# Definition:Surjection/Class Theory

< Definition:Surjection(Redirected from Definition:Surjection (Class Theory))

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## 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

- 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