PlanetMath: surjective

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surjective

(Definition)

A function $f\colon X\to Y$ is called surjective or onto if, for every $y\in Y$ , there is an $x\in X$ such that .

Equivalently, $f\colon X\to Y$ is onto when its image is all the codomain:

$\begin{displaymath}\mathrm{Im} f= Y.\end{displaymath}$

If $f\colon X\to Y$ is any function, then $f\colon X\to f(X)$ is a surjection. That is, by restricting the codomain, any function can be made into a surjection.
The composition of surjective functions (when defined) is again a surjective function.
If $f\colon X\to Y$ is a surjection and $B\subseteq Y$ , then (see this page)

$\begin{displaymath} f f^{-1}(B) = B. \end{displaymath}$

"surjective" is owned by drini. [ full author list (2) | owner history (1) ]

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Other names:

onto

Also defines:

surjection

Attachments:: definition of onto linear transformation (Definition) by Mathprof

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Cross-references: composition, codomain, image, function
There are 170 references to this entry.

This is version 3 of surjective, born on 2002-03-14, modified 2005-04-30.
Object id is 2791, canonical name is Surjective.
Accessed 31979 times total.

Classification:

03-00 (Mathematical logic and foundations :: General reference works )

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