystyle{ \mathbf{1}_r\colon \R \to \{0,1\} }$ given by $\displaystyle{ \mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if } x \ne r \\ 1, & \mbox{if } x = r. \end{cases} }$

The set of all such indicator functions, $\displaystyle{ \{ \mathbf{1}_r \}_{r\in\R} }$ , is an uncountable set indexed by $\displaystyle{ \mathbb{R} }$.

## 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; e.g., on input 1n, I can efficiently select a poly(n)-bit long element from the set.[3]