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 Template:Mvar may be indexed or labeled by means of the elements of a set Template:Mvar, then Template:Mvar is an index set. The indexing consists of a surjective function from Template:Mvar onto Template:Mvar, and the indexed collection is typically called an indexed family, often written as {A_j}_j∈J.

Examples

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

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 Template:Mvar that can sample the set efficiently; e.g., on input 1ⁿ, Template:Mvar can efficiently select a poly(n)-bit long element from the set.

See also

• Friendly-index set