Definition:Surjection/Class Theory

Definition

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:

$\forall y \in B: \exists x \in A: \map f x = y$

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

• Definition:Injection (Class Theory)

• Results about surjections can be found here.

Technical Note

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

Sources

• 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