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}j∈J.

Examples

  • An enumeration of a set S gives an index set J ⊂ N {\displaystyle J\subset \mathbb {N} }, where f : J → S is the particular enumeration of S.
  • Any countably infinite set can be (injectively) indexed by the set of natural numbers N {\displaystyle \mathbb {N} }.
  • For r ∈ R {\displaystyle r\in \mathbb {R} }, the indicator function on r is the function 1 r : R → { 0 , 1 } {\displaystyle \mathbf {1} _{r}\colon \mathbb {R} \to \{0,1\}} given by 1 r ( x ) := { 0 , if x ≠ r 1 , if x = r . {\displaystyle \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, { 1 r } r ∈ R {\displaystyle \{\mathbf {1} _{r}\}_{r\in \mathbb {R} }}, is an uncountable set indexed by R {\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.

See also