Index set - Index Sets

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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 (A_j)_j?J.

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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 indexed by ${\displaystyle \mathbb {N} }$.
• For ${\displaystyle r\in \mathbb {R} }$, the indicator function on r is the function ${\displaystyle \mathbf {1} _{r}\colon \mathbb {R} \rightarrow \{0,1\}}$ given by

The set of all the ${\displaystyle \mathbf {1} _{r}}$ functions is an uncountable set indexed by ${\displaystyle \mathbb {R} }$.

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Other uses

In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm ${\displaystyle I}$ that can sample the set efficiently; e.g., on input ${\displaystyle 1^{n}}$, ${\displaystyle I}$ can efficiently select a poly(n)-bit long element from the set.

Source of the article : Wikipedia