Index set

Not to be confused with Indexed 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 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 $J\subset {\mathbb {N}}$ , where $f : J \to S$ is the particular enumeration of $S$ .
Any countably infinite set can be indexed by $\mathbb {N}$ .
For $r\in {\mathbb {R}}$ , the indicator function on $r$ is the function ${\mathbf {1}}_{r}\colon {\mathbb {R}}\rightarrow \{0,1\}$ given by

{\mathbf {1}}_{r}(x):={\begin{cases}0,&{\mbox{if }}x\neq r\\1,&{\mbox{if }}x=r.\end{cases}}

The set of all the ${\mathbf {1}}_{r}$ functions 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 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]

References

↑ Weisstein, Eric. "Index Set". Wolfram MathWorld. Wolfram Research. Retrieved 30 December 2013.
↑ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
↑ Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3.

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Index set

Examples

Other uses

See also

References