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Milds # Index set

In mathematics, an index set is a set whose members label (or index) members of another set. 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 $J\subset \mathbb {N}$ , where f : JS is the particular enumeration of S.
• Any countably infinite set can be (injectively) indexed by the set of natural numbers $\mathbb {N}$ .
• For $r\in \mathbb {R}$ , the indicator function on r is the function $\mathbf {1} _{r}\colon \mathbb {R} \rightarrow \{0,1\}$ given by
$\mathbf {1} _{r}(x):={\begin{cases}0,&{\mbox{if }}x\neq r\\1,&{\mbox{if }}x=r.\end{cases}}$ The set of all such indicator functions, $\{\mathbf {1} _{r}\}_{r\in \mathbb {R} }$ , 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; e.g., on input 1n, I can efficiently select a poly(n)-bit long element from the set.