Index set

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 [math]\displaystyle{ J \sub \N }[/math], where f : J → S is the particular enumeration of S.
• Any countably infinite set can be (injectively) indexed by the set of natural numbers [math]\displaystyle{ \N }[/math].
• For [math]\displaystyle{ r \in \R }[/math], the indicator function on r is the function [math]\displaystyle{ \mathbf{1}_r\colon \R \to \{0,1\} }[/math] given by [math]\displaystyle{ \mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if } x \ne r \\ 1, & \mbox{if } x = r. \end{cases} }[/math]

The set of all such indicator functions, [math]\displaystyle{ \{ \mathbf{1}_r \}_{r\in\R} }[/math] , is an uncountable set indexed by [math]\displaystyle{ \mathbb{R} }[/math].

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

References

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