surjection

surjection

English

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).d͡ʒɛk.ʃən/

Noun

surjection (plural surjections)

1. (set theory) A function for which every element of the codomain is mapped to by some element of the domain; (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.

Synonyms

• surjective function
• onto function

Related terms

• bijection
• injection
• surject

Translations

function that is a many-to-one mapping

Catalan: funció exhaustiva f Chinese: Mandarin: 满射 (zh) (mǎnshè) Czech: surjekce f Estonian: sürjektsioon, surjection (et) f Finnish: surjektio (fi) French: surjection (fr) f German: Surjektion (de) f, surjektive Funktion f Greek: επίρριψη f (epírripsi) Ido: surjekto Italian: suriezione (it) f Japanese: 全射 (ja) (zensha) Korean: 전사(全射) (ko) (jeonsa) Polish: suriekcja f Portuguese: sobrejeção f Romanian: surjecție f, funcție surjectivă f Russian: сюръе́кция (ru) f (sjurʺjékcija) Serbo-Croatian: surjekcija (sh) f Spanish: función sobreyectiva f Swedish: surjektion (sv) c

References

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

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French

Etymology

Borrowing from Latin superiectiō (“a throwing over or on; (figuratively) 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

Derived terms

• surjectif