Degenerate conic: Difference between revisions

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In [[mathematics]], an '''index set''' is a set whose members label (or index) members of another set.<ref>{{cite web|last=Weisstein|first=Eric|title=Index Set|url=http://mathworld.wolfram.com/IndexSet.html|work=Wolfram MathWorld|publisher=Wolfram Research|accessdate=30 December 2013}}</ref> For instance, if the elements of a [[Set (mathematics)|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|(indexed) family]]'', often written as (''A''<sub>''j''</sub>)<sub>''j''∈''J''</sub>.
 
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; i.e., on input 1<sup>n</sup>, ''I'' can efficiently select a poly(n)-bit long element from the set.<ref>
{{cite book
  | title= Foundations of Cryptography: Volume 1, Basic Tools
  |last= Goldreich
  |first= Oded
  |year= 2001
  |publisher= Cambridge University Press
  |isbn= 0-521-79172-3
}}</ref>
 
==Examples==
 
*An [[enumeration]] of a set {{math|''S''}} gives an index set <math>J \sub \mathbb{N}</math>, where {{math|''f'' : ''J'' → ''S''}} is the particular enumeration of {{math|''S''}}.
 
*Any [[countably infinite]] set can be indexed by <math>\mathbb{N}</math>.
 
*For <math>r \in \mathbb{R}</math>, the [[indicator function]] on {{math|''r''}} is the function <math>\mathbf{1}_r\colon \mathbb{R} \rarr \{0,1\}</math> given by
 
:<math>\mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if }  x \ne r  \\ 1,  & \mbox{if } x = r. \end{cases} </math>
 
The set of all the <math>\mathbf{1}_r</math> functions <!--(which happens to be a [[basis]] for the [[vector space]] of all functions on <math>\mathbb{R}</math> over <math>\mathbb{R}</math>)--> is an [[uncountable set]] indexed by <math>\mathbb{R}</math>.
 
==See also==
* [[Friendly-index set]]
* [[Indexed family]]
 
==References==
{{Reflist}}
 
[[Category:Mathematical notation]]
[[Category:Basic concepts in set theory]]

Latest revision as of 12:14, 8 October 2014

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