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Mathematics-Online lexicon:

Map

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A map $ f$ from a set $ A$ to a set $ B$ assigns to each $ a\in A$ a unique $ b=f(a) \in B$ :

$\displaystyle f: A \longrightarrow B\,. $

The terms mapping and function are used synonymously for map; function is particularly common in real and complex analysis.

To define which elements are associated with one another, the notation

$\displaystyle a \mapsto b=f(a) $

is used where $ b$ is called the image of $ a$, and $ a$ is called the inverse image of $ b$ .

\includegraphics[width=0.5\linewidth]{abbildung_Bild}

The figure shows that not every element in $ B$ necessarily is an element of the image of $ A$; also, one element of $ B$ can be the image of several elements of $ A$ (in other words: an image $ b$ can have several inverse images, such as $ a$ and $ a'$ in the above illustration).

However, each element of $ A$ must have a unique image, that is each $ a\in A$ has to be the starting point of an arrow to an element of $ B$ .

(Authors: Höllig/Hörner/Kimmerle/Abele)
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  automatically generated 5/18/2009