Definition:Surjection/Definition 1

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Definition

Let $S$ and $T$ be sets.

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


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

$\forall y \in T: \exists x \in \Dom f: \map f x = y$

That is, if and only if $f$ is right-total.


Thus a surjection is a relation which is:

Left-total
Many-to-one
Right-total.


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