# Definition:Surjection/Also known as

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## Definition

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**.

## Sources

- 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Definition $2$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 21$: The image of a subset of the domain; surjections: Remark - 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