Index set

Not to be confused with indexed sets, or index sets in computability theory.

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 {A_j}_j∈J.

Examples

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

The set of all such indicator functions, ${\displaystyle \{\mathbf {1} _{r}\}_{r\in \mathbb {R} }}$, 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]

See also

• Friendly-index set