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.

Examples

• An enumeration of a set S gives an index set $J\subset \mathbb {N}$, where $f:J\rightarrow \mathbb {N}$ 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 $\mathbf {1} _{r}\colon \mathbb {R} \rightarrow \mathbb {R}$ 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 the $\mathbf {1} _{r}$ functions is an uncountable set indexed by $\mathbb {R}$.

See also

• Friendly-index set
• Index
• Indexed family