# Definition:Surjection/Definition 2

## Definition

Let $S$ and $T$ be sets or classes.

Let $f: S \to T$ be a mapping from $S$ to $T$.

$f: S \to T$ is a surjection if and only if:

$f \sqbrk S = T$

or, in the language and notation of direct image mappings:

$\map {f^\to} S = T$

That is, $f$ is a surjection if and only if its image equals its codomain:

$\Img f = \Cdm f$

## Also known as

The phrase $f$ is surjective is often used for $f$ is a surjection.

Authors who prefer to limit the jargon of mathematics tend to use the term an onto mapping for a surjection, and onto for surjective.

A mapping which is not surjective is thence described as into.

A surjection $f$ from $S$ to $T$ is sometimes denoted:

$f: S \twoheadrightarrow T$

to emphasize surjectivity.

The $\LaTeX$ code for $f: S \twoheadrightarrow T$ is f: S \twoheadrightarrow T .

## Also see

• Results about surjections can be found here.

## Sources

WARNING: This link is broken. Amend the page to use {{KhanAcademySecure}} and check that it links to the appropriate page.