# surjection

## English

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A surjection

### 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).[1]

### Pronunciation

• IPA(key): /sɜː(ɹ).dʒɛk.ʃən/

### Noun

surjection (plural surjections)

1. (set theory) A function that is a many-to-one mapping; (formally) Any function ${\displaystyle f:X\rightarrow Y}$  for which for every ${\displaystyle y\in Y}$ , there is at least one ${\displaystyle x\in X}$  such that ${\displaystyle f(x)=y}$ .
• 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 ${\displaystyle A\rightarrow B}$  can be identified.
• 1999, M. Pavaman Murthy, A survey of obstruction theory for projective modules of top rank, 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 ${\displaystyle J=\cap _{i}m_{i}}$  be the (irredundant) primary decomposition of ${\displaystyle J}$ . We associate to the pair ${\displaystyle (J,\omega )}$  the element ${\displaystyle \textstyle \sum _{i}(m_{i},\omega _{i})\in G}$ , where ${\displaystyle \omega _{i}}$  is the equivalence class of surjections from ${\displaystyle L/m_{i}L\oplus (A/m_{i})^{n-1}}$  to ${\displaystyle m_{i}/m_{i}^{2}}$  induced by ${\displaystyle \omega }$ .
• 2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43,
In Banach space theory, a mapping ${\displaystyle u:E\rightarrow F}$  (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from ${\displaystyle E/{\text{ker}}(u)}$  to ${\displaystyle F}$  is an isometric isomorphism. Moreover, by the classical open mapping theorem, ${\displaystyle u}$  is a surjection iff the associated mapping from ${\displaystyle E/{\text{ker}}(u)}$  to ${\displaystyle F}$  is an isomorphism.

### References

1. ^ sŭperjectĭo, Charlton T. Lewis; Charles Short [1879], A Latin Dictionary, uchicago.edu

## French

### Etymology

Borrowing from Latin superiectiō (a throwing over or on; (fig.) an exaggeration, a hyperbole). Compare injection, bijection, with the same second element but different prefixes.

### Pronunciation

• IPA(key): /syʁ.ʒɛk.sjɔ̃/
•  Audio (file)

### Noun

surjection f (plural surjections)

1. (set theory) surjection