Definition:Surjection/Definition 2
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Definition
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.
In the context of class theory, a surjection is often seen referred to as a class surjection.
Also see
- Results about surjections can be found here.
Sources
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- For a video presentation of the contents of this page, visit the Khan Academy.