# surjection

## English

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### Etymology

From French surjection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique. Ultimately borrowed from Latin superiectiō (a throwing over or on; (fig.) an exaggeration, a hyperbole).

### Pronunciation

• IPA(key): /sɜː(ɹ)ˈd͡ʒɛk.ʃən/
•  Audio (Southern England) (file)

### Noun

surjection (plural surjections)

1. A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function $f:X\rightarrow Y$ for which for every $y\in Y$ , there is at least one $x\in X$ such that $f(x)=y$ .
• 1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, Institute of Physics Publishing, page 220:
In some special cases, however, the number of surjections $A\rightarrow B$ can be identified.
• 1999, M. Pavaman Murthy, “A survey of obstruction theory for projective modules of top rank”, in Tsit-Yuen Lam, Andy R. Magid, editors, Algebra, K-theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday, American Mathematical Society, page 168:
Let $J=\cap _{i}m_{i}$ be the (irredundant) primary decomposition of $J$ . We associate to the pair $(J,\omega )$ the element $\textstyle \sum _{i}(m_{i},\omega _{i})\in G$ , where $\omega _{i}$ is the equivalence class of surjections from $L/m_{i}L\oplus (A/m_{i})^{n-1}$ to $m_{i}/m_{i}^{2}$ induced by $\omega$ .
• 2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43:
In Banach space theory, a mapping $u:E\rightarrow F$ (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from $E/{\text{ker}}(u)$ to $F$ is an isometric isomorphism. Moreover, by the classical open mapping theorem, $u$ is a surjection iff the associated mapping from $E/{\text{ker}}(u)$ to $F$ is an isomorphism.