Let $A$ and $B$ be classes.
Let $f: A \to B$ be a mapping from $A$ to $B$.
Then $f$ is a surjection if and only if:
- $\Img f = B$
where $\Img f$ denotes the image of $f$.
That is, if and only if:
- $\forall y \in B: \exists x \in A: \map f x = y$
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.
- Results about surjections can be found here.
The $\LaTeX$ code for \(f: S \twoheadrightarrow T\) is
f: S \twoheadrightarrow T .
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 9$ Functions