Index#Mathematics

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In mathematics index called (plural: indices) an element of the index set, which is used for numbering of different objects. Often the set of natural numbers is used as an index set.

In the beginnings of mathematics of the modern era was also a function value - f ( x) in modern notation - designated by subscript index x as fx. The notation ai for the terms of a sequence ( as a function over the natural numbers ) can be regarded as remnants of these older spelling. Depending on demand, occur despite confusion with the power calculation, even superscripts ai.

• 4.1 Definition
• 4.2 Axiom of Choice

Terminology

An index is an attached to a character up or down, right or left distinctive character.

In mathematics, the sign, to which the suffix is ​​attached, for a mathematical object and the index itself is preferably listed right below this sign. Depending on the mathematical department and question but, any other position of the index conceivable.

Examples

• In droves flock function parameters are usually listed as an index, while the " normal" arguments are written in the parentheses after the function name - eg

• In a matrix of its components, that is, frequently indicated the individual values ​​in the matrix. The component representation of a matrix is, for example,

• In physics, especially in the tensor physics, duplicate indexes for the shortened notation of sums are used. This convention is called Einstein summation convention.

• For stochastic processes and time series of the time parameter is often written as an index.

• In the mathematical subfield of differential geometry, the sections of a vector bundle are often referred to in index notation to have the function notation for algebraic operations between fibers of different bundles over the same point free.

Index set

Definition

A set whose elements durchindizieren elements of another set index set is called.

Note

So an index set is no particular amount, but it is much more important that you use the elements of the set to indicate other objects. In many cases the amount of natural numbers is used. However, any amount whether with a finite, countable or uncountable number of elements are used as index set, then merges mathematical objects to a family together ( here is the index quantity). Is used as the index set of the natural numbers, we speak instead of a family of a sequence.

Selection function

In mathematics, the index means of the selection function can be formally defined as a mapping from the index set in the amount of indexed objects.

Definition

Are any quantity, so you can the n- tuple

As picture

Interpret. It's called selection function.

Axiom of Choice

If you want to not be limited to a finite number of levels, but infinite ( in particular uncountable ) many consider, the existence of the just defined selection function is not clear. That is, it is at infinite index sets is not always possible to provide a concrete illustration of the selection function, and thus the existence of the point. That such a selection function does exist ensured by the axiom of choice. However, the axiom does not say about the concrete representation of the selection function.