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 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.
- An enumeration of a set S gives an index set , where f : J → S is the particular enumeration of S.
- Any countably infinite set can be indexed by .
- For , the indicator function on r is the function given by
The set of all the functions is an uncountable set indexed by .
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.