Index set
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In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called 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 (Aj)j∈J.
[edit] Examples
- An enumeration of a set S gives an index set <math>J \sub \mathbb{N}</math>, where <math>f:J \rarr \mathbb{N}</math> is the particular enumeration of S.
- Any countably infinite set can be indexed by <math>\mathbb{N}</math>.
- For <math>r \in \mathbb{R}</math>, the indicator function on r, is the function <math>\mathbf{1}_r\colon \mathbb{R} \rarr \mathbb{R}</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>.
[edit] See also
it:Famiglia (matematica) hu:Halmazrendszer no:Familie (matematikk)
